\(\int \frac {A+B x}{(c+d x)^3 (a+b x^2)^{3/2}} \, dx\) [185]

Optimal result

Integrand size = 24, antiderivative size = 292 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B c-A d}{2 \left (b c^2+a d^2\right ) (c+d x)^2 \sqrt {a+b x^2}}-\frac {2 a B d^2-b c (3 B c-5 A d)}{2 \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}-\frac {b \left (3 a \left (2 b c^2 (B c-2 A d)-a d^2 (3 B c-A d)\right )-\left (A b c \left (2 b c^2-13 a d^2\right )+a B d \left (11 b c^2-4 a d^2\right )\right ) x\right )}{2 a \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}+\frac {3 b d \left (2 b c^2 (B c-2 A d)-a d^2 (3 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{7/2}} \] Output:

1/2*(-A*d+B*c)/(a*d^2+b*c^2)/(d*x+c)^2/(b*x^2+a)^(1/2)-1/2*(2*a*B*d^2-b*c* 
(-5*A*d+3*B*c))/(a*d^2+b*c^2)^2/(d*x+c)/(b*x^2+a)^(1/2)-1/2*b*(3*a*(2*b*c^ 
2*(-2*A*d+B*c)-a*d^2*(-A*d+3*B*c))-(A*b*c*(-13*a*d^2+2*b*c^2)+a*B*d*(-4*a* 
d^2+11*b*c^2))*x)/a/(a*d^2+b*c^2)^3/(b*x^2+a)^(1/2)+3/2*b*d*(2*b*c^2*(-2*A 
*d+B*c)-a*d^2*(-A*d+3*B*c))*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)^(7/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(774\) vs. \(2(292)=584\).

Time = 10.99 (sec) , antiderivative size = 774, normalized size of antiderivative = 2.65 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-b c^2-a d^2} \left (4 b^3 c^2 x \left (\sqrt {b} x-\sqrt {a+b x^2}\right ) \left (B c x \left (2 c^2+9 c d x+6 d^2 x^2\right )+A \left (c^3-4 c^2 d x-18 c d^2 x^2-12 d^3 x^3\right )\right )+a b^2 \left (\sqrt {b} B c x \left (6 c^4+49 c^3 d x+14 c^2 d^2 x^2-54 c d^3 x^3-36 d^4 x^4\right )+3 A d \sqrt {a+b x^2} \left (2 c^4+15 c^3 d x+8 c^2 d^2 x^2-6 c d^3 x^3-4 d^4 x^4\right )+B c \sqrt {a+b x^2} \left (-2 c^4-31 c^3 d x-2 c^2 d^2 x^2+54 c d^3 x^3+36 d^4 x^4\right )+A \sqrt {b} \left (2 c^5-14 c^4 d x-81 c^3 d^2 x^2-48 c^2 d^3 x^3+18 c d^4 x^4+12 d^5 x^5\right )\right )-a^3 d^3 \left (d \sqrt {a+b x^2} (A d+B (c+2 d x))+\sqrt {b} \left (-3 A d^2 x+B \left (4 c^2+5 c d x-2 d^2 x^2\right )\right )\right )+a^2 b d \left (d \sqrt {a+b x^2} \left (3 B c \left (4 c^2+9 c d x+7 d^2 x^2\right )-A d \left (10 c^2+11 c d x+7 d^2 x^2\right )\right )+\sqrt {b} \left (B c \left (11 c^3-14 c^2 d x-54 c d^2 x^2-39 d^3 x^3\right )+A d \left (-13 c^3+4 c^2 d x+20 c d^2 x^2+13 d^3 x^3\right )\right )\right )\right )+6 b d \left (-2 b c^2 (B c-2 A d)+a d^2 (3 B c-A d)\right ) (c+d x)^2 \left (a^2+5 a b x^2+4 b^2 x^4-3 a \sqrt {b} x \sqrt {a+b x^2}-4 b^{3/2} x^3 \sqrt {a+b x^2}\right ) \arctan \left (\frac {-\sqrt {b} (c+d x)+d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{2 \left (-b c^2-a d^2\right )^{7/2} (c+d x)^2 \sqrt {a+b x^2} \left (a \left (-3 \sqrt {b} x+\sqrt {a+b x^2}\right )+4 b x^2 \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )} \] Input:

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

Output:

-1/2*(Sqrt[-(b*c^2) - a*d^2]*(4*b^3*c^2*x*(Sqrt[b]*x - Sqrt[a + b*x^2])*(B 
*c*x*(2*c^2 + 9*c*d*x + 6*d^2*x^2) + A*(c^3 - 4*c^2*d*x - 18*c*d^2*x^2 - 1 
2*d^3*x^3)) + a*b^2*(Sqrt[b]*B*c*x*(6*c^4 + 49*c^3*d*x + 14*c^2*d^2*x^2 - 
54*c*d^3*x^3 - 36*d^4*x^4) + 3*A*d*Sqrt[a + b*x^2]*(2*c^4 + 15*c^3*d*x + 8 
*c^2*d^2*x^2 - 6*c*d^3*x^3 - 4*d^4*x^4) + B*c*Sqrt[a + b*x^2]*(-2*c^4 - 31 
*c^3*d*x - 2*c^2*d^2*x^2 + 54*c*d^3*x^3 + 36*d^4*x^4) + A*Sqrt[b]*(2*c^5 - 
 14*c^4*d*x - 81*c^3*d^2*x^2 - 48*c^2*d^3*x^3 + 18*c*d^4*x^4 + 12*d^5*x^5) 
) - a^3*d^3*(d*Sqrt[a + b*x^2]*(A*d + B*(c + 2*d*x)) + Sqrt[b]*(-3*A*d^2*x 
 + B*(4*c^2 + 5*c*d*x - 2*d^2*x^2))) + a^2*b*d*(d*Sqrt[a + b*x^2]*(3*B*c*( 
4*c^2 + 9*c*d*x + 7*d^2*x^2) - A*d*(10*c^2 + 11*c*d*x + 7*d^2*x^2)) + Sqrt 
[b]*(B*c*(11*c^3 - 14*c^2*d*x - 54*c*d^2*x^2 - 39*d^3*x^3) + A*d*(-13*c^3 
+ 4*c^2*d*x + 20*c*d^2*x^2 + 13*d^3*x^3)))) + 6*b*d*(-2*b*c^2*(B*c - 2*A*d 
) + a*d^2*(3*B*c - A*d))*(c + d*x)^2*(a^2 + 5*a*b*x^2 + 4*b^2*x^4 - 3*a*Sq 
rt[b]*x*Sqrt[a + b*x^2] - 4*b^(3/2)*x^3*Sqrt[a + b*x^2])*ArcTan[(-(Sqrt[b] 
*(c + d*x)) + d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/((-(b*c^2) - a*d 
^2)^(7/2)*(c + d*x)^2*Sqrt[a + b*x^2]*(a*(-3*Sqrt[b]*x + Sqrt[a + b*x^2]) 
+ 4*b*x^2*(-(Sqrt[b]*x) + Sqrt[a + b*x^2])))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {686, 27, 688, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 679

\(\displaystyle -\frac {d \left (-\frac {\frac {\sqrt {a+b x^2} \left (A b c \left (2 b c^2-13 a d^2\right )+a B d \left (11 b c^2-4 a d^2\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}-\frac {3 a b \left (2 b c^2 (B c-2 A d)-a d^2 (3 B c-A d)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-3 a A d^2+5 a B c d+2 A b c^2\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {d \left (-\frac {\frac {3 a b \left (2 b c^2 (B c-2 A d)-a d^2 (3 B c-A d)\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}+\frac {\sqrt {a+b x^2} \left (A b c \left (2 b c^2-13 a d^2\right )+a B d \left (11 b c^2-4 a d^2\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-3 a A d^2+5 a B c d+2 A b c^2\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {d \left (-\frac {\frac {3 a b \left (2 b c^2 (B c-2 A d)-a d^2 (3 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}+\frac {\sqrt {a+b x^2} \left (A b c \left (2 b c^2-13 a d^2\right )+a B d \left (11 b c^2-4 a d^2\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-3 a A d^2+5 a B c d+2 A b c^2\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-((a*(B*c - A*d) - (A*b*c + a*B*d)*x)/(a*(b*c^2 + a*d^2)*(c + d*x)^2*Sqrt[ 
a + b*x^2])) - (d*(-1/2*((2*A*b*c^2 + 5*a*B*c*d - 3*a*A*d^2)*Sqrt[a + b*x^ 
2])/((b*c^2 + a*d^2)*(c + d*x)^2) - (((A*b*c*(2*b*c^2 - 13*a*d^2) + a*B*d* 
(11*b*c^2 - 4*a*d^2))*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x)) + (3*a* 
b*(2*b*c^2*(B*c - 2*A*d) - a*d^2*(3*B*c - A*d))*ArcTanh[(a*d - b*c*x)/(Sqr 
t[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))
 

Defintions of rubi rules used

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. \(1456\) vs. \(2(272)=544\).

Time = 1.38 (sec) , antiderivative size = 1457, normalized size of antiderivative = 4.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1457\)

Input:

int((B*x+A)/(d*x+c)^3/(b*x^2+a)^(3/2),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)^(1/2)+3*b*c*d/(a*d^2+b*c^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)*l 
n((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c 
/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))-4*b/(a*d^2+b*c^2 
)*d^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))+(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)^(1/2) 
+5/2*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)^(1/2)+3*b*c*d/(a*d^2+b*c^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*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))-3/2*b/(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^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (273) = 546\).

Time = 2.60 (sec) , antiderivative size = 2236, normalized size of antiderivative = 7.66 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(3*(2*B*a^2*b^2*c^5*d - 4*A*a^2*b^2*c^4*d^2 - 3*B*a^3*b*c^3*d^3 + A*a 
^3*b*c^2*d^4 + (2*B*a*b^3*c^3*d^3 - 4*A*a*b^3*c^2*d^4 - 3*B*a^2*b^2*c*d^5 
+ A*a^2*b^2*d^6)*x^4 + 2*(2*B*a*b^3*c^4*d^2 - 4*A*a*b^3*c^3*d^3 - 3*B*a^2* 
b^2*c^2*d^4 + A*a^2*b^2*c*d^5)*x^3 + (2*B*a*b^3*c^5*d - 4*A*a*b^3*c^4*d^2 
- B*a^2*b^2*c^3*d^3 - 3*A*a^2*b^2*c^2*d^4 - 3*B*a^3*b*c*d^5 + A*a^3*b*d^6) 
*x^2 + 2*(2*B*a^2*b^2*c^4*d^2 - 4*A*a^2*b^2*c^3*d^3 - 3*B*a^3*b*c^2*d^4 + 
A*a^3*b*c*d^5)*x)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d 
^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt( 
b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(2*B*a*b^3*c^7 - 6*A*a*b^3*c^6* 
d - 10*B*a^2*b^2*c^5*d^2 + 4*A*a^2*b^2*c^4*d^3 - 11*B*a^3*b*c^3*d^4 + 11*A 
*a^3*b*c^2*d^5 + B*a^4*c*d^6 + A*a^4*d^7 - (2*A*b^4*c^5*d^2 + 11*B*a*b^3*c 
^4*d^3 - 11*A*a*b^3*c^3*d^4 + 7*B*a^2*b^2*c^2*d^5 - 13*A*a^2*b^2*c*d^6 - 4 
*B*a^3*b*d^7)*x^3 - (4*A*b^4*c^6*d + 16*B*a*b^3*c^5*d^2 - 10*A*a*b^3*c^4*d 
^3 + 17*B*a^2*b^2*c^3*d^4 - 17*A*a^2*b^2*c^2*d^5 + B*a^3*b*c*d^6 - 3*A*a^3 
*b*d^7)*x^2 - (2*A*b^4*c^7 + 2*B*a*b^3*c^6*d + 8*A*a*b^3*c^5*d^2 + 17*B*a^ 
2*b^2*c^4*d^3 - 5*A*a^2*b^2*c^3*d^4 + 13*B*a^3*b*c^2*d^5 - 11*A*a^3*b*c*d^ 
6 - 2*B*a^4*d^7)*x)*sqrt(b*x^2 + a))/(a^2*b^4*c^10 + 4*a^3*b^3*c^8*d^2 + 6 
*a^4*b^2*c^6*d^4 + 4*a^5*b*c^4*d^6 + a^6*c^2*d^8 + (a*b^5*c^8*d^2 + 4*a^2* 
b^4*c^6*d^4 + 6*a^3*b^3*c^4*d^6 + 4*a^4*b^2*c^2*d^8 + a^5*b*d^10)*x^4 + 2* 
(a*b^5*c^9*d + 4*a^2*b^4*c^7*d^3 + 6*a^3*b^3*c^5*d^5 + 4*a^4*b^2*c^3*d^...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1390 vs. \(2 (273) = 546\).

Time = 0.14 (sec) , antiderivative size = 1390, normalized size of antiderivative = 4.76 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-15/2*B*b^3*c^4*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 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^5 + sqrt(b*x^2 + a)*a^4*d^7) + 15 
/2*A*b^3*c^3*x/(sqrt(b*x^2 + a)*a*b^3*c^6 + 3*sqrt(b*x^2 + a)*a^2*b^2*c^4* 
d^2 + 3*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^3/(sqrt(b*x^2 + a)*b^3*c^6 + 3*sqrt(b*x^2 + a)*a*b^2*c^4*d^2 + 3*sqr 
t(b*x^2 + a)*a^2*b*c^2*d^4 + sqrt(b*x^2 + a)*a^3*d^6) + 19/2*B*b^2*c^2*x/( 
sqrt(b*x^2 + a)*a*b^2*c^4*d + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^3 + sqrt(b*x^2 
 + a)*a^3*d^5) + 15/2*A*b^2*c^2/(sqrt(b*x^2 + a)*b^3*c^6/d + 3*sqrt(b*x^2 
+ a)*a*b^2*c^4*d + 3*sqrt(b*x^2 + a)*a^2*b*c^2*d^3 + sqrt(b*x^2 + a)*a^3*d 
^5) - 13/2*A*b^2*c*x/(sqrt(b*x^2 + a)*a*b^2*c^4 + 2*sqrt(b*x^2 + a)*a^2*b* 
c^2*d^2 + sqrt(b*x^2 + a)*a^3*d^4) + 5/2*B*b*c^2/(sqrt(b*x^2 + a)*b^2*c^4* 
d*x + 2*sqrt(b*x^2 + a)*a*b*c^2*d^3*x + sqrt(b*x^2 + a)*a^2*d^5*x + sqrt(b 
*x^2 + a)*b^2*c^5 + 2*sqrt(b*x^2 + a)*a*b*c^3*d^2 + sqrt(b*x^2 + a)*a^2*c* 
d^4) - 5/2*A*b*c/(sqrt(b*x^2 + a)*b^2*c^4*x + 2*sqrt(b*x^2 + a)*a*b*c^2*d^ 
2*x + sqrt(b*x^2 + a)*a^2*d^4*x + sqrt(b*x^2 + a)*b^2*c^5/d + 2*sqrt(b*x^2 
 + a)*a*b*c^3*d + sqrt(b*x^2 + a)*a^2*c*d^3) + 9/2*B*b*c/(sqrt(b*x^2 + a)* 
b^2*c^4 + 2*sqrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) - 2*B*b 
*x/(sqrt(b*x^2 + a)*a*b*c^2*d + sqrt(b*x^2 + a)*a^2*d^3) - 3/2*A*b/(sqrt(b 
*x^2 + a)*b^2*c^4/d + 2*sqrt(b*x^2 + a)*a*b*c^2*d + sqrt(b*x^2 + a)*a^2*d^ 
3) + 1/2*B*c/(sqrt(b*x^2 + a)*b*c^2*d^2*x^2 + sqrt(b*x^2 + a)*a*d^4*x^2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1048 vs. \(2 (273) = 546\).

Time = 0.15 (sec) , antiderivative size = 1048, normalized size of antiderivative = 3.59 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((A*b^6*c^9 + 3*B*a*b^5*c^8*d + 8*B*a^2*b^4*c^6*d^3 - 6*A*a^2*b^4*c^5*d^4 
+ 6*B*a^3*b^3*c^4*d^5 - 8*A*a^3*b^3*c^3*d^6 - 3*A*a^4*b^2*c*d^8 - B*a^5*b* 
d^9)*x/(a*b^6*c^12 + 6*a^2*b^5*c^10*d^2 + 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3* 
c^6*d^6 + 15*a^5*b^2*c^4*d^8 + 6*a^6*b*c^2*d^10 + a^7*d^12) - (B*a*b^5*c^9 
 - 3*A*a*b^5*c^8*d - 8*A*a^2*b^4*c^6*d^3 - 6*B*a^3*b^3*c^5*d^4 - 6*A*a^3*b 
^3*c^4*d^5 - 8*B*a^4*b^2*c^3*d^6 - 3*B*a^5*b*c*d^8 + A*a^5*b*d^9)/(a*b^6*c 
^12 + 6*a^2*b^5*c^10*d^2 + 15*a^3*b^4*c^8*d^4 + 20*a^4*b^3*c^6*d^6 + 15*a^ 
5*b^2*c^4*d^8 + 6*a^6*b*c^2*d^10 + a^7*d^12))/sqrt(b*x^2 + a) + 3*(2*B*b^2 
*c^3*d - 4*A*b^2*c^2*d^2 - 3*B*a*b*c*d^3 + A*a*b*d^4)*arctan(((sqrt(b)*x - 
 sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^3*c^6 + 3*a*b^2 
*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(-b*c^2 - a*d^2)) + (4*(sqrt(b)* 
x - sqrt(b*x^2 + a))^3*B*b^2*c^3*d^2 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A 
*b^2*c^2*d^3 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a*b*c*d^4 + (sqrt(b)*x 
- sqrt(b*x^2 + a))^3*A*a*b*d^5 + 10*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(5 
/2)*c^4*d - 14*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(5/2)*c^3*d^2 - 9*(sqrt 
(b)*x - sqrt(b*x^2 + a))^2*B*a*b^(3/2)*c^2*d^3 + 7*(sqrt(b)*x - sqrt(b*x^2 
 + a))^2*A*a*b^(3/2)*c*d^4 + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt( 
b)*d^5 - 16*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a*b^2*c^3*d^2 + 22*(sqrt(b)*x 
- sqrt(b*x^2 + a))*A*a*b^2*c^2*d^3 + 5*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2 
*b*c*d^4 + (sqrt(b)*x - sqrt(b*x^2 + a))*A*a^2*b*d^5 + 5*B*a^2*b^(3/2)*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 2940, normalized size of antiderivative = 10.07 \[ \int \frac {A+B x}{(c+d x)^3 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*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**3*b*c**2*d**4 + 6*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**3*b*c*d**5*x + 3*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** 
3*b*d**6*x**2 - 12*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**4*d**2 - 24*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**3*x - 9*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**3 - 9*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** 
4*x**2 - 18*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**4*x + 6*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*d**5*x 
**3 - 9*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*d**5*x**2 + 3*sqrt(a*d**2 + b*c**2)*log( - sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*d**6*x**4 + 
6*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**5*d - 12*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**2*x**2 + 12*sqrt(a*d 
**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c...