\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2+D x^3)}{(c+d x)^5} \, dx\) [67]

Optimal result

Integrand size = 34, antiderivative size = 542 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=-\frac {\left (8 b^3 c^7 D+4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (c^2 C d-5 B c d^2+A d^3-13 c^3 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )-d \left (A b^2 c d^3 \left (4 b c^2-a d^2\right )-12 b^3 c^6 D-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)-a b^2 c^2 d^2 \left (c C d-5 B d^2+35 c^2 D\right )\right ) x\right ) \sqrt {a+b x^2}}{8 d^4 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{3/2}}{4 d^2 \left (b c^2+a d^2\right ) (c+d x)^4}+\frac {\left (4 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (3 c^2 C d+B c d^2-5 A d^3-7 c^3 D\right )\right ) \left (a+b x^2\right )^{3/2}}{12 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^3}+\frac {\sqrt {b} D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^5}+\frac {b \left (8 b^3 c^7 D-4 a^3 d^6 (C d-5 c D)-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+a^2 b d^4 \left (c^2 C d-5 B c d^2+A d^3+35 c^3 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{8 d^5 \left (b c^2+a d^2\right )^{7/2}} \] Output:

-1/8*(8*b^3*c^7*D+4*a^3*d^6*(C*d-D*c)-a^2*b*d^4*(A*d^3-5*B*c*d^2+C*c^2*d-1 
3*D*c^3)+4*a*b^2*c^2*d^2*(A*d^3+5*D*c^3)-d*(A*b^2*c*d^3*(-a*d^2+4*b*c^2)-1 
2*b^3*c^6*D-8*a^3*d^6*D+4*a^2*b*c*d^4*(C*d-9*D*c)-a*b^2*c^2*d^2*(-5*B*d^2+ 
C*c*d+35*D*c^2))*x)*(b*x^2+a)^(1/2)/d^4/(a*d^2+b*c^2)^3/(d*x+c)^2-1/4*(A*d 
^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(3/2)/d^2/(a*d^2+b*c^2)/(d*x+c)^4+1/12 
*(4*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(-5*A*d^3+B*c*d^2+3*C*c^2*d-7*D*c^3 
))*(b*x^2+a)^(3/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)^3+b^(1/2)*D*arctanh(b^(1/2) 
*x/(b*x^2+a)^(1/2))/d^5+1/8*b*(8*b^3*c^7*D-4*a^3*d^6*(C*d-5*D*c)-4*a*b^2*c 
^2*d^2*(A*d^3-7*D*c^3)+a^2*b*d^4*(A*d^3-5*B*c*d^2+C*c^2*d+35*D*c^3))*arcta 
nh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^5/(a*d^2+b*c^2)^(7/ 
2)
 

Mathematica [A] (verified)

Time = 12.32 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\frac {-\frac {d \sqrt {a+b x^2} \left (6 \left (b c^2+a d^2\right )^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+2 \left (b c^2+a d^2\right )^2 \left (4 a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-9 c^2 C d+5 B c d^2-A d^3+13 c^3 D\right )\right ) (c+d x)-\left (b c^2+a d^2\right ) \left (-12 a^2 d^4 (C d-3 c D)+2 b^2 c^2 \left (-9 c^2 C d+B c d^2+A d^3+23 c^3 D\right )+a b d^2 \left (-35 c^2 C d+7 B c d^2-3 A d^3+87 c^3 D\right )\right ) (c+d x)^2+\left (24 a^3 d^6 D+4 a^2 b d^4 \left (-7 c C d+2 B d^2+33 c^2 D\right )-2 b^3 c^3 \left (3 c^2 C d+B c d^2+A d^3-25 c^3 D\right )+a b^2 c d^2 \left (-19 c^2 C d-9 B c d^2+13 A d^3+143 c^3 D\right )\right ) (c+d x)^3\right )}{\left (b c^2+a d^2\right )^3 (c+d x)^4}-\frac {3 b \left (8 b^3 c^7 D-4 a^3 d^6 (C d-5 c D)+4 a b^2 c^2 d^2 \left (-A d^3+7 c^3 D\right )+a^2 b d^4 \left (c^2 C d-5 B c d^2+A d^3+35 c^3 D\right )\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{7/2}}+24 \sqrt {b} D \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )+\frac {3 b \left (8 b^3 c^7 D-4 a^3 d^6 (C d-5 c D)+4 a b^2 c^2 d^2 \left (-A d^3+7 c^3 D\right )+a^2 b d^4 \left (c^2 C d-5 B c d^2+A d^3+35 c^3 D\right )\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{7/2}}}{24 d^5} \] Input:

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

Output:

(-((d*Sqrt[a + b*x^2]*(6*(b*c^2 + a*d^2)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^ 
3*D) + 2*(b*c^2 + a*d^2)^2*(4*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(-9 
*c^2*C*d + 5*B*c*d^2 - A*d^3 + 13*c^3*D))*(c + d*x) - (b*c^2 + a*d^2)*(-12 
*a^2*d^4*(C*d - 3*c*D) + 2*b^2*c^2*(-9*c^2*C*d + B*c*d^2 + A*d^3 + 23*c^3* 
D) + a*b*d^2*(-35*c^2*C*d + 7*B*c*d^2 - 3*A*d^3 + 87*c^3*D))*(c + d*x)^2 + 
 (24*a^3*d^6*D + 4*a^2*b*d^4*(-7*c*C*d + 2*B*d^2 + 33*c^2*D) - 2*b^3*c^3*( 
3*c^2*C*d + B*c*d^2 + A*d^3 - 25*c^3*D) + a*b^2*c*d^2*(-19*c^2*C*d - 9*B*c 
*d^2 + 13*A*d^3 + 143*c^3*D))*(c + d*x)^3))/((b*c^2 + a*d^2)^3*(c + d*x)^4 
)) - (3*b*(8*b^3*c^7*D - 4*a^3*d^6*(C*d - 5*c*D) + 4*a*b^2*c^2*d^2*(-(A*d^ 
3) + 7*c^3*D) + a^2*b*d^4*(c^2*C*d - 5*B*c*d^2 + A*d^3 + 35*c^3*D))*Log[c 
+ d*x])/(b*c^2 + a*d^2)^(7/2) + 24*Sqrt[b]*D*Log[b*x + Sqrt[b]*Sqrt[a + b* 
x^2]] + (3*b*(8*b^3*c^7*D - 4*a^3*d^6*(C*d - 5*c*D) + 4*a*b^2*c^2*d^2*(-(A 
*d^3) + 7*c^3*D) + a^2*b*d^4*(c^2*C*d - 5*B*c*d^2 + A*d^3 + 35*c^3*D))*Log 
[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^2)^(7/2) 
)/(24*d^5)
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2182, 25, 2182, 27, 680, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 680

\(\displaystyle \frac {\frac {-\frac {\int -\frac {2 b \left (8 D x \left (b c^2+a d^2\right )^3+a d \left (-4 b^2 D c^5-a b d^2 \left (11 D c^2+C d c-5 B d^2\right ) c+A b d^3 \left (4 b c^2-a d^2\right )+4 a^2 d^4 (C d-3 c D)\right )\right )}{(c+d x) \sqrt {b x^2+a}}dx}{4 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {b \int \frac {8 D x \left (b c^2+a d^2\right )^3+a d \left (-4 b^2 D c^5-a b d^2 \left (11 D c^2+C d c-5 B d^2\right ) c+A b d^3 \left (4 b c^2-a d^2\right )+4 a^2 d^4 (C d-3 c D)\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {b \left (\frac {8 D \left (a d^2+b c^2\right )^3 \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {\left (-4 a^3 d^6 (C d-5 c D)+a^2 b d^4 \left (A d^3-5 B c d^2+35 c^3 D+c^2 C d\right )-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+8 b^3 c^7 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {b \left (\frac {8 D \left (a d^2+b c^2\right )^3 \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {\left (-4 a^3 d^6 (C d-5 c D)+a^2 b d^4 \left (A d^3-5 B c d^2+35 c^3 D+c^2 C d\right )-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+8 b^3 c^7 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {b \left (\frac {8 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^3}{\sqrt {b} d}-\frac {\left (-4 a^3 d^6 (C d-5 c D)+a^2 b d^4 \left (A d^3-5 B c d^2+35 c^3 D+c^2 C d\right )-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+8 b^3 c^7 D\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {b \left (\frac {\left (-4 a^3 d^6 (C d-5 c D)+a^2 b d^4 \left (A d^3-5 B c d^2+35 c^3 D+c^2 C d\right )-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+8 b^3 c^7 D\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}+\frac {8 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^3}{\sqrt {b} d}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {b \left (\frac {\text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (-4 a^3 d^6 (C d-5 c D)+a^2 b d^4 \left (A d^3-5 B c d^2+35 c^3 D+c^2 C d\right )-4 a b^2 c^2 d^2 \left (A d^3-7 c^3 D\right )+8 b^3 c^7 D\right )}{d \sqrt {a d^2+b c^2}}+\frac {8 D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^3}{\sqrt {b} d}\right )}{2 d^2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (4 a^3 d^6 (C d-c D)-a^2 b d^4 \left (A d^3-5 B c d^2-13 c^3 D+c^2 C d\right )-d x \left (-8 a^3 d^6 D+4 a^2 b c d^4 (C d-9 c D)+A b^2 c d^3 \left (4 b c^2-a d^2\right )-a b^2 c^2 d^2 \left (-5 B d^2+35 c^2 D+c C d\right )-12 b^3 c^6 D\right )+4 a b^2 c^2 d^2 \left (A d^3+5 c^3 D\right )+8 b^3 c^7 D\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (4 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-5 A d^3+B c d^2-7 c^3 D+3 c^2 C d\right )\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}\)

Input:

Int[(Sqrt[a + b*x^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)^(3/2))/(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*(3*c^2 
*C*d + B*c*d^2 - 5*A*d^3 - 7*c^3*D))*(a + b*x^2)^(3/2))/(3*d^2*(b*c^2 + a* 
d^2)*(c + d*x)^3) + (-1/2*((8*b^3*c^7*D + 4*a^3*d^6*(C*d - c*D) - a^2*b*d^ 
4*(c^2*C*d - 5*B*c*d^2 + A*d^3 - 13*c^3*D) + 4*a*b^2*c^2*d^2*(A*d^3 + 5*c^ 
3*D) - d*(A*b^2*c*d^3*(4*b*c^2 - a*d^2) - 12*b^3*c^6*D - 8*a^3*d^6*D + 4*a 
^2*b*c*d^4*(C*d - 9*c*D) - a*b^2*c^2*d^2*(c*C*d - 5*B*d^2 + 35*c^2*D))*x)* 
Sqrt[a + b*x^2])/(d^2*(b*c^2 + a*d^2)*(c + d*x)^2) + (b*((8*(b*c^2 + a*d^2 
)^3*D*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) + ((8*b^3*c^7*D - 
4*a^3*d^6*(C*d - 5*c*D) - 4*a*b^2*c^2*d^2*(A*d^3 - 7*c^3*D) + a^2*b*d^4*(c 
^2*C*d - 5*B*c*d^2 + A*d^3 + 35*c^3*D))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 
+ a*d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b*c^2 + a*d^2])))/(2*d^2*(b*c^2 + a*d^ 
2)))/(d^2*(b*c^2 + a*d^2)))/(4*(b*c^2 + a*d^2))
 

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 680

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m 
+ 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* 
f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim 
p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2))   Int[(d + e*x)^(m + 2)*(a + c*x^ 
2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f 
*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, 
 g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3 
, 0]
 

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. \(4463\) vs. \(2(516)=1032\).

Time = 1.58 (sec) , antiderivative size = 4464, normalized size of antiderivative = 8.24

method	result	size
default	\(\text {Expression too large to display}\)	\(4464\)

Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x,method=_RETURNVERBOSE)
 

Output:

D/d^5*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b* 
c^2)/d^2)^(3/2)-b*c*d/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2* 
b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(a*d^2+b*c^2)/d^2/((a*d^2+b*c^2)/d 
^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1 
/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+2*b/( 
a*d^2+b*c^2)*d^2*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2 
)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) 
/d^2)^(1/2))))+(C*d-3*D*c)/d^6*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d 
)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+1/2*b*c*d/(a*d^2+b*c^2)*(-1/( 
a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 
3/2)-b*c*d/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^ 
(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/ 
d)+(a*d^2+b*c^2)/d^2)^(1/2))-(a*d^2+b*c^2)/d^2/((a*d^2+b*c^2)/d^2)^(1/2)*l 
n((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c 
/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))+2*b/(a*d^2+b*c^2 
)*d^2*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c 
^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/ 
d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1...
 

Fricas [F(-1)]

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

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**5,x)
 

Output:

Integral(sqrt(a + b*x**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. 4783 vs. \(2 (520) = 1040\).

Time = 0.23 (sec) , antiderivative size = 4783, normalized size of antiderivative = 8.82 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="maxi 
ma")
 

Output:

5/8*sqrt(b*x^2 + a)*D*b^3*c^6/(b^3*c^6*d^5*x + 3*a*b^2*c^4*d^7*x + 3*a^2*b 
*c^2*d^9*x + a^3*d^11*x + b^3*c^7*d^4 + 3*a*b^2*c^5*d^6 + 3*a^2*b*c^3*d^8 
+ a^3*c*d^10) + 5/8*(b*x^2 + a)^(3/2)*D*b^2*c^5/(b^3*c^6*d^4*x^2 + 3*a*b^2 
*c^4*d^6*x^2 + 3*a^2*b*c^2*d^8*x^2 + a^3*d^10*x^2 + 2*b^3*c^7*d^3*x + 6*a* 
b^2*c^5*d^5*x + 6*a^2*b*c^3*d^7*x + 2*a^3*c*d^9*x + b^3*c^8*d^2 + 3*a*b^2* 
c^6*d^4 + 3*a^2*b*c^4*d^6 + a^3*c^2*d^8) - 5/8*sqrt(b*x^2 + a)*C*b^3*c^5/( 
b^3*c^6*d^4*x + 3*a*b^2*c^4*d^6*x + 3*a^2*b*c^2*d^8*x + a^3*d^10*x + b^3*c 
^7*d^3 + 3*a*b^2*c^5*d^5 + 3*a^2*b*c^3*d^7 + a^3*c*d^9) - 5/8*sqrt(b*x^2 + 
 a)*D*b^3*c^5/(b^3*c^6*d^4 + 3*a*b^2*c^4*d^6 + 3*a^2*b*c^2*d^8 + a^3*d^10) 
 - 5/8*(b*x^2 + a)^(3/2)*C*b^2*c^4/(b^3*c^6*d^3*x^2 + 3*a*b^2*c^4*d^5*x^2 
+ 3*a^2*b*c^2*d^7*x^2 + a^3*d^9*x^2 + 2*b^3*c^7*d^2*x + 6*a*b^2*c^5*d^4*x 
+ 6*a^2*b*c^3*d^6*x + 2*a^3*c*d^8*x + b^3*c^8*d + 3*a*b^2*c^6*d^3 + 3*a^2* 
b*c^4*d^5 + a^3*c^2*d^7) + 5/8*sqrt(b*x^2 + a)*B*b^3*c^4/(b^3*c^6*d^3*x + 
3*a*b^2*c^4*d^5*x + 3*a^2*b*c^2*d^7*x + a^3*d^9*x + b^3*c^7*d^2 + 3*a*b^2* 
c^5*d^4 + 3*a^2*b*c^3*d^6 + a^3*c*d^8) + 5/8*sqrt(b*x^2 + a)*C*b^3*c^4/(b^ 
3*c^6*d^3 + 3*a*b^2*c^4*d^5 + 3*a^2*b*c^2*d^7 + a^3*d^9) + 5/8*(b*x^2 + a) 
^(3/2)*B*b^2*c^3/(b^3*c^6*d^2*x^2 + 3*a*b^2*c^4*d^4*x^2 + 3*a^2*b*c^2*d^6* 
x^2 + a^3*d^8*x^2 + 2*b^3*c^7*d*x + 6*a*b^2*c^5*d^3*x + 6*a^2*b*c^3*d^5*x 
+ 2*a^3*c*d^7*x + b^3*c^8 + 3*a*b^2*c^6*d^2 + 3*a^2*b*c^4*d^4 + a^3*c^2*d^ 
6) - 5/8*sqrt(b*x^2 + a)*A*b^3*c^3/(b^3*c^6*d^2*x + 3*a*b^2*c^4*d^4*x +...
 

Giac [F(-2)]

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

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x, algorithm="giac 
")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^5,x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^5, x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x)
 

Output:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^5,x)