\(\int \frac {(A+B x) (a+b x^2)^{3/2}}{(c+d x)^3} \, dx\) [161]

Optimal result

Integrand size = 24, antiderivative size = 238 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {3 \left (a B d^2+2 b c (2 B c-A d)+b d (2 B c-A d) x\right ) \sqrt {a+b x^2}}{2 d^4 (c+d x)}+\frac {(2 B c-A d+B d x) \left (a+b x^2\right )^{3/2}}{2 d^2 (c+d x)^2}+\frac {3 \sqrt {b} \left (a B d^2+2 b c (2 B c-A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^5}+\frac {3 b \left (2 b c^2 (2 B c-A d)+a d^2 (3 B c-A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (A d+B (c+2 d x))+b \left (-A d \left (6 c^2+9 c d x+2 d^2 x^2\right )+B \left (12 c^3+18 c^2 d x+4 c d^2 x^2-d^3 x^3\right )\right )\right )}{(c+d x)^2}-\frac {6 b \left (2 b c^2 (2 B c-A d)+a d^2 (3 B c-A d)\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+3 \sqrt {b} \left (a B d^2+2 b c (2 B c-A d)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^5} \] Input:

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

Output:

-1/2*((d*Sqrt[a + b*x^2]*(a*d^2*(A*d + B*(c + 2*d*x)) + b*(-(A*d*(6*c^2 + 
9*c*d*x + 2*d^2*x^2)) + B*(12*c^3 + 18*c^2*d*x + 4*c*d^2*x^2 - d^3*x^3)))) 
/(c + d*x)^2 - (6*b*(2*b*c^2*(2*B*c - A*d) + a*d^2*(3*B*c - A*d))*ArcTan[( 
Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]])/Sqrt[-(b*c 
^2) - a*d^2] + 3*Sqrt[b]*(a*B*d^2 + 2*b*c*(2*B*c - A*d))*Log[-(Sqrt[b]*x) 
+ Sqrt[a + b*x^2]])/d^5
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {681, 27, 681, 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.

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {3 \left (-\frac {b \left (\frac {\left (a d^2 (3 B c-A d)+2 b c^2 (2 B c-A d)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a B d^2+2 b c (2 B c-A d)\right )}{\sqrt {b} d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (a B d^2+b d x (2 B c-A d)+2 b c (2 B c-A d)\right )}{d^2 (c+d x)}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} (-A d+2 B c+B d x)}{2 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {3 \left (-\frac {b \left (-\frac {\left (a d^2 (3 B c-A d)+2 b c^2 (2 B c-A d)\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a B d^2+2 b c (2 B c-A d)\right )}{\sqrt {b} d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (a B d^2+b d x (2 B c-A d)+2 b c (2 B c-A d)\right )}{d^2 (c+d x)}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} (-A d+2 B c+B d x)}{2 d^2 (c+d x)^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*B*c - A*d + B*d*x)*(a + b*x^2)^(3/2))/(2*d^2*(c + d*x)^2) + (3*(-(((a* 
B*d^2 + 2*b*c*(2*B*c - A*d) + b*d*(2*B*c - A*d)*x)*Sqrt[a + b*x^2])/(d^2*( 
c + d*x))) - (b*(-(((a*B*d^2 + 2*b*c*(2*B*c - A*d))*ArcTanh[(Sqrt[b]*x)/Sq 
rt[a + b*x^2]])/(Sqrt[b]*d)) - ((2*b*c^2*(2*B*c - 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])])/(d*Sqrt[ 
b*c^2 + a*d^2])))/d^2))/(2*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 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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1004\) vs. \(2(214)=428\).

Time = 1.52 (sec) , antiderivative size = 1005, normalized size of antiderivative = 4.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(1005\)
default	\(\text {Expression too large to display}\)	\(2352\)

Input:

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

Output:

1/2*b*(B*d*x+2*A*d-6*B*c)*(b*x^2+a)^(1/2)/d^4-1/2/d^4*(2/d^3*(4*A*a*b*c*d^ 
3+4*A*b^2*c^3*d-B*a^2*d^4-6*B*a*b*c^2*d^2-5*B*b^2*c^4)*(-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)))-2*(A*a^2*d^5+2*A*a*b*c^2*d^3+A*b^2*c^4*d-B*a^2*c*d^4 
-2*B*a*b*c^3*d^2-B*b^2*c^5)/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)+3/2*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)-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)))+1/2*b/(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)))+3* 
b^(1/2)*(2*A*b*c*d-B*a*d^2-4*B*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+4*b/ 
d^2*(A*a*d^3+3*A*b*c^2*d-3*B*a*c*d^2-5*B*b*c^3)/((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. \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

integrate((B*x+A)*(b*x**2+a)**(3/2)/(d*x+c)**3,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. 776 vs. \(2 (215) = 430\).

Time = 0.12 (sec) , antiderivative size = 776, normalized size of antiderivative = 3.26 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-3/2*sqrt(b*x^2 + a)*B*b^2*c^3/(b*c^2*d^4 + a*d^6) + 3/2*sqrt(b*x^2 + a)*B 
*b^2*c^2*x/(b*c^2*d^3 + a*d^5) - 1/2*(b*x^2 + a)^(3/2)*B*b*c^2/(b*c^2*d^3* 
x + a*d^5*x + b*c^3*d^2 + a*c*d^4) + 3/2*sqrt(b*x^2 + a)*A*b^2*c^2/(b*c^2* 
d^3 + a*d^5) - 3/2*sqrt(b*x^2 + a)*A*b^2*c*x/(b*c^2*d^2 + a*d^4) + 1/2*(b* 
x^2 + a)^(5/2)*B*c/(b*c^2*d^2*x^2 + a*d^4*x^2 + 2*b*c^3*d*x + 2*a*c*d^3*x 
+ b*c^4 + a*c^2*d^2) + 1/2*(b*x^2 + a)^(3/2)*A*b*c/(b*c^2*d^2*x + a*d^4*x 
+ b*c^3*d + a*c*d^3) - 1/2*(b*x^2 + a)^(3/2)*B*b*c/(b*c^2*d^2 + a*d^4) - 1 
/2*(b*x^2 + a)^(5/2)*A/(b*c^2*d*x^2 + a*d^3*x^2 + 2*b*c^3*x + 2*a*c*d^2*x 
+ b*c^4/d + a*c^2*d) + 1/2*(b*x^2 + a)^(3/2)*A*b/(b*c^2*d + a*d^3) - (b*x^ 
2 + a)^(3/2)*B/(d^3*x + c*d^2) + 3/2*sqrt(b*x^2 + a)*B*b*x/d^3 + 6*B*b^(3/ 
2)*c^2*arcsinh(b*x/sqrt(a*b))/d^5 - 3*A*b^(3/2)*c*arcsinh(b*x/sqrt(a*b))/d 
^4 + 3/2*B*a*sqrt(b)*arcsinh(b*x/sqrt(a*b))/d^3 - 3/2*B*b^2*c^3*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^6) + 3/2*A*b^2*c^2*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^5) - 9/2*B*sqrt(a + b*c^2 
/d^2)*b*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x 
+ c)))/d^4 + 3/2*A*sqrt(a + b*c^2/d^2)*b*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x 
+ c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^3 - 9/2*sqrt(b*x^2 + a)*B*b*c/d^4 
+ 3/2*sqrt(b*x^2 + a)*A*b/d^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (215) = 430\).

Time = 0.19 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.80 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {B b x}{d^{3}} - \frac {2 \, {\left (3 \, B b c d^{8} - A b d^{9}\right )}}{d^{12}}\right )} - \frac {3 \, {\left (4 \, B b^{\frac {3}{2}} c^{2} - 2 \, A b^{\frac {3}{2}} c d + B a \sqrt {b} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, d^{5}} - \frac {3 \, {\left (4 \, B b^{2} c^{3} - 2 \, A b^{2} c^{2} d + 3 \, B a b c d^{2} - A a b d^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} d^{5}} - \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B b^{2} c^{3} d - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b^{2} c^{2} d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B a b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a b d^{4} + 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {5}{2}} c^{4} - 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {5}{2}} c^{3} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a b^{\frac {3}{2}} c^{2} d^{2} + 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} c d^{3} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} d^{4} - 20 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a b^{2} c^{3} d + 14 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b^{2} c^{2} d^{2} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b d^{4} + 7 \, B a^{2} b^{\frac {3}{2}} c^{2} d^{2} - 5 \, A a^{2} b^{\frac {3}{2}} c d^{3} + 2 \, B a^{3} \sqrt {b} d^{4}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2} d^{5}} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*(B*b*x/d^3 - 2*(3*B*b*c*d^8 - A*b*d^9)/d^12) - 3/2*(4* 
B*b^(3/2)*c^2 - 2*A*b^(3/2)*c*d + B*a*sqrt(b)*d^2)*log(abs(-sqrt(b)*x + sq 
rt(b*x^2 + a)))/d^5 - 3*(4*B*b^2*c^3 - 2*A*b^2*c^2*d + 3*B*a*b*c*d^2 - A*a 
*b*d^3)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 
- a*d^2))/(sqrt(-b*c^2 - a*d^2)*d^5) - (8*(sqrt(b)*x - sqrt(b*x^2 + a))^3* 
B*b^2*c^3*d - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b^2*c^2*d^2 + 3*(sqrt(b) 
*x - sqrt(b*x^2 + a))^3*B*a*b*c*d^3 - (sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a* 
b*d^4 + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(5/2)*c^4 - 10*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^2*A*b^(5/2)*c^3*d - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B* 
a*b^(3/2)*c^2*d^2 + 5*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2)*c*d^3 - 
2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)*d^4 - 20*(sqrt(b)*x - sqrt 
(b*x^2 + a))*B*a*b^2*c^3*d + 14*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a*b^2*c^2* 
d^2 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2*b*c*d^3 - (sqrt(b)*x - sqrt(b* 
x^2 + a))*A*a^2*b*d^4 + 7*B*a^2*b^(3/2)*c^2*d^2 - 5*A*a^2*b^(3/2)*c*d^3 + 
2*B*a^3*sqrt(b)*d^4)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - 
sqrt(b*x^2 + a))*sqrt(b)*c - a*d)^2*d^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 2274, normalized size of antiderivative = 9.55 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(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*c**2*d**3 + 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*c*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*d**5 
*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*b**2*c**4*d + 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*b**2*c**3*d**2*x - 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* 
b**2*c**3*d**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*b**2*c**2*d**3*x**2 - 36*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**2*d** 
3*x - 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*b**2*c*d**4*x**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)*b**3*c**5 - 48*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** 
4*d*x - 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)*b**3*c**3*d**2*x**2 - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x 
)*a**2*b*c**2*d**3 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**4*x 
 - 6*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**5*x**2 - 12*sqrt(a*d**2 
+ b*c**2)*log(c + d*x)*a*b**2*c**4*d - 24*sqrt(a*d**2 + b*c**2)*log(c +...