\(\int \frac {1}{x^2 (c+d x) (a+b x^2)^{5/2}} \, dx\) [1284]

Optimal result

Integrand size = 22, antiderivative size = 217 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=-\frac {b (a d+b c x)}{3 a^2 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}-\frac {b \left (3 a d \left (b c^2+2 a d^2\right )+b c \left (5 b c^2+8 a d^2\right ) x\right )}{3 a^3 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {\sqrt {a+b x^2}}{a^3 c x}-\frac {d^6 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^2 \left (b c^2+a d^2\right )^{5/2}}+\frac {d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c^2} \] Output:

-1/3*b*(b*c*x+a*d)/a^2/(a*d^2+b*c^2)/(b*x^2+a)^(3/2)-1/3*b*(3*a*d*(2*a*d^2 
+b*c^2)+b*c*(8*a*d^2+5*b*c^2)*x)/a^3/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)-(b*x^ 
2+a)^(1/2)/a^3/c/x-d^6*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^ 
(1/2))/c^2/(a*d^2+b*c^2)^(5/2)+d*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)/ 
c^2
 

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=-\frac {\frac {c \left (3 a^4 d^4+8 b^4 c^4 x^4+a^3 b d^2 \left (6 c^2+7 c d x+6 d^2 x^2\right )+a b^3 c^2 x^2 \left (12 c^2+3 c d x+14 d^2 x^2\right )+a^2 b^2 \left (3 c^4+4 c^3 d x+21 c^2 d^2 x^2+6 c d^3 x^3+3 d^4 x^4\right )\right )}{a^3 \left (b c^2+a d^2\right )^2 x \left (a+b x^2\right )^{3/2}}+\frac {6 d^6 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {6 d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}}{3 c^2} \] Input:

Integrate[1/(x^2*(c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

-1/3*((c*(3*a^4*d^4 + 8*b^4*c^4*x^4 + a^3*b*d^2*(6*c^2 + 7*c*d*x + 6*d^2*x 
^2) + a*b^3*c^2*x^2*(12*c^2 + 3*c*d*x + 14*d^2*x^2) + a^2*b^2*(3*c^4 + 4*c 
^3*d*x + 21*c^2*d^2*x^2 + 6*c*d^3*x^3 + 3*d^4*x^4)))/(a^3*(b*c^2 + a*d^2)^ 
2*x*(a + b*x^2)^(3/2)) + (6*d^6*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)^(5/2) + (6*d*ArcTanh[(Sqr 
t[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(5/2))/c^2
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.41, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 2009}

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.

\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right )^{5/2} (c+d x)} \, dx\)

\(\Big \downarrow \) 617

\(\displaystyle \int \left (\frac {d^2}{c^2 \left (a+b x^2\right )^{5/2} (c+d x)}-\frac {d}{c^2 x \left (a+b x^2\right )^{5/2}}+\frac {1}{c x^2 \left (a+b x^2\right )^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2} c^2}-\frac {8 b x}{3 a^3 c \sqrt {a+b x^2}}+\frac {d^2 \left (3 a^2 d^3+b c x \left (5 a d^2+2 b c^2\right )\right )}{3 a^2 c^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}-\frac {d}{a^2 c^2 \sqrt {a+b x^2}}-\frac {4 b x}{3 a^2 c \left (a+b x^2\right )^{3/2}}-\frac {d^6 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^2 \left (a d^2+b c^2\right )^{5/2}}+\frac {d^2 (a d+b c x)}{3 a c^2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )}-\frac {d}{3 a c^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{a c x \left (a+b x^2\right )^{3/2}}\)

Input:

Int[1/(x^2*(c + d*x)*(a + b*x^2)^(5/2)),x]
 

Output:

-1/3*d/(a*c^2*(a + b*x^2)^(3/2)) - 1/(a*c*x*(a + b*x^2)^(3/2)) - (4*b*x)/( 
3*a^2*c*(a + b*x^2)^(3/2)) + (d^2*(a*d + b*c*x))/(3*a*c^2*(b*c^2 + a*d^2)* 
(a + b*x^2)^(3/2)) - d/(a^2*c^2*Sqrt[a + b*x^2]) - (8*b*x)/(3*a^3*c*Sqrt[a 
 + b*x^2]) + (d^2*(3*a^2*d^3 + b*c*(2*b*c^2 + 5*a*d^2)*x))/(3*a^2*c^2*(b*c 
^2 + a*d^2)^2*Sqrt[a + b*x^2]) - (d^6*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + 
a*d^2]*Sqrt[a + b*x^2])])/(c^2*(b*c^2 + a*d^2)^(5/2)) + (d*ArcTanh[Sqrt[a 
+ b*x^2]/Sqrt[a]])/(a^(5/2)*c^2)
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(688\) vs. \(2(195)=390\).

Time = 0.42 (sec) , antiderivative size = 689, normalized size of antiderivative = 3.18

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}}{a^{3} c x}-\frac {-\frac {d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}+\frac {b c \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{4 \sqrt {-a b}\, d -4 b c}-\frac {b c \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{4 \left (\sqrt {-a b}\, d +b c \right )}-\frac {b c \left (4 \sqrt {-a b}\, d -3 b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}\, d -b c \right )^{2} a \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {b c \left (4 \sqrt {-a b}\, d +3 b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}\, d +b c \right )^{2} a \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} a^{2} d^{5} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{a^{2} c}\)	\(689\)
default	\(\frac {-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}}{c}+\frac {d \left (\frac {d^{2}}{3 \left (a \,d^{2}+b \,c^{2}\right ) \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}}}+\frac {b c d \left (\frac {\frac {4 b \left (x +\frac {c}{d}\right )}{3}-\frac {4 b c}{3 d}}{\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \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}}}+\frac {16 b \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{3 {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right )}^{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}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {d^{2} \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}}\right )}{c^{2}}-\frac {d \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{c^{2}}\)	\(729\)

Input:

int(1/x^2/(d*x+c)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-(b*x^2+a)^(1/2)/a^3/c/x-1/a^2/c*(-d/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a) 
^(1/2))/x)+1/4*b*c/((-a*b)^(1/2)*d-b*c)*(1/3/(-a*b)^(1/2)/(x+(-a*b)^(1/2)/ 
b)^2*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)-1/3/ 
a/(x+(-a*b)^(1/2)/b)*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2 
)/b))^(1/2))-1/4*b*c/((-a*b)^(1/2)*d+b*c)*(-1/3/(-a*b)^(1/2)/(x-(-a*b)^(1/ 
2)/b)^2*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)-1 
/3/a/(x-(-a*b)^(1/2)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^( 
1/2)/b))^(1/2))-1/4*b*c*(4*(-a*b)^(1/2)*d-3*b*c)/((-a*b)^(1/2)*d-b*c)^2/a/ 
(x+(-a*b)^(1/2)/b)*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/ 
b))^(1/2)+1/4*b*c*(4*(-a*b)^(1/2)*d+3*b*c)/((-a*b)^(1/2)*d+b*c)^2/a/(x-(-a 
*b)^(1/2)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1 
/2)+b^2*a^2*d^5/((-a*b)^(1/2)*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2/c/((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. 705 vs. \(2 (196) = 392\).

Time = 1.85 (sec) , antiderivative size = 2890, normalized size of antiderivative = 13.32 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(a^3*b^2*d^6*x^5 + 2*a^4*b*d^6*x^3 + a^5*d^6*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*s 
qrt(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 
)) + 3*((b^5*c^6*d + 3*a*b^4*c^4*d^3 + 3*a^2*b^3*c^2*d^5 + a^3*b^2*d^7)*x^ 
5 + 2*(a*b^4*c^6*d + 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^2*d^5 + a^4*b*d^7)*x^ 
3 + (a^2*b^3*c^6*d + 3*a^3*b^2*c^4*d^3 + 3*a^4*b*c^2*d^5 + a^5*d^7)*x)*sqr 
t(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(3*a^2*b^3*c^ 
7 + 9*a^3*b^2*c^5*d^2 + 9*a^4*b*c^3*d^4 + 3*a^5*c*d^6 + (8*b^5*c^7 + 22*a* 
b^4*c^5*d^2 + 17*a^2*b^3*c^3*d^4 + 3*a^3*b^2*c*d^6)*x^4 + 3*(a*b^4*c^6*d + 
 3*a^2*b^3*c^4*d^3 + 2*a^3*b^2*c^2*d^5)*x^3 + 3*(4*a*b^4*c^7 + 11*a^2*b^3* 
c^5*d^2 + 9*a^3*b^2*c^3*d^4 + 2*a^4*b*c*d^6)*x^2 + (4*a^2*b^3*c^6*d + 11*a 
^3*b^2*c^4*d^3 + 7*a^4*b*c^2*d^5)*x)*sqrt(b*x^2 + a))/((a^3*b^5*c^8 + 3*a^ 
4*b^4*c^6*d^2 + 3*a^5*b^3*c^4*d^4 + a^6*b^2*c^2*d^6)*x^5 + 2*(a^4*b^4*c^8 
+ 3*a^5*b^3*c^6*d^2 + 3*a^6*b^2*c^4*d^4 + a^7*b*c^2*d^6)*x^3 + (a^5*b^3*c^ 
8 + 3*a^6*b^2*c^6*d^2 + 3*a^7*b*c^4*d^4 + a^8*c^2*d^6)*x), -1/6*(6*(a^3*b^ 
2*d^6*x^5 + 2*a^4*b*d^6*x^3 + a^5*d^6*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*((b^5*c^6*d + 3*a*b^4*c^4*d^3 + 3*a^2*b^3*c^2*d^5 + 
 a^3*b^2*d^7)*x^5 + 2*(a*b^4*c^6*d + 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^2*d^5 
 + a^4*b*d^7)*x^3 + (a^2*b^3*c^6*d + 3*a^3*b^2*c^4*d^3 + 3*a^4*b*c^2*d^...
 

Sympy [F]

\[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )}\, dx \] Input:

integrate(1/x**2/(d*x+c)/(b*x**2+a)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(1/(x**2*(a + b*x**2)**(5/2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x + c\right )} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (196) = 392\).

Time = 0.17 (sec) , antiderivative size = 1100, normalized size of antiderivative = 5.07 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/x^2/(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-2*d^6*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - 
a*d^2))/((b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4)*sqrt(-b*c^2 - a*d^2)) - 1 
/3*((((5*a^5*b^11*c^15 + 38*a^6*b^10*c^13*d^2 + 123*a^7*b^9*c^11*d^4 + 220 
*a^8*b^8*c^9*d^6 + 235*a^9*b^7*c^7*d^8 + 150*a^10*b^6*c^5*d^10 + 53*a^11*b 
^5*c^3*d^12 + 8*a^12*b^4*c*d^14)*x/(a^8*b^9*c^16 + 8*a^9*b^8*c^14*d^2 + 28 
*a^10*b^7*c^12*d^4 + 56*a^11*b^6*c^10*d^6 + 70*a^12*b^5*c^8*d^8 + 56*a^13* 
b^4*c^6*d^10 + 28*a^14*b^3*c^4*d^12 + 8*a^15*b^2*c^2*d^14 + a^16*b*d^16) + 
 3*(a^6*b^10*c^14*d + 8*a^7*b^9*c^12*d^3 + 27*a^8*b^8*c^10*d^5 + 50*a^9*b^ 
7*c^8*d^7 + 55*a^10*b^6*c^6*d^9 + 36*a^11*b^5*c^4*d^11 + 13*a^12*b^4*c^2*d 
^13 + 2*a^13*b^3*d^15)/(a^8*b^9*c^16 + 8*a^9*b^8*c^14*d^2 + 28*a^10*b^7*c^ 
12*d^4 + 56*a^11*b^6*c^10*d^6 + 70*a^12*b^5*c^8*d^8 + 56*a^13*b^4*c^6*d^10 
 + 28*a^14*b^3*c^4*d^12 + 8*a^15*b^2*c^2*d^14 + a^16*b*d^16))*x + 3*(2*a^6 
*b^10*c^15 + 15*a^7*b^9*c^13*d^2 + 48*a^8*b^8*c^11*d^4 + 85*a^9*b^7*c^9*d^ 
6 + 90*a^10*b^6*c^7*d^8 + 57*a^11*b^5*c^5*d^10 + 20*a^12*b^4*c^3*d^12 + 3* 
a^13*b^3*c*d^14)/(a^8*b^9*c^16 + 8*a^9*b^8*c^14*d^2 + 28*a^10*b^7*c^12*d^4 
 + 56*a^11*b^6*c^10*d^6 + 70*a^12*b^5*c^8*d^8 + 56*a^13*b^4*c^6*d^10 + 28* 
a^14*b^3*c^4*d^12 + 8*a^15*b^2*c^2*d^14 + a^16*b*d^16))*x + (4*a^7*b^9*c^1 
4*d + 31*a^8*b^8*c^12*d^3 + 102*a^9*b^7*c^10*d^5 + 185*a^10*b^6*c^8*d^7 + 
200*a^11*b^5*c^6*d^9 + 129*a^12*b^4*c^4*d^11 + 46*a^13*b^3*c^2*d^13 + 7*a^ 
14*b^2*d^15)/(a^8*b^9*c^16 + 8*a^9*b^8*c^14*d^2 + 28*a^10*b^7*c^12*d^4 ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^2+a\right )}^{5/2}\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (d x +c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

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

Output:

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