\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2+D x^3)}{c+d x} \, dx\) [75]

Optimal result

Integrand size = 34, antiderivative size = 501 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {\left (b c^2+a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{d^6}-\frac {\left (a^2 d^3 D+6 a b d \left (c C d-B d^2-c^2 D\right )+8 b^2 c \left (c^2 C-B c d+A d^2-\frac {c^3 D}{d}\right )\right ) x \sqrt {a+b x^2}}{16 b d^4}+\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{3/2}}{3 d^4}-\frac {\left (a d^2 D+6 b \left (c C d-B d^2-c^2 D\right )\right ) x \left (a+b x^2\right )^{3/2}}{24 b d^3}+\frac {(6 C d-11 c D) \left (a+b x^2\right )^{5/2}}{30 b d^2}+\frac {D (c+d x) \left (a+b x^2\right )^{5/2}}{6 b d^2}-\frac {\left (a^3 d^6 D+6 a^2 b d^4 \left (c C d-B d^2-c^2 D\right )+16 b^3 c^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+24 a b^2 c d^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2} d^7}-\frac {\left (b c^2+a d^2\right )^{3/2} \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^7} \] Output:

(a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(1/2)/d^6-1/16*(a^2* 
d^3*D+6*a*b*d*(-B*d^2+C*c*d-D*c^2)+8*b^2*c*(C*c^2-B*c*d+A*d^2-c^3*D/d))*x* 
(b*x^2+a)^(1/2)/b/d^4+1/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(3/2)/d^ 
4-1/24*(a*d^2*D+6*b*(-B*d^2+C*c*d-D*c^2))*x*(b*x^2+a)^(3/2)/b/d^3+1/30*(6* 
C*d-11*D*c)*(b*x^2+a)^(5/2)/b/d^2+1/6*D*(d*x+c)*(b*x^2+a)^(5/2)/b/d^2-1/16 
*(a^3*d^6*D+6*a^2*b*d^4*(-B*d^2+C*c*d-D*c^2)+16*b^3*c^3*(A*d^3-B*c*d^2+C*c 
^2*d-D*c^3)+24*a*b^2*c*d^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3))*arctanh(b^(1/2)* 
x/(b*x^2+a)^(1/2))/b^(3/2)/d^7-(a*d^2+b*c^2)^(3/2)*(A*d^3-B*c*d^2+C*c^2*d- 
D*c^3)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^7
 

Mathematica [A] (verified)

Time = 3.50 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (3 a^2 d^4 (16 C d-16 c D+5 d D x)+2 a b d^2 \left (-160 c^3 D+5 c^2 d (32 C+15 D x)-c d^2 \left (160 B+75 C x+48 D x^2\right )+d^3 \left (160 A+x \left (75 B+48 C x+35 D x^2\right )\right )\right )-4 b^2 \left (60 c^5 D-30 c^4 d (2 C+D x)+10 c^3 d^2 (6 B+x (3 C+2 D x))-5 c^2 d^3 (12 A+x (6 B+x (4 C+3 D x)))+c d^4 x (30 A+x (20 B+3 x (5 C+4 D x)))-d^5 x^2 (20 A+x (15 B+2 x (6 C+5 D x)))\right )\right )}{b}+480 \left (-b c^2-a d^2\right )^{3/2} \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )-\frac {15 \left (-a^3 d^6 D+6 a^2 b d^4 \left (-c C d+B d^2+c^2 D\right )+16 b^3 c^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )+24 a b^2 c d^2 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{240 d^7} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(3*a^2*d^4*(16*C*d - 16*c*D + 5*d*D*x) + 2*a*b*d^2*(-1 
60*c^3*D + 5*c^2*d*(32*C + 15*D*x) - c*d^2*(160*B + 75*C*x + 48*D*x^2) + d 
^3*(160*A + x*(75*B + 48*C*x + 35*D*x^2))) - 4*b^2*(60*c^5*D - 30*c^4*d*(2 
*C + D*x) + 10*c^3*d^2*(6*B + x*(3*C + 2*D*x)) - 5*c^2*d^3*(12*A + x*(6*B 
+ x*(4*C + 3*D*x))) + c*d^4*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) - d^5* 
x^2*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))))))/b + 480*(-(b*c^2) - a*d^2)^(3 
/2)*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)*ArcTan[(Sqrt[b]*(c + d*x) - d*S 
qrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] - (15*(-(a^3*d^6*D) + 6*a^2*b*d^4* 
(-(c*C*d) + B*d^2 + c^2*D) + 16*b^3*c^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^ 
3*D) + 24*a*b^2*c*d^2*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))*Log[-(Sqrt[b 
]*x) + Sqrt[a + b*x^2]])/b^(3/2))/(240*d^7)
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 497, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {2185, 2185, 27, 682, 25, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b (6 C d-11 c D) x^2 d^2+(6 A b d-a c D) d^2+\left (-5 b D c^2+6 b B d^2-a d^2 D\right ) x d\right )}{c+d x}dx}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int \frac {5 b d^3 \left (d (6 A b d-a c D)-\left (a D d^2+6 b \left (-D c^2+C d c-B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{5 b d^2}+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {\left (d (6 A b d-a c D)-\left (a D d^2+6 b \left (-D c^2+C d c-B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {\int \frac {3 b \left (a d \left (a^2 c D d^4-2 a b \left (-5 D c^3+5 C d c^2-5 B d^2 c+8 A d^3\right ) d^2-8 b^2 c^2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right )\right )+\left (a^3 D d^6+6 a^2 b \left (-D c^2+C d c-B d^2\right ) d^4+24 a b^2 c \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) d^2+16 b^3 c^3 \left (-D c^3+C d c^2-B d^2 c+A d^3\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \int \frac {a d \left (a^2 c D d^4-2 a b \left (-5 D c^3+5 C d c^2-5 B d^2 c+8 A d^3\right ) d^2-8 b^2 c^2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right )\right )+\left (a^3 D d^6+6 a^2 b \left (-D c^2+C d c-B d^2\right ) d^4+24 a b^2 c \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) d^2+16 b^3 c^3 \left (-D c^3+C d c^2-B d^2 c+A d^3\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \left (\frac {\left (a^3 d^6 D+6 a^2 b d^4 \left (-B d^2+c^2 (-D)+c C d\right )+24 a b^2 c d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+16 b^3 c^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {16 b \left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \left (\frac {\left (a^3 d^6 D+6 a^2 b d^4 \left (-B d^2+c^2 (-D)+c C d\right )+24 a b^2 c d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+16 b^3 c^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {16 b \left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 d^6 D+6 a^2 b d^4 \left (-B d^2+c^2 (-D)+c C d\right )+24 a b^2 c d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+16 b^3 c^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )\right )}{\sqrt {b} d}-\frac {16 b \left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \left (\frac {16 b \left (a d^2+b c^2\right )^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 d^6 D+6 a^2 b d^4 \left (-B d^2+c^2 (-D)+c C d\right )+24 a b^2 c d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+16 b^3 c^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )\right )}{\sqrt {b} d}\right )}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d \left (\frac {\left (a+b x^2\right )^{3/2} \left (8 b \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{4 d^2}-\frac {\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^3 d^6 D+6 a^2 b d^4 \left (-B d^2+c^2 (-D)+c C d\right )+24 a b^2 c d^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )+16 b^3 c^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )\right )}{\sqrt {b} d}+\frac {16 b \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d}\right )}{2 d^2}-\frac {\sqrt {a+b x^2} \left (48 b \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )-d x \left (4 b c d^2 (6 A b d-a c D)+\left (3 a d^2+4 b c^2\right ) \left (a d^2 D+6 b \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )\right )}{2 d^2}}{4 d^2}\right )+\frac {1}{5} d \left (a+b x^2\right )^{5/2} (6 C d-11 c D)}{6 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)}{6 b d^2}\)

Input:

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

Output:

(D*(c + d*x)*(a + b*x^2)^(5/2))/(6*b*d^2) + ((d*(6*C*d - 11*c*D)*(a + b*x^ 
2)^(5/2))/5 + d*(((8*b*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) - d*(a*d^2*D + 
6*b*(c*C*d - B*d^2 - c^2*D))*x)*(a + b*x^2)^(3/2))/(4*d^2) - (-1/2*((48*b* 
(b*c^2 + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) - d*(4*b*c*d^2*(6*A*b* 
d - a*c*D) + (4*b*c^2 + 3*a*d^2)*(a*d^2*D + 6*b*(c*C*d - B*d^2 - c^2*D)))* 
x)*Sqrt[a + b*x^2])/d^2 + (3*(((a^3*d^6*D + 6*a^2*b*d^4*(c*C*d - B*d^2 - c 
^2*D) + 16*b^3*c^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D) + 24*a*b^2*c*d^2*(c 
^2*C*d - B*c*d^2 + A*d^3 - c^3*D))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/( 
Sqrt[b]*d) + (16*b*(b*c^2 + a*d^2)^(3/2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3* 
D)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/d))/(2*d^ 
2))/(4*d^2)))/(6*b*d^3)
 

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 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 = 1.41 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {B \,d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+D c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\frac {d \left (C d -D c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+D d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )-C c d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{d^{3}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (\frac {\left (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}}\right )^{\frac {3}{2}}}{3}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{8 b^{\frac {3}{2}}}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\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}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\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}}}}\right )}{d^{2}}\right )}{d^{4}}\)	\(796\)

Input:

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

Output:

1/d^3*(B*d^2*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^( 
1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))+D*c^2*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1 
/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))+1/5*d*( 
C*d-D*c)/b*(b*x^2+a)^(5/2)+D*d^2*(1/6*x*(b*x^2+a)^(5/2)/b-1/6*a/b*(1/4*x*( 
b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b* 
x^2+a)^(1/2)))))-C*c*d*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2) 
+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+(A*d^3-B*c*d^2+C*c^2*d-D*c 
^3)/d^4*(1/3*(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*( 
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)))+(a*d 
^2+b*c^2)/d^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))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 971 vs. \(2 (463) = 926\).

Time = 0.17 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{c+d x} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*D*b*c^4*x/d^5 - 1/2*sqrt(b*x^2 + a)*C*b*c^3*x/d^4 + 1/ 
4*(b*x^2 + a)^(3/2)*D*c^2*x/d^3 + 3/8*sqrt(b*x^2 + a)*D*a*c^2*x/d^3 + 1/2* 
sqrt(b*x^2 + a)*B*b*c^2*x/d^3 - 1/4*(b*x^2 + a)^(3/2)*C*c*x/d^2 - 3/8*sqrt 
(b*x^2 + a)*C*a*c*x/d^2 - 1/2*sqrt(b*x^2 + a)*A*b*c*x/d^2 + 1/4*(b*x^2 + a 
)^(3/2)*B*x/d + 3/8*sqrt(b*x^2 + a)*B*a*x/d + 1/6*(b*x^2 + a)^(5/2)*D*x/(b 
*d) - 1/24*(b*x^2 + a)^(3/2)*D*a*x/(b*d) - 1/16*sqrt(b*x^2 + a)*D*a^2*x/(b 
*d) + D*b^(3/2)*c^6*arcsinh(b*x/sqrt(a*b))/d^7 - C*b^(3/2)*c^5*arcsinh(b*x 
/sqrt(a*b))/d^6 + 3/2*D*a*sqrt(b)*c^4*arcsinh(b*x/sqrt(a*b))/d^5 + B*b^(3/ 
2)*c^4*arcsinh(b*x/sqrt(a*b))/d^5 - 3/2*C*a*sqrt(b)*c^3*arcsinh(b*x/sqrt(a 
*b))/d^4 - A*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b))/d^4 + 3/8*D*a^2*c^2*arcsin 
h(b*x/sqrt(a*b))/(sqrt(b)*d^3) + 3/2*B*a*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b) 
)/d^3 - 3/8*C*a^2*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - 3/2*A*a*sqrt(b) 
*c*arcsinh(b*x/sqrt(a*b))/d^2 - 1/16*D*a^3*arcsinh(b*x/sqrt(a*b))/(b^(3/2) 
*d) + 3/8*B*a^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) - D*(a + b*c^2/d^2)^(3/ 
2)*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c 
)))/d^4 + C*(a + b*c^2/d^2)^(3/2)*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c 
)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^3 - B*(a + b*c^2/d^2)^(3/2)*c*arcsinh 
(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^2 + A*(a 
 + b*c^2/d^2)^(3/2)*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b 
)*abs(d*x + c)))/d - sqrt(b*x^2 + a)*D*b*c^5/d^6 + sqrt(b*x^2 + a)*C*b*...
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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