\(\int \frac {A+B x}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [194]

Optimal result

Integrand size = 24, antiderivative size = 436 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {B c-A d}{2 \left (b c^2+a d^2\right ) (c+d x)^2 \left (a+b x^2\right )^{3/2}}-\frac {2 a B d^2-b c (5 B c-7 A d)}{2 \left (b c^2+a d^2\right )^2 (c+d x) \left (a+b x^2\right )^{3/2}}-\frac {b \left (5 a \left (2 b c^2 (2 B c-3 A d)-a d^2 (3 B c-A d)\right )-\left (A b c \left (2 b c^2-33 a d^2\right )+a B d \left (27 b c^2-8 a d^2\right )\right ) x\right )}{6 a \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}-\frac {b \left (15 a^2 d^2 \left (2 b c^2 (2 B c-3 A d)-a d^2 (3 B c-A d)\right )-\left (A b c \left (4 b^2 c^4+28 a b c^2 d^2-81 a^2 d^4\right )-a B d \left (6 b^2 c^4-83 a b c^2 d^2+16 a^2 d^4\right )\right ) x\right )}{6 a^2 \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}-\frac {5 b d^3 \left (a d^2 (3 B c-A d)-b \left (4 B c^3-6 A c^2 d\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

1/2*(-A*d+B*c)/(a*d^2+b*c^2)/(d*x+c)^2/(b*x^2+a)^(3/2)-1/2*(2*a*B*d^2-b*c* 
(-7*A*d+5*B*c))/(a*d^2+b*c^2)^2/(d*x+c)/(b*x^2+a)^(3/2)-1/6*b*(5*a*(2*b*c^ 
2*(-3*A*d+2*B*c)-a*d^2*(-A*d+3*B*c))-(A*b*c*(-33*a*d^2+2*b*c^2)+a*B*d*(-8* 
a*d^2+27*b*c^2))*x)/a/(a*d^2+b*c^2)^3/(b*x^2+a)^(3/2)-1/6*b*(15*a^2*d^2*(2 
*b*c^2*(-3*A*d+2*B*c)-a*d^2*(-A*d+3*B*c))-(A*b*c*(-81*a^2*d^4+28*a*b*c^2*d 
^2+4*b^2*c^4)-a*B*d*(16*a^2*d^4-83*a*b*c^2*d^2+6*b^2*c^4))*x)/a^2/(a*d^2+b 
*c^2)^4/(b*x^2+a)^(1/2)-5/2*b*d^3*(a*d^2*(-A*d+3*B*c)-b*(-6*A*c^2*d+4*B*c^ 
3))*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 = 13.93 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-\frac {3 d^4 (-B c+A d) \left (b c^2+a d^2\right )}{(c+d x)^2}-\frac {3 d^4 \left (2 a B d^2+b c (-9 B c+11 A d)\right )}{c+d x}+\frac {2 b \left (b c^2+a d^2\right ) \left (A b^2 c^3 x-a^2 d^2 (-3 B c+A d+B d x)+a b c (-B c (c-3 d x)+3 A d (c-d x))\right )}{a \left (a+b x^2\right )^2}+\frac {2 b \left (2 A b^3 c^5 x+a b^2 c^3 d (-3 B c+14 A d) x+a^3 d^4 (18 B c-6 A d-5 B d x)+2 a^2 b c d^2 (3 A d (5 c-4 d x)+B c (-9 c+14 d x))\right )}{a^2 \left (a+b x^2\right )}\right )}{\left (b c^2+a d^2\right )^4}-\frac {15 b d^3 \left (2 b c^2 (2 B c-3 A d)+a d^2 (-3 B c+A d)\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}+\frac {15 b d^3 \left (2 b c^2 (2 B c-3 A d)+a d^2 (-3 B c+A d)\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*((-3*d^4*(-(B*c) + A*d)*(b*c^2 + a*d^2))/(c + d*x)^2 - ( 
3*d^4*(2*a*B*d^2 + b*c*(-9*B*c + 11*A*d)))/(c + d*x) + (2*b*(b*c^2 + a*d^2 
)*(A*b^2*c^3*x - a^2*d^2*(-3*B*c + A*d + B*d*x) + a*b*c*(-(B*c*(c - 3*d*x) 
) + 3*A*d*(c - d*x))))/(a*(a + b*x^2)^2) + (2*b*(2*A*b^3*c^5*x + a*b^2*c^3 
*d*(-3*B*c + 14*A*d)*x + a^3*d^4*(18*B*c - 6*A*d - 5*B*d*x) + 2*a^2*b*c*d^ 
2*(3*A*d*(5*c - 4*d*x) + B*c*(-9*c + 14*d*x))))/(a^2*(a + b*x^2))))/(b*c^2 
 + a*d^2)^4 - (15*b*d^3*(2*b*c^2*(2*B*c - 3*A*d) + a*d^2*(-3*B*c + A*d))*L 
og[c + d*x])/(b*c^2 + a*d^2)^(9/2) + (15*b*d^3*(2*b*c^2*(2*B*c - 3*A*d) + 
a*d^2*(-3*B*c + A*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)^(9/2))/6
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {686, 25, 27, 686, 27, 688, 25, 679, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

\(\displaystyle \frac {-\frac {d \left (\frac {\sqrt {a+b x^2} \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}-\frac {\int -\frac {2 a d \left (A b c \left (2 b c^2-33 a d^2\right )+a B d \left (27 b c^2-8 a d^2\right )\right )+b \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right )\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (3 b c^2-4 a d^2\right )-A b c \left (9 a d^2+2 b c^2\right )\right )+a d \left (-5 a A d^2+7 a B c d+2 A b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {d \left (\frac {\int \frac {2 a d \left (A b c \left (2 b c^2-33 a d^2\right )+a B d \left (27 b c^2-8 a d^2\right )\right )+b \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right )\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (3 b c^2-4 a d^2\right )-A b c \left (9 a d^2+2 b c^2\right )\right )+a d \left (-5 a A d^2+7 a B c d+2 A b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {-\frac {d \left (\frac {\frac {15 a^2 b d^2 \left (2 b c^2 (2 B c-3 A d)-a d^2 (3 B c-A d)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {\sqrt {a+b x^2} \left (A b c \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )-a B d \left (16 a^2 d^4-83 a b c^2 d^2+6 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (3 b c^2-4 a d^2\right )-A b c \left (9 a d^2+2 b c^2\right )\right )+a d \left (-5 a A d^2+7 a B c d+2 A b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {d \left (\frac {-\frac {15 a^2 b d^2 \left (2 b c^2 (2 B c-3 A d)-a d^2 (3 B c-A d)\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}-\frac {\sqrt {a+b x^2} \left (A b c \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )-a B d \left (16 a^2 d^4-83 a b c^2 d^2+6 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (3 b c^2-4 a d^2\right )-A b c \left (9 a d^2+2 b c^2\right )\right )+a d \left (-5 a A d^2+7 a B c d+2 A b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {d \left (\frac {-\frac {15 a^2 b d^2 \left (2 b c^2 (2 B c-3 A d)-a d^2 (3 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} \left (A b c \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )-a B d \left (16 a^2 d^4-83 a b c^2 d^2+6 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} \left (a B c d \left (6 b c^2-29 a d^2\right )-A \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (3 b c^2-4 a d^2\right )-A b c \left (9 a d^2+2 b c^2\right )\right )+a d \left (-5 a A d^2+7 a B c d+2 A b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/3*(a*(B*c - A*d) - (A*b*c + a*B*d)*x)/(a*(b*c^2 + a*d^2)*(c + d*x)^2*(a 
 + b*x^2)^(3/2)) + (-((a*d*(2*A*b*c^2 + 7*a*B*c*d - 5*a*A*d^2) + (a*B*d*(3 
*b*c^2 - 4*a*d^2) - A*b*c*(2*b*c^2 + 9*a*d^2))*x)/(a*(b*c^2 + a*d^2)*(c + 
d*x)^2*Sqrt[a + b*x^2])) - (d*(((a*B*c*d*(6*b*c^2 - 29*a*d^2) - A*(4*b^2*c 
^4 + 24*a*b*c^2*d^2 - 15*a^2*d^4))*Sqrt[a + b*x^2])/(2*(b*c^2 + a*d^2)*(c 
+ d*x)^2) + (-(((A*b*c*(4*b^2*c^4 + 28*a*b*c^2*d^2 - 81*a^2*d^4) - a*B*d*( 
6*b^2*c^4 - 83*a*b*c^2*d^2 + 16*a^2*d^4))*Sqrt[a + b*x^2])/((b*c^2 + a*d^2 
)*(c + d*x))) - (15*a^2*b*d^2*(2*b*c^2*(2*B*c - 3*A*d) - a*d^2*(3*B*c - A* 
d))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + 
 a*d^2)^(3/2))/(2*(b*c^2 + a*d^2))))/(a*(b*c^2 + a*d^2)))/(3*a*(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 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 688

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)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2510\) vs. \(2(412)=824\).

Time = 1.44 (sec) , antiderivative size = 2511, normalized size of antiderivative = 5.76

method	result	size
default	\(\text {Expression too large to display}\)	\(2511\)

Input:

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

Output:

B/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)^(3/2)+5*b*c*d/(a*d^2+b*c^2)*(1/3/(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)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+ 
c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*( 
x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^ 
2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2) 
^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/(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)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/( 
4*b*(a*d^2+b*c^2)/d^2-4*b^2*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)-1/(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))))-4*b/(a*d^2+b*c^2)*d^2*(2/ 
3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2 
-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4* 
b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2)))+(A*d-B*c)/d^4*(-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)+7/2*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)+5*b*c*d/(a*d^2+b*c^2)*(1/3/(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)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1934 vs. \(2 (413) = 826\).

Time = 4.94 (sec) , antiderivative size = 3894, normalized size of antiderivative = 8.93 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2543 vs. \(2 (413) = 826\).

Time = 0.24 (sec) , antiderivative size = 2543, normalized size of antiderivative = 5.83 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-35/2*B*b^3*c^4*x/(sqrt(b*x^2 + a)*a*b^4*c^8/d + 4*sqrt(b*x^2 + a)*a^2*b^3 
*c^6*d + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^3 + 4*sqrt(b*x^2 + a)*a^4*b*c^2*d 
^5 + sqrt(b*x^2 + a)*a^5*d^7) - 35/6*B*b^3*c^4*x/((b*x^2 + a)^(3/2)*a*b^3* 
c^6*d + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^3 + 3*(b*x^2 + a)^(3/2)*a^3*b*c^ 
2*d^5 + (b*x^2 + a)^(3/2)*a^4*d^7) - 35/3*B*b^3*c^4*x/(sqrt(b*x^2 + a)*a^2 
*b^3*c^6*d + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^3 + 3*sqrt(b*x^2 + a)*a^4*b*c 
^2*d^5 + sqrt(b*x^2 + a)*a^5*d^7) + 35/6*A*b^3*c^3*x/((b*x^2 + a)^(3/2)*a* 
b^3*c^6 + 3*(b*x^2 + a)^(3/2)*a^2*b^2*c^4*d^2 + 3*(b*x^2 + a)^(3/2)*a^3*b* 
c^2*d^4 + (b*x^2 + a)^(3/2)*a^4*d^6) + 35/2*A*b^3*c^3*x/(4*sqrt(b*x^2 + a) 
*a^2*b^3*c^6 + sqrt(b*x^2 + a)*a*b^4*c^8/d^2 + 6*sqrt(b*x^2 + a)*a^3*b^2*c 
^4*d^2 + 4*sqrt(b*x^2 + a)*a^4*b*c^2*d^4 + sqrt(b*x^2 + a)*a^5*d^6) + 35/3 
*A*b^3*c^3*x/(sqrt(b*x^2 + a)*a^2*b^3*c^6 + 3*sqrt(b*x^2 + a)*a^3*b^2*c^4* 
d^2 + 3*sqrt(b*x^2 + a)*a^4*b*c^2*d^4 + sqrt(b*x^2 + a)*a^5*d^6) - 35/6*B* 
b^2*c^3/((b*x^2 + a)^(3/2)*b^3*c^6 + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d^2 + 3 
*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^4 + (b*x^2 + a)^(3/2)*a^3*d^6) - 35/2*B*b^2 
*c^3/(4*sqrt(b*x^2 + a)*a*b^3*c^6 + sqrt(b*x^2 + a)*b^4*c^8/d^2 + 6*sqrt(b 
*x^2 + a)*a^2*b^2*c^4*d^2 + 4*sqrt(b*x^2 + a)*a^3*b*c^2*d^4 + sqrt(b*x^2 + 
 a)*a^4*d^6) + 15/2*B*b^2*c^2*x/(sqrt(b*x^2 + a)*a*b^3*c^6/d + 3*sqrt(b*x^ 
2 + a)*a^2*b^2*c^4*d + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^3 + sqrt(b*x^2 + a)*a 
^4*d^5) + 43/6*B*b^2*c^2*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d + 2*(b*x^2 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3353 vs. \(2 (413) = 826\).

Time = 0.23 (sec) , antiderivative size = 3353, normalized size of antiderivative = 7.69 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

5*(4*B*b^2*c^3*d^3 - 6*A*b^2*c^2*d^4 - 3*B*a*b*c*d^5 + A*a*b*d^6)*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)) + 1/3*((((2*A*b^18*c^29 - 3*B*a*b^17*c^28*d + 38*A*a*b^1 
7*c^27*d^2 - 8*B*a^2*b^16*c^26*d^3 + 276*A*a^2*b^16*c^25*d^4 + 133*B*a^3*b 
^15*c^24*d^5 + 1076*A*a^3*b^15*c^23*d^6 + 1128*B*a^4*b^14*c^22*d^7 + 2486* 
A*a^4*b^14*c^21*d^8 + 4345*B*a^5*b^13*c^20*d^9 + 3234*A*a^5*b^13*c^19*d^10 
 + 10384*B*a^6*b^12*c^18*d^11 + 1056*A*a^6*b^12*c^17*d^12 + 16929*B*a^7*b^ 
11*c^16*d^13 - 4488*A*a^7*b^11*c^15*d^14 + 19536*B*a^8*b^10*c^14*d^15 - 10 
098*A*a^8*b^10*c^13*d^16 + 16071*B*a^9*b^9*c^12*d^17 - 11638*A*a^9*b^9*c^1 
1*d^18 + 9240*B*a^10*b^8*c^10*d^19 - 8668*A*a^10*b^8*c^9*d^20 + 3487*B*a^1 
1*b^7*c^8*d^21 - 4332*A*a^11*b^7*c^7*d^22 + 712*B*a^12*b^6*c^6*d^23 - 1414 
*A*a^12*b^6*c^5*d^24 + 3*B*a^13*b^5*c^4*d^25 - 274*A*a^13*b^5*c^3*d^26 - 3 
2*B*a^14*b^4*c^2*d^27 - 24*A*a^14*b^4*c*d^28 - 5*B*a^15*b^3*d^29)*x/(a^2*b 
^17*c^32 + 16*a^3*b^16*c^30*d^2 + 120*a^4*b^15*c^28*d^4 + 560*a^5*b^14*c^2 
6*d^6 + 1820*a^6*b^13*c^24*d^8 + 4368*a^7*b^12*c^22*d^10 + 8008*a^8*b^11*c 
^20*d^12 + 11440*a^9*b^10*c^18*d^14 + 12870*a^10*b^9*c^16*d^16 + 11440*a^1 
1*b^8*c^14*d^18 + 8008*a^12*b^7*c^12*d^20 + 4368*a^13*b^6*c^10*d^22 + 1820 
*a^14*b^5*c^8*d^24 + 560*a^15*b^4*c^6*d^26 + 120*a^16*b^3*c^4*d^28 + 16*a^ 
17*b^2*c^2*d^30 + a^18*b*d^32) - 6*(3*B*a^2*b^16*c^27*d^2 - 5*A*a^2*b^1...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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