\(\int \frac {x^3 \sqrt {a+b x^2} (A+B x+C x^2)}{(c+d x)^2} \, dx\) [38]

Optimal result

Integrand size = 32, antiderivative size = 492 \[ \int \frac {x^3 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=-\frac {\left (2 a^2 C d^4-5 a b d^2 \left (3 c^2 C-2 B c d+A d^2\right )-15 b^2 c^2 \left (5 c^2 C-4 B c d+3 A d^2\right )\right ) \sqrt {a+b x^2}}{15 b^2 d^6}-\frac {\left (a d^2 (2 c C-B d)+4 b c \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) x \sqrt {a+b x^2}}{8 b d^5}+\frac {\left (a C d^2+5 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) x^2 \sqrt {a+b x^2}}{15 b d^4}-\frac {(2 c C-B d) x^3 \sqrt {a+b x^2}}{4 d^3}+\frac {C x^4 \sqrt {a+b x^2}}{5 d^2}+\frac {c^3 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d^6 (c+d x)}+\frac {\left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )-8 b^2 c^3 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2} d^7}-\frac {c^2 \left (a d^2 \left (5 c^2 C-4 B c d+3 A d^2\right )+b c^2 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^7 \sqrt {b c^2+a d^2}} \] Output:

-1/15*(2*a^2*C*d^4-5*a*b*d^2*(A*d^2-2*B*c*d+3*C*c^2)-15*b^2*c^2*(3*A*d^2-4 
*B*c*d+5*C*c^2))*(b*x^2+a)^(1/2)/b^2/d^6-1/8*(a*d^2*(-B*d+2*C*c)+4*b*c*(2* 
A*d^2-3*B*c*d+4*C*c^2))*x*(b*x^2+a)^(1/2)/b/d^5+1/15*(a*C*d^2+5*b*(A*d^2-2 
*B*c*d+3*C*c^2))*x^2*(b*x^2+a)^(1/2)/b/d^4-1/4*(-B*d+2*C*c)*x^3*(b*x^2+a)^ 
(1/2)/d^3+1/5*C*x^4*(b*x^2+a)^(1/2)/d^2+c^3*(A*d^2-B*c*d+C*c^2)*(b*x^2+a)^ 
(1/2)/d^6/(d*x+c)+1/8*(a^2*d^4*(-B*d+2*C*c)-4*a*b*c*d^2*(2*A*d^2-3*B*c*d+4 
*C*c^2)-8*b^2*c^3*(4*A*d^2-5*B*c*d+6*C*c^2))*arctanh(b^(1/2)*x/(b*x^2+a)^( 
1/2))/b^(3/2)/d^7-c^2*(a*d^2*(3*A*d^2-4*B*c*d+5*C*c^2)+b*c^2*(4*A*d^2-5*B* 
c*d+6*C*c^2))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^ 
7/(a*d^2+b*c^2)^(1/2)
 

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-16 a^2 C d^4 (c+d x)+a b d^2 (c+d x) \left (120 c^2 C-10 c d (8 B+3 C x)+d^2 \left (40 A+15 B x+8 C x^2\right )\right )+2 b^2 \left (360 c^5 C-60 c^4 d (5 B-3 C x)+30 c^3 d^2 (8 A-x (5 B+2 C x))+10 c^2 d^3 x (12 A+x (5 B+3 C x))+d^5 x^3 (20 A+3 x (5 B+4 C x))-c d^4 x^2 (40 A+x (25 B+18 C x))\right )\right )}{b^2 (c+d x)}-\frac {240 c^2 \left (a d^2 \left (5 c^2 C-4 B c d+3 A d^2\right )+b c^2 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {15 \left (a^2 d^4 (-2 c C+B d)+4 a b c d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+8 b^2 c^3 \left (6 c^2 C-5 B c d+4 A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{120 d^7} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(-16*a^2*C*d^4*(c + d*x) + a*b*d^2*(c + d*x)*(120*c^2* 
C - 10*c*d*(8*B + 3*C*x) + d^2*(40*A + 15*B*x + 8*C*x^2)) + 2*b^2*(360*c^5 
*C - 60*c^4*d*(5*B - 3*C*x) + 30*c^3*d^2*(8*A - x*(5*B + 2*C*x)) + 10*c^2* 
d^3*x*(12*A + x*(5*B + 3*C*x)) + d^5*x^3*(20*A + 3*x*(5*B + 4*C*x)) - c*d^ 
4*x^2*(40*A + x*(25*B + 18*C*x)))))/(b^2*(c + d*x)) - (240*c^2*(a*d^2*(5*c 
^2*C - 4*B*c*d + 3*A*d^2) + b*c^2*(6*c^2*C - 5*B*c*d + 4*A*d^2))*ArcTan[(S 
qrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/Sqrt[-(b*c^ 
2) - a*d^2] + (15*(a^2*d^4*(-2*c*C + B*d) + 4*a*b*c*d^2*(4*c^2*C - 3*B*c*d 
 + 2*A*d^2) + 8*b^2*c^3*(6*c^2*C - 5*B*c*d + 4*A*d^2))*Log[-(Sqrt[b]*x) + 
Sqrt[a + b*x^2]])/b^(3/2))/(120*d^7)
 

Rubi [A] (verified)

Time = 3.50 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.23, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2182, 25, 2185, 25, 2185, 25, 2185, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {\int -\frac {\sqrt {b x^2+a} \left (-b \left (b c^2+a d^2\right ) \left (8 a C d^2-b \left (102 C c^2-55 B d c+20 A d^2\right )\right ) x^2 d^4+5 a b c \left (a (2 c C-B d) d^2+b c \left (6 C c^2-5 B d c+4 A d^2\right )\right ) d^4+b \left (a^2 (2 c C-5 B d) d^4+4 a b c \left (6 c^2 C-5 A d^2\right ) d^2-3 b^2 c^3 \left (6 C c^2-15 B d c+20 A d^2\right )\right ) x d^3\right )}{c+d x}dx}{4 b d^3}+\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {\int \frac {\sqrt {b x^2+a} \left (-b \left (b c^2+a d^2\right ) \left (8 a C d^2-b \left (102 C c^2-55 B d c+20 A d^2\right )\right ) x^2 d^4+5 a b c \left (a (2 c C-B d) d^2+b c \left (6 C c^2-5 B d c+4 A d^2\right )\right ) d^4+b \left (a^2 (2 c C-5 B d) d^4+4 a b c \left (6 c^2 C-5 A d^2\right ) d^2-3 b^2 c^3 \left (6 C c^2-15 B d c+20 A d^2\right )\right ) x d^3\right )}{c+d x}dx}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {\frac {\int \frac {15 b^2 d^5 \left (a c d \left (a (2 c C-B d) d^2+b c \left (6 C c^2-5 B d c+4 A d^2\right )\right )+\left (a^2 (2 c C-B d) d^4-a b c \left (14 C c^2-11 B d c+8 A d^2\right ) d^2-4 b^2 c^3 \left (6 C c^2-5 B d c+4 A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{3 b d^2}-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \int \frac {\left (a c d \left (a (2 c C-B d) d^2+b c \left (6 C c^2-5 B d c+4 A d^2\right )\right )+\left (a^2 (2 c C-B d) d^4-a b c \left (14 C c^2-11 B d c+8 A d^2\right ) d^2-4 b^2 c^3 \left (6 C c^2-5 B d c+4 A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\int \frac {b \left (b c^2+a d^2\right ) \left (a c d \left (a (2 c C-B d) d^2+4 b c \left (6 C c^2-5 B d c+4 A d^2\right )\right )+\left (a^2 (2 c C-B d) d^4-4 a b c \left (4 C c^2-3 B d c+2 A d^2\right ) d^2-8 b^2 c^3 \left (6 C c^2-5 B d c+4 A d^2\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \int \frac {a c d \left (a (2 c C-B d) d^2+4 b c \left (6 C c^2-5 B d c+4 A d^2\right )\right )+\left (a^2 (2 c C-B d) d^4-4 a b c \left (4 C c^2-3 B d c+2 A d^2\right ) d^2-8 b^2 c^3 \left (6 C c^2-5 B d c+4 A d^2\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )-8 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}+\frac {8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )-8 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}+\frac {8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )-8 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )}{\sqrt {b} d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )-8 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )}{\sqrt {b} d}-\frac {8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}{5 b d^4}-\frac {\frac {1}{4} \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right ) (18 c C-5 B d)-\frac {5 b d^3 \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (2 c C-B d)-4 a b c d^2 \left (2 A d^2-3 B c d+4 c^2 C\right )-8 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )}{\sqrt {b} d}-\frac {8 b c^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )}{d \sqrt {a d^2+b c^2}}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a^2 d^4 (2 c C-B d)-a b c d^2 \left (8 A d^2-11 B c d+14 c^2 C\right )-4 b^2 c^3 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )+8 b c^2 \left (a d^2 \left (3 A d^2-4 B c d+5 c^2 C\right )+b c^2 \left (4 A d^2-5 B c d+6 c^2 C\right )\right )\right )}{2 d^2}\right )-\frac {1}{3} d^3 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right ) \left (8 a C d^2-b \left (20 A d^2-55 B c d+102 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}}{a d^2+b c^2}+\frac {c^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{d^4 (c+d x) \left (a d^2+b c^2\right )}\)

Input:

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

Output:

(c^3*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/2))/(d^4*(b*c^2 + a*d^2)*(c + 
d*x)) + ((C*(b*c^2 + a*d^2)*(c + d*x)^2*(a + b*x^2)^(3/2))/(5*b*d^4) - ((( 
18*c*C - 5*B*d)*(b*c^2 + a*d^2)*(c + d*x)*(a + b*x^2)^(3/2))/4 - (-1/3*(d^ 
3*(b*c^2 + a*d^2)*(8*a*C*d^2 - b*(102*c^2*C - 55*B*c*d + 20*A*d^2))*(a + b 
*x^2)^(3/2)) + 5*b*d^3*(((8*b*c^2*(a*d^2*(5*c^2*C - 4*B*c*d + 3*A*d^2) + b 
*c^2*(6*c^2*C - 5*B*c*d + 4*A*d^2)) + d*(a^2*d^4*(2*c*C - B*d) - 4*b^2*c^3 
*(6*c^2*C - 5*B*c*d + 4*A*d^2) - a*b*c*d^2*(14*c^2*C - 11*B*c*d + 8*A*d^2) 
)*x)*Sqrt[a + b*x^2])/(2*d^2) + ((b*c^2 + a*d^2)*(((a^2*d^4*(2*c*C - B*d) 
- 4*a*b*c*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) - 8*b^2*c^3*(6*c^2*C - 5*B*c*d 
 + 4*A*d^2))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (8*b*c^2* 
(a*d^2*(5*c^2*C - 4*B*c*d + 3*A*d^2) + b*c^2*(6*c^2*C - 5*B*c*d + 4*A*d^2) 
)*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)))/(4*b*d^3))/(5*b*d^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 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]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.61

method	result	size
risch	\(\frac {\left (24 C \,d^{4} b^{2} x^{4}+30 B \,b^{2} d^{4} x^{3}-60 C \,b^{2} c \,d^{3} x^{3}+40 A \,b^{2} d^{4} x^{2}-80 B \,b^{2} c \,d^{3} x^{2}+8 C a b \,d^{4} x^{2}+120 C \,b^{2} c^{2} d^{2} x^{2}-120 A \,b^{2} c \,d^{3} x +15 B a b \,d^{4} x +180 B \,b^{2} c^{2} d^{2} x -30 C a b c \,d^{3} x -240 C \,b^{2} c^{3} d x +40 A a b \,d^{4}+360 A \,b^{2} c^{2} d^{2}-80 B a b c \,d^{3}-480 B \,b^{2} c^{3} d -16 a^{2} C \,d^{4}+120 C a b \,c^{2} d^{2}+600 C \,b^{2} c^{4}\right ) \sqrt {b \,x^{2}+a}}{120 b^{2} d^{6}}-\frac {\frac {\left (8 A a b c \,d^{4}+32 A \,b^{2} c^{3} d^{2}+a^{2} B \,d^{5}-12 B a b \,c^{2} d^{3}-40 B \,b^{2} c^{4} d -2 C \,a^{2} c \,d^{4}+16 C a b \,c^{3} d^{2}+48 C \,b^{2} c^{5}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {8 b \,c^{2} \left (3 A a \,d^{4}+5 A b \,c^{2} d^{2}-4 B a c \,d^{3}-6 c^{3} B b d +5 C a \,c^{2} d^{2}+7 c^{4} C b \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {8 b \,c^{3} \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{8 b \,d^{6}}\)	\(790\)
default	\(\text {Expression too large to display}\)	\(1110\)

Input:

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

Output:

1/120*(24*C*b^2*d^4*x^4+30*B*b^2*d^4*x^3-60*C*b^2*c*d^3*x^3+40*A*b^2*d^4*x 
^2-80*B*b^2*c*d^3*x^2+8*C*a*b*d^4*x^2+120*C*b^2*c^2*d^2*x^2-120*A*b^2*c*d^ 
3*x+15*B*a*b*d^4*x+180*B*b^2*c^2*d^2*x-30*C*a*b*c*d^3*x-240*C*b^2*c^3*d*x+ 
40*A*a*b*d^4+360*A*b^2*c^2*d^2-80*B*a*b*c*d^3-480*B*b^2*c^3*d-16*C*a^2*d^4 
+120*C*a*b*c^2*d^2+600*C*b^2*c^4)*(b*x^2+a)^(1/2)/b^2/d^6-1/8/b/d^6*((8*A* 
a*b*c*d^4+32*A*b^2*c^3*d^2+B*a^2*d^5-12*B*a*b*c^2*d^3-40*B*b^2*c^4*d-2*C*a 
^2*c*d^4+16*C*a*b*c^3*d^2+48*C*b^2*c^5)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^ 
(1/2)+8*b*c^2/d^2*(3*A*a*d^4+5*A*b*c^2*d^2-4*B*a*c*d^3-6*B*b*c^3*d+5*C*a*c 
^2*d^2+7*C*b*c^4)/((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))+8*b*c^3*(A*a*d^4+A*b*c^2*d^2-B*a*c*d^3-B*b*c^3 
*d+C*a*c^2*d^2+C*b*c^4)/d^3*(-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)^(1/2)-b*c*d/(a*d^2+b*c^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))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.79 \[ \int \frac {x^3 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

sqrt(b*x^2 + a)*C*c^5/(d^7*x + c*d^6) - sqrt(b*x^2 + a)*B*c^4/(d^6*x + c*d 
^5) + sqrt(b*x^2 + a)*A*c^3/(d^5*x + c*d^4) + 1/5*(b*x^2 + a)^(3/2)*C*x^2/ 
(b*d^2) - 2*sqrt(b*x^2 + a)*C*c^3*x/d^5 + 3/2*sqrt(b*x^2 + a)*B*c^2*x/d^4 
- sqrt(b*x^2 + a)*A*c*x/d^3 - 1/2*(b*x^2 + a)^(3/2)*C*c*x/(b*d^3) + 1/4*sq 
rt(b*x^2 + a)*C*a*c*x/(b*d^3) + 1/4*(b*x^2 + a)^(3/2)*B*x/(b*d^2) - 1/8*sq 
rt(b*x^2 + a)*B*a*x/(b*d^2) - 6*C*sqrt(b)*c^5*arcsinh(b*x/sqrt(a*b))/d^7 + 
 5*B*sqrt(b)*c^4*arcsinh(b*x/sqrt(a*b))/d^6 - 2*C*a*c^3*arcsinh(b*x/sqrt(a 
*b))/(sqrt(b)*d^5) - 4*A*sqrt(b)*c^3*arcsinh(b*x/sqrt(a*b))/d^5 + 3/2*B*a* 
c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^4) + 1/4*C*a^2*c*arcsinh(b*x/sqrt(a* 
b))/(b^(3/2)*d^3) - A*a*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) - 1/8*B*a^2 
*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^2) + C*b*c^6*arcsinh(b*c*x/(sqrt(a*b)*a 
bs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^8) - B 
*b*c^5*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c 
)))/(sqrt(a + b*c^2/d^2)*d^7) + 5*C*sqrt(a + b*c^2/d^2)*c^4*arcsinh(b*c*x/ 
(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^6 + A*b*c^4*arc 
sinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt( 
a + b*c^2/d^2)*d^6) - 4*B*sqrt(a + b*c^2/d^2)*c^3*arcsinh(b*c*x/(sqrt(a*b) 
*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^5 + 3*A*sqrt(a + b*c^2/d^ 
2)*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c 
)))/d^4 + 5*sqrt(b*x^2 + a)*C*c^4/d^6 - 4*sqrt(b*x^2 + a)*B*c^3/d^5 + 3...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 37.47 (sec) , antiderivative size = 3172, normalized size of antiderivative = 6.45 \[ \int \frac {x^3 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(720*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a* 
d + b*c*x)*a**2*b**2*c**3*d**4 + 720*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b* 
x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**2*d**5*x + 960*sqr 
t(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c* 
x)*a*b**3*c**5*d**2 + 960*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt( 
a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**4*d**3*x - 960*sqrt(a*d**2 + b*c 
**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**4 
*d**3 - 960*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c** 
2) - a*d + b*c*x)*a*b**3*c**3*d**4*x + 1200*sqrt(a*d**2 + b*c**2)*log(sqrt 
(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**6*d**2 + 1200* 
sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b 
*c*x)*a*b**2*c**5*d**3*x - 1200*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2) 
*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**4*c**6*d - 1200*sqrt(a*d**2 + b*c 
**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**4*c**5*d 
**2*x + 1440*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c* 
*2) - a*d + b*c*x)*b**3*c**8 + 1440*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x 
**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**3*c**7*d*x - 720*sqrt(a*d**2 
+ b*c**2)*log(c + d*x)*a**2*b**2*c**3*d**4 - 720*sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*a**2*b**2*c**2*d**5*x - 960*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a 
*b**3*c**5*d**2 - 960*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**3*c**4*d*...