\(\int \frac {x}{(c+d x) (a+b x^2)^{5/2}} \, dx\) [1281]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=-\frac {c-d x}{3 \left (b c^2+a d^2\right ) \left (a+b x^2\right )^{3/2}}-\frac {d \left (3 a c d+\left (b c^2-2 a d^2\right ) x\right )}{3 a \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}+\frac {c d^3 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\frac {-b^2 c^2 d x^3+a^2 d^2 (-4 c+3 d x)-a b \left (c^3+3 c d^2 x^2-2 d^3 x^3\right )}{3 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}+\frac {2 c d^3 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

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

Output:

(-(b^2*c^2*d*x^3) + a^2*d^2*(-4*c + 3*d*x) - a*b*(c^3 + 3*c*d^2*x^2 - 2*d^ 
3*x^3))/(3*a*(b*c^2 + a*d^2)^2*(a + b*x^2)^(3/2)) + (2*c*d^3*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)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {593, 25, 686, 27, 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.

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

\(\Big \downarrow \) 593

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/3*(c - d*x)/((b*c^2 + a*d^2)*(a + b*x^2)^(3/2)) - (d*((3*a*c*d + (b*c^2 
 - 2*a*d^2)*x)/(a*(b*c^2 + a*d^2)*Sqrt[a + b*x^2]) - (3*c*d^2*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)))/ 
(3*(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 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(130)=260\).

Time = 0.34 (sec) , antiderivative size = 639, normalized size of antiderivative = 4.44

method	result	size
default	\(\frac {\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}}{d}-\frac {c \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 )}{d^{2}}\)	\(639\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (132) = 264\).

Time = 0.20 (sec) , antiderivative size = 816, normalized size of antiderivative = 5.67 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a b^{2} c d^{3} x^{4} + 2 \, a^{2} b c d^{3} x^{2} + a^{3} c d^{3}\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} + 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (a b^{2} c^{5} + 5 \, a^{2} b c^{3} d^{2} + 4 \, a^{3} c d^{4} + {\left (b^{3} c^{4} d - a b^{2} c^{2} d^{3} - 2 \, a^{2} b d^{5}\right )} x^{3} + 3 \, {\left (a b^{2} c^{3} d^{2} + a^{2} b c d^{4}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} d^{3} + a^{3} d^{5}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6} + {\left (a b^{5} c^{6} + 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (a^{2} b^{4} c^{6} + 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} + a^{5} b d^{6}\right )} x^{2}\right )}}, \frac {3 \, {\left (a b^{2} c d^{3} x^{4} + 2 \, a^{2} b c d^{3} x^{2} + a^{3} c d^{3}\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - {\left (a b^{2} c^{5} + 5 \, a^{2} b c^{3} d^{2} + 4 \, a^{3} c d^{4} + {\left (b^{3} c^{4} d - a b^{2} c^{2} d^{3} - 2 \, a^{2} b d^{5}\right )} x^{3} + 3 \, {\left (a b^{2} c^{3} d^{2} + a^{2} b c d^{4}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} d^{3} + a^{3} d^{5}\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{3} c^{6} + 3 \, a^{4} b^{2} c^{4} d^{2} + 3 \, a^{5} b c^{2} d^{4} + a^{6} d^{6} + {\left (a b^{5} c^{6} + 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (a^{2} b^{4} c^{6} + 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} + a^{5} b d^{6}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (132) = 264\).

Time = 0.09 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.31 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx=-\frac {b c^{2} x}{\frac {\sqrt {b x^{2} + a} a b^{2} c^{4}}{d} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d + \sqrt {b x^{2} + a} a^{3} d^{3}} - \frac {b c^{2} x}{3 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} a b c^{2} d + {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{3}\right )}} - \frac {2 \, b c^{2} x}{3 \, {\left (\sqrt {b x^{2} + a} a^{2} b c^{2} d + \sqrt {b x^{2} + a} a^{3} d^{3}\right )}} - \frac {c}{3 \, {\left ({\left (b x^{2} + a\right )}^{\frac {3}{2}} b c^{2} + {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d^{2}\right )}} - \frac {c}{2 \, \sqrt {b x^{2} + a} a b c^{2} + \frac {\sqrt {b x^{2} + a} b^{2} c^{4}}{d^{2}} + \sqrt {b x^{2} + a} a^{2} d^{2}} + \frac {2 \, x}{3 \, \sqrt {b x^{2} + a} a^{2} d} + \frac {x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d} - \frac {c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{2}} \] Input:

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

Output:

-b*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 + sq 
rt(b*x^2 + a)*a^3*d^3) - 1/3*b*c^2*x/((b*x^2 + a)^(3/2)*a*b*c^2*d + (b*x^2 
 + a)^(3/2)*a^2*d^3) - 2/3*b*c^2*x/(sqrt(b*x^2 + a)*a^2*b*c^2*d + sqrt(b*x 
^2 + a)*a^3*d^3) - 1/3*c/((b*x^2 + a)^(3/2)*b*c^2 + (b*x^2 + a)^(3/2)*a*d^ 
2) - c/(2*sqrt(b*x^2 + a)*a*b*c^2 + sqrt(b*x^2 + a)*b^2*c^4/d^2 + sqrt(b*x 
^2 + a)*a^2*d^2) + 2/3*x/(sqrt(b*x^2 + a)*a^2*d) + 1/3*x/((b*x^2 + a)^(3/2 
)*a*d) - 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)^(5/2)*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (132) = 264\).

Time = 0.16 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.76 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2*c*d^3*arctan(((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - 
 a*d^2))/((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)*sqrt(-b*c^2 - a*d^2)) - 1/3* 
((((b^9*c^14*d + 4*a*b^8*c^12*d^3 + 3*a^2*b^7*c^10*d^5 - 10*a^3*b^6*c^8*d^ 
7 - 25*a^4*b^5*c^6*d^9 - 24*a^5*b^4*c^4*d^11 - 11*a^6*b^3*c^2*d^13 - 2*a^7 
*b^2*d^15)*x/(a*b^9*c^16 + 8*a^2*b^8*c^14*d^2 + 28*a^3*b^7*c^12*d^4 + 56*a 
^4*b^6*c^10*d^6 + 70*a^5*b^5*c^8*d^8 + 56*a^6*b^4*c^6*d^10 + 28*a^7*b^3*c^ 
4*d^12 + 8*a^8*b^2*c^2*d^14 + a^9*b*d^16) + 3*(a*b^8*c^13*d^2 + 6*a^2*b^7* 
c^11*d^4 + 15*a^3*b^6*c^9*d^6 + 20*a^4*b^5*c^7*d^8 + 15*a^5*b^4*c^5*d^10 + 
 6*a^6*b^3*c^3*d^12 + a^7*b^2*c*d^14)/(a*b^9*c^16 + 8*a^2*b^8*c^14*d^2 + 2 
8*a^3*b^7*c^12*d^4 + 56*a^4*b^6*c^10*d^6 + 70*a^5*b^5*c^8*d^8 + 56*a^6*b^4 
*c^6*d^10 + 28*a^7*b^3*c^4*d^12 + 8*a^8*b^2*c^2*d^14 + a^9*b*d^16))*x - 3* 
(a^2*b^7*c^12*d^3 + 6*a^3*b^6*c^10*d^5 + 15*a^4*b^5*c^8*d^7 + 20*a^5*b^4*c 
^6*d^9 + 15*a^6*b^3*c^4*d^11 + 6*a^7*b^2*c^2*d^13 + a^8*b*d^15)/(a*b^9*c^1 
6 + 8*a^2*b^8*c^14*d^2 + 28*a^3*b^7*c^12*d^4 + 56*a^4*b^6*c^10*d^6 + 70*a^ 
5*b^5*c^8*d^8 + 56*a^6*b^4*c^6*d^10 + 28*a^7*b^3*c^4*d^12 + 8*a^8*b^2*c^2* 
d^14 + a^9*b*d^16))*x + (a*b^8*c^15 + 10*a^2*b^7*c^13*d^2 + 39*a^3*b^6*c^1 
1*d^4 + 80*a^4*b^5*c^9*d^6 + 95*a^5*b^4*c^7*d^8 + 66*a^6*b^3*c^5*d^10 + 25 
*a^7*b^2*c^3*d^12 + 4*a^8*b*c*d^14)/(a*b^9*c^16 + 8*a^2*b^8*c^14*d^2 + 28* 
a^3*b^7*c^12*d^4 + 56*a^4*b^6*c^10*d^6 + 70*a^5*b^5*c^8*d^8 + 56*a^6*b^4*c 
^6*d^10 + 28*a^7*b^3*c^4*d^12 + 8*a^8*b^2*c^2*d^14 + a^9*b*d^16))/(b*x^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 3768, normalized size of antiderivative = 26.17 \[ \int \frac {x}{(c+d x) \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqr 
t(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)* 
sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b*c**2*d + 12*sqrt(b)*s 
qrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b 
*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 
+ b*c**2)*c - a*d**2 - 2*b*c**2))*a*b**2*c**2*d*x**2 + 6*sqrt(b)*sqrt(2*sq 
rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)*a 
tan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2 
)*c - a*d**2 - 2*b*c**2))*b**3*c**2*d*x**4 + 6*sqrt(2*sqrt(b)*sqrt(a*d**2 
+ b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/s 
qrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**3*b*c*d**3 
+ 6*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt 
(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d 
**2 - 2*b*c**2))*a**2*b**2*c**3*d + 12*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2 
)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sq 
rt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b**2*c*d**3*x**2 
+ 12*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqr 
t(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a* 
d**2 - 2*b*c**2))*a*b**3*c**3*d*x**2 + 6*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c* 
*2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt...