\(\int \frac {(a+b x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^5} \, dx\) [93]

Optimal result

Integrand size = 34, antiderivative size = 864 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\frac {b \left (7 a d^2 (C d-5 c D)+3 b \left (15 c^2 C d-5 B c d^2+A d^3-35 c^3 D\right )\right ) \sqrt {a+b x^2}}{3 d^8}+\frac {b \left (9 a d^2 D-4 b \left (5 c C d-B d^2-15 c^2 D\right )\right ) x \sqrt {a+b x^2}}{8 d^7}+\frac {b^2 (C d-5 c D) x^2 \sqrt {a+b x^2}}{3 d^6}+\frac {b^2 D x^3 \sqrt {a+b x^2}}{4 d^5}-\frac {\left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{4 d^8 (c+d x)^4}+\frac {\left (b c^2+a d^2\right ) \left (4 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (25 c^2 C d-21 B c d^2+17 A d^3-29 c^3 D\right )\right ) \sqrt {a+b x^2}}{12 d^8 (c+d x)^3}-\frac {\left (12 a^2 d^4 (C d-3 c D)+a b d^2 \left (155 c^2 C d-79 B c d^2+27 A d^3-255 c^3 D\right )+2 b^2 c^2 \left (101 c^2 C d-69 B c d^2+43 A d^3-139 c^3 D\right )\right ) \sqrt {a+b x^2}}{24 d^8 (c+d x)^2}-\frac {\left (24 a^3 d^6 D-4 a^2 b d^4 \left (55 c C d-14 B d^2-141 c^2 D\right )-a b^2 c d^2 \left (843 c^2 C d-383 B c d^2+139 A d^3-1591 c^3 D\right )-2 b^3 c^3 \left (319 c^2 C d-171 B c d^2+77 A d^3-533 c^3 D\right )\right ) \sqrt {a+b x^2}}{24 d^8 \left (b c^2+a d^2\right ) (c+d x)}+\frac {5 \sqrt {b} \left (3 a^2 d^4 D-4 a b d^2 \left (5 c C d-B d^2-15 c^2 D\right )-8 b^2 c \left (7 c^2 C d-3 B c d^2+A d^3-14 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 d^9}-\frac {5 b \left (4 a^3 d^6 (C d-5 c D)+4 a b^2 c^2 d^2 \left (26 c^2 C d-10 B c d^2+3 A d^3-57 c^3 D\right )+3 a^2 b d^4 \left (17 c^2 C d-5 B c d^2+A d^3-45 c^3 D\right )+8 b^3 c^4 \left (7 c^2 C d-3 B c d^2+A d^3-14 c^3 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{8 d^9 \left (b c^2+a d^2\right )^{3/2}} \] Output:

1/3*b*(7*a*d^2*(C*d-5*D*c)+3*b*(A*d^3-5*B*c*d^2+15*C*c^2*d-35*D*c^3))*(b*x 
^2+a)^(1/2)/d^8+1/8*b*(9*a*d^2*D-4*b*(-B*d^2+5*C*c*d-15*D*c^2))*x*(b*x^2+a 
)^(1/2)/d^7+1/3*b^2*(C*d-5*D*c)*x^2*(b*x^2+a)^(1/2)/d^6+1/4*b^2*D*x^3*(b*x 
^2+a)^(1/2)/d^5-1/4*(a*d^2+b*c^2)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a 
)^(1/2)/d^8/(d*x+c)^4+1/12*(a*d^2+b*c^2)*(4*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2) 
+b*c*(17*A*d^3-21*B*c*d^2+25*C*c^2*d-29*D*c^3))*(b*x^2+a)^(1/2)/d^8/(d*x+c 
)^3-1/24*(12*a^2*d^4*(C*d-3*D*c)+a*b*d^2*(27*A*d^3-79*B*c*d^2+155*C*c^2*d- 
255*D*c^3)+2*b^2*c^2*(43*A*d^3-69*B*c*d^2+101*C*c^2*d-139*D*c^3))*(b*x^2+a 
)^(1/2)/d^8/(d*x+c)^2-1/24*(24*a^3*d^6*D-4*a^2*b*d^4*(-14*B*d^2+55*C*c*d-1 
41*D*c^2)-a*b^2*c*d^2*(139*A*d^3-383*B*c*d^2+843*C*c^2*d-1591*D*c^3)-2*b^3 
*c^3*(77*A*d^3-171*B*c*d^2+319*C*c^2*d-533*D*c^3))*(b*x^2+a)^(1/2)/d^8/(a* 
d^2+b*c^2)/(d*x+c)+5/8*b^(1/2)*(3*a^2*d^4*D-4*a*b*d^2*(-B*d^2+5*C*c*d-15*D 
*c^2)-8*b^2*c*(A*d^3-3*B*c*d^2+7*C*c^2*d-14*D*c^3))*arctanh(b^(1/2)*x/(b*x 
^2+a)^(1/2))/d^9-5/8*b*(4*a^3*d^6*(C*d-5*D*c)+4*a*b^2*c^2*d^2*(3*A*d^3-10* 
B*c*d^2+26*C*c^2*d-57*D*c^3)+3*a^2*b*d^4*(A*d^3-5*B*c*d^2+17*C*c^2*d-45*D* 
c^3)+8*b^3*c^4*(A*d^3-3*B*c*d^2+7*C*c^2*d-14*D*c^3))*arctanh((-b*c*x+a*d)/ 
(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^9/(a*d^2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 13.43 (sec) , antiderivative size = 986, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\frac {8 b d \left (7 a d^2 (C d-5 c D)-3 b \left (-15 c^2 C d+5 B c d^2-A d^3+35 c^3 D\right )\right ) \sqrt {a+b x^2}+3 b d^2 \left (9 a d^2 D+4 b \left (-5 c C d+B d^2+15 c^2 D\right )\right ) x \sqrt {a+b x^2}+8 b^2 d^3 (C d-5 c D) x^2 \sqrt {a+b x^2}+6 b^2 d^4 D x^3 \sqrt {a+b x^2}-\frac {6 d \left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{(c+d x)^4}+\frac {2 d \left (b c^2+a d^2\right ) \left (-4 a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (25 c^2 C d-21 B c d^2+17 A d^3-29 c^3 D\right )\right ) \sqrt {a+b x^2}}{(c+d x)^3}-\frac {d \left (12 a^2 d^4 (C d-3 c D)+a b d^2 \left (155 c^2 C d-79 B c d^2+27 A d^3-255 c^3 D\right )+2 b^2 c^2 \left (101 c^2 C d-69 B c d^2+43 A d^3-139 c^3 D\right )\right ) \sqrt {a+b x^2}}{(c+d x)^2}+\frac {d \left (-24 a^3 d^6 D-4 a^2 b d^4 \left (-55 c C d+14 B d^2+141 c^2 D\right )+a b^2 c d^2 \left (843 c^2 C d-383 B c d^2+139 A d^3-1591 c^3 D\right )-2 b^3 c^3 \left (-319 c^2 C d+171 B c d^2-77 A d^3+533 c^3 D\right )\right ) \sqrt {a+b x^2}}{\left (b c^2+a d^2\right ) (c+d x)}-\frac {15 b \left (-4 a^3 d^6 (C d-5 c D)-3 a^2 b d^4 \left (17 c^2 C d-5 B c d^2+A d^3-45 c^3 D\right )+8 b^3 c^4 \left (-7 c^2 C d+3 B c d^2-A d^3+14 c^3 D\right )+4 a b^2 c^2 d^2 \left (-26 c^2 C d+10 B c d^2-3 A d^3+57 c^3 D\right )\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{3/2}}+15 \sqrt {b} \left (3 a^2 d^4 D+4 a b d^2 \left (-5 c C d+B d^2+15 c^2 D\right )+8 b^2 c \left (-7 c^2 C d+3 B c d^2-A d^3+14 c^3 D\right )\right ) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )+\frac {15 b \left (-4 a^3 d^6 (C d-5 c D)-3 a^2 b d^4 \left (17 c^2 C d-5 B c d^2+A d^3-45 c^3 D\right )+8 b^3 c^4 \left (-7 c^2 C d+3 B c d^2-A d^3+14 c^3 D\right )+4 a b^2 c^2 d^2 \left (-26 c^2 C d+10 B c d^2-3 A d^3+57 c^3 D\right )\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{3/2}}}{24 d^9} \] Input:

Integrate[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^5,x]
 

Output:

(8*b*d*(7*a*d^2*(C*d - 5*c*D) - 3*b*(-15*c^2*C*d + 5*B*c*d^2 - A*d^3 + 35* 
c^3*D))*Sqrt[a + b*x^2] + 3*b*d^2*(9*a*d^2*D + 4*b*(-5*c*C*d + B*d^2 + 15* 
c^2*D))*x*Sqrt[a + b*x^2] + 8*b^2*d^3*(C*d - 5*c*D)*x^2*Sqrt[a + b*x^2] + 
6*b^2*d^4*D*x^3*Sqrt[a + b*x^2] - (6*d*(b*c^2 + a*d^2)^2*(c^2*C*d - B*c*d^ 
2 + A*d^3 - c^3*D)*Sqrt[a + b*x^2])/(c + d*x)^4 + (2*d*(b*c^2 + a*d^2)*(-4 
*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(25*c^2*C*d - 21*B*c*d^2 + 17*A* 
d^3 - 29*c^3*D))*Sqrt[a + b*x^2])/(c + d*x)^3 - (d*(12*a^2*d^4*(C*d - 3*c* 
D) + a*b*d^2*(155*c^2*C*d - 79*B*c*d^2 + 27*A*d^3 - 255*c^3*D) + 2*b^2*c^2 
*(101*c^2*C*d - 69*B*c*d^2 + 43*A*d^3 - 139*c^3*D))*Sqrt[a + b*x^2])/(c + 
d*x)^2 + (d*(-24*a^3*d^6*D - 4*a^2*b*d^4*(-55*c*C*d + 14*B*d^2 + 141*c^2*D 
) + a*b^2*c*d^2*(843*c^2*C*d - 383*B*c*d^2 + 139*A*d^3 - 1591*c^3*D) - 2*b 
^3*c^3*(-319*c^2*C*d + 171*B*c*d^2 - 77*A*d^3 + 533*c^3*D))*Sqrt[a + b*x^2 
])/((b*c^2 + a*d^2)*(c + d*x)) - (15*b*(-4*a^3*d^6*(C*d - 5*c*D) - 3*a^2*b 
*d^4*(17*c^2*C*d - 5*B*c*d^2 + A*d^3 - 45*c^3*D) + 8*b^3*c^4*(-7*c^2*C*d + 
 3*B*c*d^2 - A*d^3 + 14*c^3*D) + 4*a*b^2*c^2*d^2*(-26*c^2*C*d + 10*B*c*d^2 
 - 3*A*d^3 + 57*c^3*D))*Log[c + d*x])/(b*c^2 + a*d^2)^(3/2) + 15*Sqrt[b]*( 
3*a^2*d^4*D + 4*a*b*d^2*(-5*c*C*d + B*d^2 + 15*c^2*D) + 8*b^2*c*(-7*c^2*C* 
d + 3*B*c*d^2 - A*d^3 + 14*c^3*D))*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]] + (1 
5*b*(-4*a^3*d^6*(C*d - 5*c*D) - 3*a^2*b*d^4*(17*c^2*C*d - 5*B*c*d^2 + A*d^ 
3 - 45*c^3*D) + 8*b^3*c^4*(-7*c^2*C*d + 3*B*c*d^2 - A*d^3 + 14*c^3*D) +...
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 1175, normalized size of antiderivative = 1.36, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2182, 25, 2182, 25, 27, 681, 27, 681, 27, 682, 27, 719, 224, 219, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx\)

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{5/2} \left (4 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (4 C d-4 c D)-b \left (\frac {7 D c^3}{d^2}-\frac {7 C c^2}{d}+3 B c-3 A d\right )\right ) x+4 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^4}dx}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{5/2} \left (4 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (4 a (C d-c D)-b \left (\frac {7 D c^3}{d^2}-\frac {7 C c^2}{d}+3 B c-3 A d\right )\right ) x+4 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^4}dx}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^{7/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-A d^3-3 B c d^2-11 c^3 D+7 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}-\frac {\int -\frac {\left (3 d \left (A b d \left (4 b c^2+3 a d^2\right )+a \left (4 a (C d-2 c D) d^2+b c \left (-7 D c^2+3 C d c+B d^2\right )\right )\right )+4 \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{d^2 (c+d x)^3}dx}{3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (3 d \left (A b d \left (4 b c^2+3 a d^2\right )+a \left (4 a (C d-2 c D) d^2+b c \left (-7 D c^2+3 C d c+B d^2\right )\right )\right )+4 \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{d^2 (c+d x)^3}dx}{3 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-A d^3-3 B c d^2-11 c^3 D+7 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (3 d \left (A b d \left (4 b c^2+3 a d^2\right )+a \left (4 a (C d-2 c D) d^2+b c \left (-7 D c^2+3 C d c+B d^2\right )\right )\right )+4 \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{(c+d x)^3}dx}{3 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-A d^3-3 B c d^2-11 c^3 D+7 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {-\frac {5 \int -\frac {8 \left (2 a d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right )+3 b \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{(c+d x)^2}dx}{16 d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (3 \left (2 a^2 d^4 (2 C d-7 c D)+a b d^2 \left (3 A d^3-7 B c d^2-43 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (A d^3-3 B c d^2-14 c^3 D+7 c^2 C d\right )\right )-2 d x \left (3 a^2 d^4 D-2 a b d^2 \left (-2 B d^2-9 c^2 D+4 c C d\right )-b^2 c \left (-A d^3-3 B c d^2-14 c^3 D+7 c^2 C d\right )\right )\right )}{2 d^2 (c+d x)^2}}{3 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-A d^3-3 B c d^2-11 c^3 D+7 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {5 \int \frac {\left (2 a d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right )+3 b \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{(c+d x)^2}dx}{2 d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (3 \left (2 a^2 d^4 (2 C d-7 c D)+a b d^2 \left (3 A d^3-7 B c d^2-43 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (A d^3-3 B c d^2-14 c^3 D+7 c^2 C d\right )\right )-2 d x \left (3 a^2 d^4 D-2 a b d^2 \left (-2 B d^2-9 c^2 D+4 c C d\right )-b^2 c \left (-A d^3-3 B c d^2-14 c^3 D+7 c^2 C d\right )\right )\right )}{2 d^2 (c+d x)^2}}{3 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-A d^3-3 B c d^2-11 c^3 D+7 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}-\frac {\int -\frac {6 b \left (a d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )+2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{2 d^2}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \int \frac {\left (a d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )+2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\int \frac {2 b \left (b c^2+a d^2\right ) \left (a d \left (a^2 (4 C d-17 c D) d^4+a b \left (-72 D c^3+31 C d c^2-11 B d^2 c+3 A d^3\right ) d^2+4 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )+\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \int \frac {a d \left (a^2 (4 C d-17 c D) d^4+a b \left (-72 D c^3+31 C d c^2-11 B d^2 c+3 A d^3\right ) d^2+4 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )+\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}+\frac {\left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}+\frac {\left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}-\frac {\left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}\right )}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (4 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-11 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) \left (b x^2+a\right )^{7/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\frac {5 \left (\frac {3 b \left (\frac {\sqrt {b x^2+a} \left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+\left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x d+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )}{d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (b c^2+a d^2\right ) \left (3 a^2 D d^4-4 a b \left (-15 D c^2+5 C d c-B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}-\frac {\left (4 a^3 (C d-5 c D) d^6+3 a^2 b \left (-45 D c^3+17 C d c^2-5 B d^2 c+A d^3\right ) d^4+4 a b^2 c^2 \left (-57 D c^3+26 C d c^2-10 B d^2 c+3 A d^3\right ) d^2+8 b^3 c^4 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{d \sqrt {b c^2+a d^2}}\right )}{d^2}\right )}{d^2}-\frac {\left (2 \left (3 a^3 D d^6-2 a^2 b \left (-23 D c^2+8 C d c-2 B d^2\right ) d^4-a b^2 c \left (-100 D c^3+45 C d c^2-17 B d^2 c+5 A d^3\right ) d^2-4 b^3 c^3 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-b d \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}\right )}{2 d^2}-\frac {\left (3 \left (2 a^2 (2 C d-7 c D) d^4+a b \left (-43 D c^3+19 C d c^2-7 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-14 D c^3+7 C d c^2-3 B d^2 c+A d^3\right )\right )-2 d \left (3 a^2 D d^4-2 a b \left (-9 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-14 D c^3+7 C d c^2-3 B d^2 c-A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 (c+d x)^2}}{3 d^2 \left (b c^2+a d^2\right )}}{4 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}\)

Input:

Int[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^5,x]
 

Output:

-1/4*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(7/2))/(d^2*(b*c^2 + 
 a*d^2)*(c + d*x)^4) + (((4*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2 
*C*d - 3*B*c*d^2 - A*d^3 - 11*c^3*D))*(a + b*x^2)^(7/2))/(3*d^2*(b*c^2 + a 
*d^2)*(c + d*x)^3) + (-1/2*((3*(2*a^2*d^4*(2*C*d - 7*c*D) + a*b*d^2*(19*c^ 
2*C*d - 7*B*c*d^2 + 3*A*d^3 - 43*c^3*D) + 2*b^2*c^2*(7*c^2*C*d - 3*B*c*d^2 
 + A*d^3 - 14*c^3*D)) - 2*d*(3*a^2*d^4*D - 2*a*b*d^2*(4*c*C*d - 2*B*d^2 - 
9*c^2*D) - b^2*c*(7*c^2*C*d - 3*B*c*d^2 - A*d^3 - 14*c^3*D))*x)*(a + b*x^2 
)^(5/2))/(d^2*(c + d*x)^2) + (5*(-(((2*(3*a^3*d^6*D - 2*a^2*b*d^4*(8*c*C*d 
 - 2*B*d^2 - 23*c^2*D) - a*b^2*c*d^2*(45*c^2*C*d - 17*B*c*d^2 + 5*A*d^3 - 
100*c^3*D) - 4*b^3*c^3*(7*c^2*C*d - 3*B*c*d^2 + A*d^3 - 14*c^3*D)) - b*d*( 
2*a^2*d^4*(2*C*d - 7*c*D) + a*b*d^2*(19*c^2*C*d - 7*B*c*d^2 + 3*A*d^3 - 43 
*c^3*D) + 2*b^2*c^2*(7*c^2*C*d - 3*B*c*d^2 + A*d^3 - 14*c^3*D))*x)*(a + b* 
x^2)^(3/2))/(d^2*(c + d*x))) + (3*b*(((4*a^3*d^6*(C*d - 5*c*D) + 4*a*b^2*c 
^2*d^2*(26*c^2*C*d - 10*B*c*d^2 + 3*A*d^3 - 57*c^3*D) + 3*a^2*b*d^4*(17*c^ 
2*C*d - 5*B*c*d^2 + A*d^3 - 45*c^3*D) + 8*b^3*c^4*(7*c^2*C*d - 3*B*c*d^2 + 
 A*d^3 - 14*c^3*D) + d*(3*a^3*d^6*D - 2*a^2*b*d^4*(8*c*C*d - 2*B*d^2 - 23* 
c^2*D) - a*b^2*c*d^2*(45*c^2*C*d - 17*B*c*d^2 + 5*A*d^3 - 100*c^3*D) - 4*b 
^3*c^3*(7*c^2*C*d - 3*B*c*d^2 + A*d^3 - 14*c^3*D))*x)*Sqrt[a + b*x^2])/d^2 
 + ((b*c^2 + a*d^2)*(((b*c^2 + a*d^2)*(3*a^2*d^4*D - 4*a*b*d^2*(5*c*C*d - 
B*d^2 - 15*c^2*D) - 8*b^2*c*(7*c^2*C*d - 3*B*c*d^2 + A*d^3 - 14*c^3*D))...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(13260\) vs. \(2(816)=1632\).

Time = 1.69 (sec) , antiderivative size = 13261, normalized size of antiderivative = 15.35

method	result	size
default	\(\text {Expression too large to display}\)	\(13261\)

Input:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{5}}\, dx \] Input:

integrate((b*x**2+a)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**5,x)
 

Output:

Integral((a + b*x**2)**(5/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9069 vs. \(2 (821) = 1642\).

Time = 0.61 (sec) , antiderivative size = 9069, normalized size of antiderivative = 10.50 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="maxi 
ma")
 

Output:

5/16*D*b^6*c^10*arcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^9 + 3*a*b^(5/2)*c^4* 
d^11 + 3*a^2*b^(3/2)*c^2*d^13 + a^3*sqrt(b)*d^15) - 5/16*C*b^6*c^9*arcsinh 
(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^8 + 3*a*b^(5/2)*c^4*d^10 + 3*a^2*b^(3/2)*c^ 
2*d^12 + a^3*sqrt(b)*d^14) + 5/16*D*a*b^5*c^8*arcsinh(b*x/sqrt(a*b))/(b^(7 
/2)*c^6*d^7 + 3*a*b^(5/2)*c^4*d^9 + 3*a^2*b^(3/2)*c^2*d^11 + a^3*sqrt(b)*d 
^13) + 5/16*B*b^6*c^8*arcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^7 + 3*a*b^(5/2 
)*c^4*d^9 + 3*a^2*b^(3/2)*c^2*d^11 + a^3*sqrt(b)*d^13) - 5/16*sqrt(b*x^2 + 
 a)*D*b^5*c^8*x/(b^3*c^6*d^7 + 3*a*b^2*c^4*d^9 + 3*a^2*b*c^2*d^11 + a^3*d^ 
13) - 5/16*C*a*b^5*c^7*arcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^6 + 3*a*b^(5/ 
2)*c^4*d^8 + 3*a^2*b^(3/2)*c^2*d^10 + a^3*sqrt(b)*d^12) - 5/16*A*b^6*c^7*a 
rcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^6 + 3*a*b^(5/2)*c^4*d^8 + 3*a^2*b^(3/ 
2)*c^2*d^10 + a^3*sqrt(b)*d^12) - 105/16*D*b^5*c^8*arcsinh(b*x/sqrt(a*b))/ 
(b^(5/2)*c^4*d^9 + 2*a*b^(3/2)*c^2*d^11 + a^2*sqrt(b)*d^13) + 5/16*sqrt(b* 
x^2 + a)*C*b^5*c^7*x/(b^3*c^6*d^6 + 3*a*b^2*c^4*d^8 + 3*a^2*b*c^2*d^10 + a 
^3*d^12) + 5/16*B*a*b^5*c^6*arcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^5 + 3*a* 
b^(5/2)*c^4*d^7 + 3*a^2*b^(3/2)*c^2*d^9 + a^3*sqrt(b)*d^11) + 85/16*C*b^5* 
c^7*arcsinh(b*x/sqrt(a*b))/(b^(5/2)*c^4*d^8 + 2*a*b^(3/2)*c^2*d^10 + a^2*s 
qrt(b)*d^12) + 5/24*(b*x^2 + a)^(3/2)*D*b^4*c^7/(b^3*c^6*d^6 + 3*a*b^2*c^4 
*d^8 + 3*a^2*b*c^2*d^10 + a^3*d^12) - 5/24*(b*x^2 + a)^(3/2)*D*b^4*c^6*x/( 
b^3*c^6*d^5 + 3*a*b^2*c^4*d^7 + 3*a^2*b*c^2*d^9 + a^3*d^11) - 5/16*sqrt...
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="giac 
")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^5} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^5,x)
 

Output:

int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^5, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{5}}d x \] Input:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x)
 

Output:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x)