\(\int \frac {x^2}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [1241]

Optimal result

Integrand size = 22, antiderivative size = 151 \[ \int \frac {x^2}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 a c d+\left (b c^2-a d^2\right ) x}{\left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {c^2 d \sqrt {a+b x^2}}{\left (b c^2+a d^2\right )^2 (c+d x)}-\frac {c \left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}} \] Output:

-(2*a*c*d+(-a*d^2+b*c^2)*x)/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)-c^2*d*(b*x^2+a 
)^(1/2)/(a*d^2+b*c^2)^2/(d*x+c)-c*(-2*a*d^2+b*c^2)*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)^(5/2)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-b c^2 x (c+2 d x)+a d \left (-3 c^2-c d x+d^2 x^2\right )}{\left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}-\frac {2 c \left (b c^2-2 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

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

Output:

(-(b*c^2*x*(c + 2*d*x)) + a*d*(-3*c^2 - c*d*x + d^2*x^2))/((b*c^2 + a*d^2) 
^2*(c + d*x)*Sqrt[a + b*x^2]) - (2*c*(b*c^2 - 2*a*d^2)*ArcTan[(Sqrt[-(b*c^ 
2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + b*x^2])])/(-(b*c^2) - a*d^2 
)^(5/2)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {601, 25, 27, 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.

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

\(\Big \downarrow \) 601

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((2*a*c*d + (b*c^2 - a*d^2)*x)/((b*c^2 + a*d^2)^2*Sqrt[a + b*x^2])) + (c* 
(-((c*d*Sqrt[a + b*x^2])/(c + d*x)) - ((b*c^2 - 2*a*d^2)*ArcTanh[(a*d - b* 
c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/Sqrt[b*c^2 + a*d^2]))/(b*c^2 
+ a*d^2)^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 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(851\) vs. \(2(141)=282\).

Time = 0.40 (sec) , antiderivative size = 852, normalized size of antiderivative = 5.64

method	result	size
default	\(\frac {x}{d^{2} \sqrt {b \,x^{2}+a}\, a}+\frac {c^{2} \left (-\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {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}}}}+\frac {3 b c d \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\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}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\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}}}}-\frac {d^{2} \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 )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\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}}}}\right )}{d^{4}}-\frac {2 c \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\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}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\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}}}}-\frac {d^{2} \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}}\)	\(852\)

Input:

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

Output:

1/d^2/(b*x^2+a)^(1/2)/a*x+c^2/d^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)+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)) 
-2*c/d^3*(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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (142) = 284\).

Time = 0.19 (sec) , antiderivative size = 909, normalized size of antiderivative = 6.02 \[ \int \frac {x^2}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*((a*b*c^4 - 2*a^2*c^2*d^2 + (b^2*c^3*d - 2*a*b*c*d^3)*x^3 + (b^2*c^4 
 - 2*a*b*c^2*d^2)*x^2 + (a*b*c^3*d - 2*a^2*c*d^3)*x)*sqrt(b*c^2 + a*d^2)*l 
og((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*(3*a*b*c^4*d + 3*a^2*c^2*d^3 + (2*b^2*c^4*d + a*b*c^2*d^3 - a^2*d^5)*x 
^2 + (b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4)*x)*sqrt(b*x^2 + a))/(a*b^3*c^7 
+ 3*a^2*b^2*c^5*d^2 + 3*a^3*b*c^3*d^4 + a^4*c*d^6 + (b^4*c^6*d + 3*a*b^3*c 
^4*d^3 + 3*a^2*b^2*c^2*d^5 + a^3*b*d^7)*x^3 + (b^4*c^7 + 3*a*b^3*c^5*d^2 + 
 3*a^2*b^2*c^3*d^4 + a^3*b*c*d^6)*x^2 + (a*b^3*c^6*d + 3*a^2*b^2*c^4*d^3 + 
 3*a^3*b*c^2*d^5 + a^4*d^7)*x), -((a*b*c^4 - 2*a^2*c^2*d^2 + (b^2*c^3*d - 
2*a*b*c*d^3)*x^3 + (b^2*c^4 - 2*a*b*c^2*d^2)*x^2 + (a*b*c^3*d - 2*a^2*c*d^ 
3)*x)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt( 
b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + (3*a*b*c^4*d + 
 3*a^2*c^2*d^3 + (2*b^2*c^4*d + a*b*c^2*d^3 - a^2*d^5)*x^2 + (b^2*c^5 + 2* 
a*b*c^3*d^2 + a^2*c*d^4)*x)*sqrt(b*x^2 + a))/(a*b^3*c^7 + 3*a^2*b^2*c^5*d^ 
2 + 3*a^3*b*c^3*d^4 + a^4*c*d^6 + (b^4*c^6*d + 3*a*b^3*c^4*d^3 + 3*a^2*b^2 
*c^2*d^5 + a^3*b*d^7)*x^3 + (b^4*c^7 + 3*a*b^3*c^5*d^2 + 3*a^2*b^2*c^3*d^4 
 + a^3*b*c*d^6)*x^2 + (a*b^3*c^6*d + 3*a^2*b^2*c^4*d^3 + 3*a^3*b*c^2*d^5 + 
 a^4*d^7)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (142) = 284\).

Time = 0.08 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.71 \[ \int \frac {x^2}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} c^{4} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} d^{2} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{4} + \sqrt {b x^{2} + a} a^{3} d^{6}} + \frac {3 \, b c^{3}}{\sqrt {b x^{2} + a} b^{2} c^{4} d + 2 \, \sqrt {b x^{2} + a} a b c^{2} d^{3} + \sqrt {b x^{2} + a} a^{2} d^{5}} - \frac {4 \, b c^{2} x}{\sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} - \frac {c^{2}}{\sqrt {b x^{2} + a} b c^{2} d^{2} x + \sqrt {b x^{2} + a} a d^{4} x + \sqrt {b x^{2} + a} b c^{3} d + \sqrt {b x^{2} + a} a c d^{3}} - \frac {2 \, c}{\sqrt {b x^{2} + a} b c^{2} d + \sqrt {b x^{2} + a} a d^{3}} + \frac {x}{\sqrt {b x^{2} + a} a d^{2}} + \frac {3 \, b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{5}} - \frac {2 \, c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} \] Input:

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

Output:

3*b^2*c^4*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^2 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d 
^4 + sqrt(b*x^2 + a)*a^3*d^6) + 3*b*c^3/(sqrt(b*x^2 + a)*b^2*c^4*d + 2*sqr 
t(b*x^2 + a)*a*b*c^2*d^3 + sqrt(b*x^2 + a)*a^2*d^5) - 4*b*c^2*x/(sqrt(b*x^ 
2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) - c^2/(sqrt(b*x^2 + a)*b*c^2 
*d^2*x + sqrt(b*x^2 + a)*a*d^4*x + sqrt(b*x^2 + a)*b*c^3*d + sqrt(b*x^2 + 
a)*a*c*d^3) - 2*c/(sqrt(b*x^2 + a)*b*c^2*d + sqrt(b*x^2 + a)*a*d^3) + x/(s 
qrt(b*x^2 + a)*a*d^2) + 3*b*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a 
*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5/2)*d^5) - 2*c*arcsinh(b*c 
*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^ 
2)^(3/2)*d^3)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 1009, normalized size of antiderivative = 6.68 \[ \int \frac {x^2}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*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*c**2*d**2 + 2*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*c*d**3*x - 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*c**4 
- 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*c**3*d*x + 2*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*c**2*d**2*x**2 + 2*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*c 
*d**3*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)*b**2*c**4*x**2 - 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**2*c**3*d*x**3 - 2*sqrt( 
a*d**2 + b*c**2)*log(c + d*x)*a**2*c**2*d**2 - 2*sqrt(a*d**2 + b*c**2)*log 
(c + d*x)*a**2*c*d**3*x + sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**4 + sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**3*d*x - 2*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*a*b*c**2*d**2*x**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c 
*d**3*x**3 + sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**4*x**2 + sqrt(a*d* 
*2 + b*c**2)*log(c + d*x)*b**2*c**3*d*x**3 - 3*sqrt(a + b*x**2)*a**2*c**2* 
d**3 - sqrt(a + b*x**2)*a**2*c*d**4*x + sqrt(a + b*x**2)*a**2*d**5*x**2 - 
3*sqrt(a + b*x**2)*a*b*c**4*d - 2*sqrt(a + b*x**2)*a*b*c**3*d**2*x - sqrt( 
a + b*x**2)*a*b*c**2*d**3*x**2 - sqrt(a + b*x**2)*b**2*c**5*x - 2*sqrt(...