\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2+D x^3)}{(c+d x)^6} \, dx\) [68]

Optimal result

Integrand size = 34, antiderivative size = 525 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {\left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-6 B d)-4 a^2 d^3 D-a b d \left (6 c C d-B d^2-3 c^2 D\right )\right )\right ) (a d-b c x) \sqrt {a+b x^2}}{8 \left (b c^2+a d^2\right )^4 (c+d x)^2}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{3/2}}{5 d^2 \left (b c^2+a d^2\right ) (c+d x)^5}+\frac {\left (5 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (3 c^2 C d+2 B c d^2-7 A d^3-8 c^3 D\right )\right ) \left (a+b x^2\right )^{3/2}}{20 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^4}-\frac {\left (20 a^2 d^4 (C d-3 c D)-b^2 c^2 \left (3 c^2 C d+2 B c d^2-27 A d^3+12 c^3 D\right )-a b d^2 \left (18 c^2 C d-33 B c d^2+8 A d^3+37 c^3 D\right )\right ) \left (a+b x^2\right )^{3/2}}{60 d^2 \left (b c^2+a d^2\right )^3 (c+d x)^3}-\frac {a b \left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-6 B d)-4 a^2 d^3 D-a b d \left (6 c C d-B d^2-3 c^2 D\right )\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{8 \left (b c^2+a d^2\right )^{9/2}} \] Output:

-1/8*(A*b^2*c*(-3*a*d^2+4*b*c^2)-a*(b^2*c^2*(-6*B*d+C*c)-4*a^2*d^3*D-a*b*d 
*(-B*d^2+6*C*c*d-3*D*c^2)))*(-b*c*x+a*d)*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^4/( 
d*x+c)^2-1/5*(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)^5+1/20*(5*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(-7*A*d^3+2*B*c*d^ 
2+3*C*c^2*d-8*D*c^3))*(b*x^2+a)^(3/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)^4-1/60*( 
20*a^2*d^4*(C*d-3*D*c)-b^2*c^2*(-27*A*d^3+2*B*c*d^2+3*C*c^2*d+12*D*c^3)-a* 
b*d^2*(8*A*d^3-33*B*c*d^2+18*C*c^2*d+37*D*c^3))*(b*x^2+a)^(3/2)/d^2/(a*d^2 
+b*c^2)^3/(d*x+c)^3-1/8*a*b*(A*b^2*c*(-3*a*d^2+4*b*c^2)-a*(b^2*c^2*(-6*B*d 
+C*c)-4*a^2*d^3*D-a*b*d*(-B*d^2+6*C*c*d-3*D*c^2)))*arctanh((-b*c*x+a*d)/(a 
*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2)^(9/2)
 

Mathematica [A] (verified)

Time = 12.13 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=-\frac {\sqrt {a+b x^2} \left (24 \left (b c^2+a d^2\right )^4 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+6 \left (b c^2+a d^2\right )^3 \left (5 a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-11 c^2 C d+6 B c d^2-A d^3+16 c^3 D\right )\right ) (c+d x)-2 \left (b c^2+a d^2\right )^2 \left (-20 a^2 d^4 (C d-3 c D)+b^2 c^2 \left (-27 c^2 C d+2 B c d^2+3 A d^3+72 c^3 D\right )+a b d^2 \left (-54 c^2 C d+9 B c d^2-4 A d^3+139 c^3 D\right )\right ) (c+d x)^2+\left (b c^2+a d^2\right ) \left (60 a^3 d^6 D+5 a^2 b d^4 \left (-10 c C d+3 B d^2+57 c^2 D\right )+2 b^3 c^3 \left (-3 c^2 C d-2 B c d^2-3 A d^3+48 c^3 D\right )+a b^2 c d^2 \left (-21 c^2 C d-24 B c d^2+29 A d^3+286 c^3 D\right )\right ) (c+d x)^3-b \left (20 a^3 d^6 (-2 C d+9 c D)+2 b^3 c^4 \left (3 c^2 C d+2 B c d^2+3 A d^3+12 c^3 D\right )+a b^2 c^2 d^2 \left (27 c^2 C d+28 B c d^2-83 A d^3+98 c^3 D\right )+a^2 b d^4 \left (86 c^2 C d-81 B c d^2+16 A d^3+149 c^3 D\right )\right ) (c+d x)^4\right )}{120 \left (b c^2 d+a d^3\right )^4 (c+d x)^5}+\frac {a b \left (A b^2 c \left (4 b c^2-3 a d^2\right )+a \left (b^2 c^2 (-c C+6 B d)+4 a^2 d^3 D-a b d \left (-6 c C d+B d^2+3 c^2 D\right )\right )\right ) \log (c+d x)}{8 \left (b c^2+a d^2\right )^{9/2}}-\frac {a b \left (A b^2 c \left (4 b c^2-3 a d^2\right )+a \left (b^2 c^2 (-c C+6 B d)+4 a^2 d^3 D-a b d \left (-6 c C d+B d^2+3 c^2 D\right )\right )\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{8 \left (b c^2+a d^2\right )^{9/2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^6,x]
 

Output:

-1/120*(Sqrt[a + b*x^2]*(24*(b*c^2 + a*d^2)^4*(c^2*C*d - B*c*d^2 + A*d^3 - 
 c^3*D) + 6*(b*c^2 + a*d^2)^3*(5*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c* 
(-11*c^2*C*d + 6*B*c*d^2 - A*d^3 + 16*c^3*D))*(c + d*x) - 2*(b*c^2 + a*d^2 
)^2*(-20*a^2*d^4*(C*d - 3*c*D) + b^2*c^2*(-27*c^2*C*d + 2*B*c*d^2 + 3*A*d^ 
3 + 72*c^3*D) + a*b*d^2*(-54*c^2*C*d + 9*B*c*d^2 - 4*A*d^3 + 139*c^3*D))*( 
c + d*x)^2 + (b*c^2 + a*d^2)*(60*a^3*d^6*D + 5*a^2*b*d^4*(-10*c*C*d + 3*B* 
d^2 + 57*c^2*D) + 2*b^3*c^3*(-3*c^2*C*d - 2*B*c*d^2 - 3*A*d^3 + 48*c^3*D) 
+ a*b^2*c*d^2*(-21*c^2*C*d - 24*B*c*d^2 + 29*A*d^3 + 286*c^3*D))*(c + d*x) 
^3 - b*(20*a^3*d^6*(-2*C*d + 9*c*D) + 2*b^3*c^4*(3*c^2*C*d + 2*B*c*d^2 + 3 
*A*d^3 + 12*c^3*D) + a*b^2*c^2*d^2*(27*c^2*C*d + 28*B*c*d^2 - 83*A*d^3 + 9 
8*c^3*D) + a^2*b*d^4*(86*c^2*C*d - 81*B*c*d^2 + 16*A*d^3 + 149*c^3*D))*(c 
+ d*x)^4))/((b*c^2*d + a*d^3)^4*(c + d*x)^5) + (a*b*(A*b^2*c*(4*b*c^2 - 3* 
a*d^2) + a*(b^2*c^2*(-(c*C) + 6*B*d) + 4*a^2*d^3*D - a*b*d*(-6*c*C*d + B*d 
^2 + 3*c^2*D)))*Log[c + d*x])/(8*(b*c^2 + a*d^2)^(9/2)) - (a*b*(A*b^2*c*(4 
*b*c^2 - 3*a*d^2) + a*(b^2*c^2*(-(c*C) + 6*B*d) + 4*a^2*d^3*D - a*b*d*(-6* 
c*C*d + B*d^2 + 3*c^2*D)))*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + 
b*x^2]])/(8*(b*c^2 + a*d^2)^(9/2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2182, 25, 2182, 25, 27, 679, 486, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx\)

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}-\frac {\int -\frac {\left (4 \left (A b \left (5 b c^2-2 a d^2\right )+a \left (5 a d (C d-2 c D)-b c \left (\frac {3 D c^2}{d}+2 C c-7 B d\right )\right )\right ) d^2+\left (20 a^2 D d^4+5 a b \left (5 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (12 D c^3+3 C d c^2+2 B d^2 c-7 A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{d^2 (c+d x)^4}dx}{4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (4 \left (A b \left (5 b c^2-2 a d^2\right )+a \left (5 a d (C d-2 c D)-b c \left (\frac {3 D c^2}{d}+2 C c-7 B d\right )\right )\right ) d^2+\left (20 a^2 D d^4+5 a b \left (5 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (12 D c^3+3 C d c^2+2 B d^2 c-7 A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{d^2 (c+d x)^4}dx}{4 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (4 \left (A b \left (5 b c^2-2 a d^2\right )+a \left (5 a d (C d-2 c D)-b c \left (\frac {3 D c^2}{d}+2 C c-7 B d\right )\right )\right ) d^2+\left (20 a^2 D d^4+5 a b \left (5 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (12 D c^3+3 C d c^2+2 B d^2 c-7 A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{(c+d x)^4}dx}{4 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\frac {5 d^2 \left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (-4 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+6 c C d\right )+b^2 c^2 (c C-6 B d)\right )\right ) \int \frac {\sqrt {b x^2+a}}{(c+d x)^3}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (20 a^2 d^4 (C d-3 c D)-a b d^2 \left (8 A d^3-33 B c d^2+37 c^3 D+18 c^2 C d\right )-b^2 c^2 \left (-27 A d^3+2 B c d^2+12 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {5 d^2 \left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (-4 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+6 c C d\right )+b^2 c^2 (c C-6 B d)\right )\right ) \left (\frac {a b \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (20 a^2 d^4 (C d-3 c D)-a b d^2 \left (8 A d^3-33 B c d^2+37 c^3 D+18 c^2 C d\right )-b^2 c^2 \left (-27 A d^3+2 B c d^2+12 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {5 d^2 \left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (-4 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+6 c C d\right )+b^2 c^2 (c C-6 B d)\right )\right ) \left (-\frac {a b \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (20 a^2 d^4 (C d-3 c D)-a b d^2 \left (8 A d^3-33 B c d^2+37 c^3 D+18 c^2 C d\right )-b^2 c^2 \left (-27 A d^3+2 B c d^2+12 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {5 d^2 \left (-\frac {a b \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 \left (a d^2+b c^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right ) \left (A b^2 c \left (4 b c^2-3 a d^2\right )-a \left (-4 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+6 c C d\right )+b^2 c^2 (c C-6 B d)\right )\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (20 a^2 d^4 (C d-3 c D)-a b d^2 \left (8 A d^3-33 B c d^2+37 c^3 D+18 c^2 C d\right )-b^2 c^2 \left (-27 A d^3+2 B c d^2+12 c^3 D+3 c^2 C d\right )\right )}{3 (c+d x)^3 \left (a d^2+b c^2\right )}}{4 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (5 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-7 A d^3+2 B c d^2-8 c^3 D+3 c^2 C d\right )\right )}{4 d^2 (c+d x)^4 \left (a d^2+b c^2\right )}}{5 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{5 d^2 (c+d x)^5 \left (a d^2+b c^2\right )}\)

Input:

Int[(Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^6,x]
 

Output:

-1/5*((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)^5) + (((5*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(3*c^2 
*C*d + 2*B*c*d^2 - 7*A*d^3 - 8*c^3*D))*(a + b*x^2)^(3/2))/(4*d^2*(b*c^2 + 
a*d^2)*(c + d*x)^4) + (-1/3*((20*a^2*d^4*(C*d - 3*c*D) - b^2*c^2*(3*c^2*C* 
d + 2*B*c*d^2 - 27*A*d^3 + 12*c^3*D) - a*b*d^2*(18*c^2*C*d - 33*B*c*d^2 + 
8*A*d^3 + 37*c^3*D))*(a + b*x^2)^(3/2))/((b*c^2 + a*d^2)*(c + d*x)^3) + (5 
*d^2*(A*b^2*c*(4*b*c^2 - 3*a*d^2) - a*(b^2*c^2*(c*C - 6*B*d) - 4*a^2*d^3*D 
 - a*b*d*(6*c*C*d - B*d^2 - 3*c^2*D)))*(-1/2*((a*d - b*c*x)*Sqrt[a + b*x^2 
])/((b*c^2 + a*d^2)*(c + d*x)^2) - (a*b*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 
+ a*d^2]*Sqrt[a + b*x^2])])/(2*(b*c^2 + a*d^2)^(3/2))))/(b*c^2 + a*d^2))/( 
4*d^2*(b*c^2 + a*d^2)))/(5*(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 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 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 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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. \(7009\) vs. \(2(501)=1002\).

Time = 1.65 (sec) , antiderivative size = 7010, normalized size of antiderivative = 13.35

method	result	size
default	\(\text {Expression too large to display}\)	\(7010\)

Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{6}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**6,x)
 

Output:

Integral(sqrt(a + b*x**2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**6, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8174 vs. \(2 (498) = 996\).

Time = 0.33 (sec) , antiderivative size = 8174, normalized size of antiderivative = 15.57 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm="maxi 
ma")
 

Output:

7/8*sqrt(b*x^2 + a)*D*b^4*c^7/(b^4*c^8*d^5*x + 4*a*b^3*c^6*d^7*x + 6*a^2*b 
^2*c^4*d^9*x + 4*a^3*b*c^2*d^11*x + a^4*d^13*x + b^4*c^9*d^4 + 4*a*b^3*c^7 
*d^6 + 6*a^2*b^2*c^5*d^8 + 4*a^3*b*c^3*d^10 + a^4*c*d^12) + 7/8*(b*x^2 + a 
)^(3/2)*D*b^3*c^6/(b^4*c^8*d^4*x^2 + 4*a*b^3*c^6*d^6*x^2 + 6*a^2*b^2*c^4*d 
^8*x^2 + 4*a^3*b*c^2*d^10*x^2 + a^4*d^12*x^2 + 2*b^4*c^9*d^3*x + 8*a*b^3*c 
^7*d^5*x + 12*a^2*b^2*c^5*d^7*x + 8*a^3*b*c^3*d^9*x + 2*a^4*c*d^11*x + b^4 
*c^10*d^2 + 4*a*b^3*c^8*d^4 + 6*a^2*b^2*c^6*d^6 + 4*a^3*b*c^4*d^8 + a^4*c^ 
2*d^10) - 7/8*sqrt(b*x^2 + a)*C*b^4*c^6/(b^4*c^8*d^4*x + 4*a*b^3*c^6*d^6*x 
 + 6*a^2*b^2*c^4*d^8*x + 4*a^3*b*c^2*d^10*x + a^4*d^12*x + b^4*c^9*d^3 + 4 
*a*b^3*c^7*d^5 + 6*a^2*b^2*c^5*d^7 + 4*a^3*b*c^3*d^9 + a^4*c*d^11) - 7/8*s 
qrt(b*x^2 + a)*D*b^4*c^6/(b^4*c^8*d^4 + 4*a*b^3*c^6*d^6 + 6*a^2*b^2*c^4*d^ 
8 + 4*a^3*b*c^2*d^10 + a^4*d^12) - 7/8*(b*x^2 + a)^(3/2)*C*b^3*c^5/(b^4*c^ 
8*d^3*x^2 + 4*a*b^3*c^6*d^5*x^2 + 6*a^2*b^2*c^4*d^7*x^2 + 4*a^3*b*c^2*d^9* 
x^2 + a^4*d^11*x^2 + 2*b^4*c^9*d^2*x + 8*a*b^3*c^7*d^4*x + 12*a^2*b^2*c^5* 
d^6*x + 8*a^3*b*c^3*d^8*x + 2*a^4*c*d^10*x + b^4*c^10*d + 4*a*b^3*c^8*d^3 
+ 6*a^2*b^2*c^6*d^5 + 4*a^3*b*c^4*d^7 + a^4*c^2*d^9) + 7/8*sqrt(b*x^2 + a) 
*B*b^4*c^5/(b^4*c^8*d^3*x + 4*a*b^3*c^6*d^5*x + 6*a^2*b^2*c^4*d^7*x + 4*a^ 
3*b*c^2*d^9*x + a^4*d^11*x + b^4*c^9*d^2 + 4*a*b^3*c^7*d^4 + 6*a^2*b^2*c^5 
*d^6 + 4*a^3*b*c^3*d^8 + a^4*c*d^10) + 7/8*sqrt(b*x^2 + a)*C*b^4*c^5/(b^4* 
c^8*d^3 + 4*a*b^3*c^6*d^5 + 6*a^2*b^2*c^4*d^7 + 4*a^3*b*c^2*d^9 + a^4*d...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5885 vs. \(2 (498) = 996\).

Time = 0.47 (sec) , antiderivative size = 5885, normalized size of antiderivative = 11.21 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x, algorithm="giac 
")
 

Output:

-1/4*(C*a^2*b^3*c^3 - 4*A*a*b^4*c^3 + 3*D*a^3*b^2*c^2*d - 6*B*a^2*b^3*c^2* 
d - 6*C*a^3*b^2*c*d^2 + 3*A*a^2*b^3*c*d^2 - 4*D*a^4*b*d^3 + B*a^3*b^2*d^3) 
*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2 
))/((b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4 
*d^8)*sqrt(-b*c^2 - a*d^2)) + 1/60*(120*(sqrt(b)*x - sqrt(b*x^2 + a))^9*D* 
b^5*c^8*d^4 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^9*D*a*b^4*c^6*d^6 + 720*(s 
qrt(b)*x - sqrt(b*x^2 + a))^9*D*a^2*b^3*c^4*d^8 + 15*(sqrt(b)*x - sqrt(b*x 
^2 + a))^9*C*a^2*b^3*c^3*d^9 - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a*b^4* 
c^3*d^9 + 525*(sqrt(b)*x - sqrt(b*x^2 + a))^9*D*a^3*b^2*c^2*d^10 - 90*(sqr 
t(b)*x - sqrt(b*x^2 + a))^9*B*a^2*b^3*c^2*d^10 - 90*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^9*C*a^3*b^2*c*d^11 + 45*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*a^2*b^3* 
c*d^11 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*D*a^4*b*d^12 + 15*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^9*B*a^3*b^2*d^12 + 480*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D 
*b^(11/2)*c^9*d^3 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*b^(11/2)*c^8*d^4 
 + 1920*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a*b^(9/2)*c^7*d^5 + 480*(sqrt(b) 
*x - sqrt(b*x^2 + a))^8*C*a*b^(9/2)*c^6*d^6 + 2880*(sqrt(b)*x - sqrt(b*x^2 
 + a))^8*D*a^2*b^(7/2)*c^5*d^7 + 855*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^2 
*b^(7/2)*c^4*d^8 - 540*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a*b^(9/2)*c^4*d^8 
 + 2325*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^3*b^(5/2)*c^3*d^9 - 810*(sqrt( 
b)*x - sqrt(b*x^2 + a))^8*B*a^2*b^(7/2)*c^3*d^9 - 330*(sqrt(b)*x - sqrt...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^6} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^6,x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^6, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^6} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{6}}d x \] Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x)
 

Output:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^6,x)