\(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x^5} \, dx\) [181]

Optimal result

Integrand size = 21, antiderivative size = 173 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\frac {b d^2 \sqrt {a+\frac {b}{c+d x^2}}}{c^3}+\frac {(9 b+4 a c) d \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{8 c^3 x^2}-\frac {(b+a c) \left (c+d x^2\right )^2 \sqrt {a+\frac {b}{c+d x^2}}}{4 c^3 x^4}-\frac {3 b (5 b+4 a c) d^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{8 c^{7/2} \sqrt {b+a c}} \] Output:

b*d^2*(a+b/(d*x^2+c))^(1/2)/c^3+1/8*(4*a*c+9*b)*d*(d*x^2+c)*(a+b/(d*x^2+c) 
)^(1/2)/c^3/x^2-1/4*(a*c+b)*(d*x^2+c)^2*(a+b/(d*x^2+c))^(1/2)/c^3/x^4-3/8* 
b*(4*a*c+5*b)*d^2*arctanh(c^(1/2)*(a+b/(d*x^2+c))^(1/2)/(a*c+b)^(1/2))/c^( 
7/2)/(a*c+b)^(1/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-2 b c^2-2 a c^3+5 b c d x^2+15 b d^2 x^4+2 a c d^2 x^4\right )}{8 c^3 x^4}+\frac {3 b (5 b+4 a c) d^2 \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{8 c^{7/2} \sqrt {-b-a c}} \] Input:

Integrate[(a + b/(c + d*x^2))^(3/2)/x^5,x]
 

Output:

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-2*b*c^2 - 2*a*c^3 + 5*b*c*d*x^2 + 
 15*b*d^2*x^4 + 2*a*c*d^2*x^4))/(8*c^3*x^4) + (3*b*(5*b + 4*a*c)*d^2*ArcTa 
n[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[-b - a*c]])/(8*c^(7 
/2)*Sqrt[-b - a*c])
 

Rubi [A] (warning: unable to verify)

Time = 0.77 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 2053, 2052, 25, 27, 360, 1471, 27, 299, 221}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx\)

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2053

\(\displaystyle \frac {1}{2} \int \frac {\left (\frac {a d x^2+b+a c}{d x^2+c}\right )^{3/2}}{x^6}dx^2\)

\(\Big \downarrow \) 2052

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 360

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

\(\Big \downarrow \) 1471

\(\displaystyle b d^2 \left (\frac {\frac {(4 a c+9 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {\int \frac {(b+a c) \left (8 c x^4+7 b+4 a c\right )}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{2 (a c+b)}}{4 c^3}-\frac {b (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^3 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b d^2 \left (\frac {\frac {(4 a c+9 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}-\frac {1}{2} \int \frac {8 c x^4+7 b+4 a c}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{4 c^3}-\frac {b (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^3 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 299

\(\displaystyle b d^2 \left (\frac {\frac {1}{2} \left (8 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}-3 (4 a c+5 b) \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\right )+\frac {(4 a c+9 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}}{4 c^3}-\frac {b (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^3 \left (a c+b-c x^4\right )^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle b d^2 \left (\frac {\frac {1}{2} \left (8 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}-\frac {3 (4 a c+5 b) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{\sqrt {c} \sqrt {a c+b}}\right )+\frac {(4 a c+9 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{2 \left (a c+b-c x^4\right )}}{4 c^3}-\frac {b (a c+b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 c^3 \left (a c+b-c x^4\right )^2}\right )\)

Input:

Int[(a + b/(c + d*x^2))^(3/2)/x^5,x]
 

Output:

b*d^2*(-1/4*(b*(b + a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(c^3*(b + 
a*c - c*x^4)^2) + (((9*b + 4*a*c)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/( 
2*(b + a*c - c*x^4)) + (8*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)] - (3*(5*b 
+ 4*a*c)*ArcTanh[(Sqrt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + 
a*c]])/(Sqrt[c]*Sqrt[b + a*c]))/2)/(4*c^3))
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2052

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S 
ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d)   Subst[Int[x^(q* 
(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* 
x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] 
&& IntegerQ[m]
 

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {\left (d \,x^{2}+c \right ) \left (-2 a d \,x^{2} c -7 b d \,x^{2}+2 a \,c^{2}+2 b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{8 c^{3} x^{4}}+\frac {d^{2} b \left (-\frac {\left (12 a c +15 b \right ) \ln \left (\frac {2 a \,c^{2}+2 b c +\left (2 a c d +b d \right ) x^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}}{x^{2}}\right )}{2 \sqrt {a \,c^{2}+b c}}+\frac {8 a d \,x^{2}+8 a c +8 b}{\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{8 c^{3} \left (a d \,x^{2}+a c +b \right )}\)	\(274\)
default	\(\text {Expression too large to display}\)	\(1653\)

Input:

int((a+b/(d*x^2+c))^(3/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/8*(d*x^2+c)*(-2*a*c*d*x^2-7*b*d*x^2+2*a*c^2+2*b*c)/c^3/x^4*((a*d*x^2+a* 
c+b)/(d*x^2+c))^(1/2)+1/8/c^3*d^2*b*(-1/2*(12*a*c+15*b)/(a*c^2+b*c)^(1/2)* 
ln((2*a*c^2+2*b*c+(2*a*c*d+b*d)*x^2+2*(a*c^2+b*c)^(1/2)*(a*c^2+b*c+(2*a*c* 
d+b*d)*x^2+a*d^2*x^4)^(1/2))/x^2)+8*(a*d*x^2+a*c+b)/(a*d^2*x^4+2*a*c*d*x^2 
+b*d*x^2+a*c^2+b*c)^(1/2))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*((d*x^2+c)*(a 
*d*x^2+a*c+b))^(1/2)/(a*d*x^2+a*c+b)
 

Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.22 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\left [\frac {3 \, {\left (4 \, a b c + 5 \, b^{2}\right )} \sqrt {a c^{2} + b c} d^{2} x^{4} \log \left (\frac {{\left (8 \, a^{2} c^{2} + 8 \, a b c + b^{2}\right )} d^{2} x^{4} + 8 \, a^{2} c^{4} + 16 \, a b c^{3} + 8 \, b^{2} c^{2} + 8 \, {\left (2 \, a^{2} c^{3} + 3 \, a b c^{2} + b^{2} c\right )} d x^{2} - 4 \, {\left ({\left (2 \, a c + b\right )} d^{2} x^{4} + 2 \, a c^{3} + {\left (4 \, a c^{2} + 3 \, b c\right )} d x^{2} + 2 \, b c^{2}\right )} \sqrt {a c^{2} + b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{x^{4}}\right ) - 4 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 17 \, a b c^{2} + 15 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - 5 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a c^{5} + b c^{4}\right )} x^{4}}, \frac {3 \, {\left (4 \, a b c + 5 \, b^{2}\right )} \sqrt {-a c^{2} - b c} d^{2} x^{4} \arctan \left (\frac {{\left ({\left (2 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + 2 \, b c\right )} \sqrt {-a c^{2} - b c} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} c^{3} + 2 \, a b c^{2} + {\left (a^{2} c^{2} + a b c\right )} d x^{2} + b^{2} c\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{5} - {\left (2 \, a^{2} c^{3} + 17 \, a b c^{2} + 15 \, b^{2} c\right )} d^{2} x^{4} + 4 \, a b c^{4} + 2 \, b^{2} c^{3} - 5 \, {\left (a b c^{3} + b^{2} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a c^{5} + b c^{4}\right )} x^{4}}\right ] \] Input:

integrate((a+b/(d*x^2+c))^(3/2)/x^5,x, algorithm="fricas")
 

Output:

[1/32*(3*(4*a*b*c + 5*b^2)*sqrt(a*c^2 + b*c)*d^2*x^4*log(((8*a^2*c^2 + 8*a 
*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 
3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a*c + b)*d^2*x^4 + 2*a*c^3 + (4*a*c^2 + 3 
*b*c)*d*x^2 + 2*b*c^2)*sqrt(a*c^2 + b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + 
 c)))/x^4) - 4*(2*a^2*c^5 - (2*a^2*c^3 + 17*a*b*c^2 + 15*b^2*c)*d^2*x^4 + 
4*a*b*c^4 + 2*b^2*c^3 - 5*(a*b*c^3 + b^2*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + 
 b)/(d*x^2 + c)))/((a*c^5 + b*c^4)*x^4), 1/16*(3*(4*a*b*c + 5*b^2)*sqrt(-a 
*c^2 - b*c)*d^2*x^4*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt( 
-a*c^2 - b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*c^3 + 2*a*b*c^2 + 
 (a^2*c^2 + a*b*c)*d*x^2 + b^2*c)) - 2*(2*a^2*c^5 - (2*a^2*c^3 + 17*a*b*c^ 
2 + 15*b^2*c)*d^2*x^4 + 4*a*b*c^4 + 2*b^2*c^3 - 5*(a*b*c^3 + b^2*c^2)*d*x^ 
2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/((a*c^5 + b*c^4)*x^4)]
 

Sympy [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:

integrate((a+b/(d*x**2+c))**(3/2)/x**5,x)
 

Output:

Integral(((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)/x**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (151) = 302\).

Time = 0.14 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\frac {b d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c^{3}} + \frac {3 \, {\left (4 \, a b c + 5 \, b^{2}\right )} d^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{16 \, \sqrt {{\left (a c + b\right )} c} c^{3}} - \frac {{\left (4 \, a b c^{2} + 9 \, b^{2} c\right )} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c^{2} + 11 \, a b^{2} c + 7 \, b^{3}\right )} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{2} c^{5} + 2 \, a b c^{4} + b^{2} c^{3} + \frac {{\left (a d x^{2} + a c + b\right )}^{2} c^{5}}{{\left (d x^{2} + c\right )}^{2}} - \frac {2 \, {\left (a c^{5} + b c^{4}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}\right )}} \] Input:

integrate((a+b/(d*x^2+c))^(3/2)/x^5,x, algorithm="maxima")
 

Output:

b*d^2*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/c^3 + 3/16*(4*a*b*c + 5*b^2)*d 
^2*log((c*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)) - sqrt((a*c + b)*c))/(c*sq 
rt((a*d*x^2 + a*c + b)/(d*x^2 + c)) + sqrt((a*c + b)*c)))/(sqrt((a*c + b)* 
c)*c^3) - 1/8*((4*a*b*c^2 + 9*b^2*c)*d^2*((a*d*x^2 + a*c + b)/(d*x^2 + c)) 
^(3/2) - (4*a^2*b*c^2 + 11*a*b^2*c + 7*b^3)*d^2*sqrt((a*d*x^2 + a*c + b)/( 
d*x^2 + c)))/(a^2*c^5 + 2*a*b*c^4 + b^2*c^3 + (a*d*x^2 + a*c + b)^2*c^5/(d 
*x^2 + c)^2 - 2*(a*c^5 + b*c^4)*(a*d*x^2 + a*c + b)/(d*x^2 + c))
 

Giac [F]

\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:

integrate((a+b/(d*x^2+c))^(3/2)/x^5,x, algorithm="giac")
 

Output:

undef
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^5} \,d x \] Input:

int((a + b/(c + d*x^2))^(3/2)/x^5,x)
 

Output:

int((a + b/(c + d*x^2))^(3/2)/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 640, normalized size of antiderivative = 3.70 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^5} \, dx=\frac {-2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} c^{5}+2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} c^{3} d^{2} x^{4}-4 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b \,c^{4}+5 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b \,c^{3} d \,x^{2}+17 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a b \,c^{2} d^{2} x^{4}-2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b^{2} c^{3}+5 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b^{2} c^{2} d \,x^{2}+15 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b^{2} c \,d^{2} x^{4}+12 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a b \,c^{2} d^{2} x^{4}+12 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a b c \,d^{3} x^{6}+15 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{2} c \,d^{2} x^{4}+15 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b^{2} d^{3} x^{6}-12 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a b \,c^{2} d^{2} x^{4}-12 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a b c \,d^{3} x^{6}-15 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b^{2} c \,d^{2} x^{4}-15 \sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b^{2} d^{3} x^{6}}{8 c^{4} x^{4} \left (a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )} \] Input:

int((a+b/(d*x^2+c))^(3/2)/x^5,x)
 

Output:

( - 2*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a**2*c**5 + 2*sqrt(c + d*x 
**2)*sqrt(a*c + a*d*x**2 + b)*a**2*c**3*d**2*x**4 - 4*sqrt(c + d*x**2)*sqr 
t(a*c + a*d*x**2 + b)*a*b*c**4 + 5*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + 
b)*a*b*c**3*d*x**2 + 17*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a*b*c**2 
*d**2*x**4 - 2*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b**2*c**3 + 5*sqr 
t(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b**2*c**2*d*x**2 + 15*sqrt(c + d*x* 
*2)*sqrt(a*c + a*d*x**2 + b)*b**2*c*d**2*x**4 + 12*sqrt(c)*sqrt(a*c + b)*l 
og(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2)*a*c 
 - sqrt(c)*sqrt(c + d*x**2)*b)*a*b*c**2*d**2*x**4 + 12*sqrt(c)*sqrt(a*c + 
b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2) 
*a*c - sqrt(c)*sqrt(c + d*x**2)*b)*a*b*c*d**3*x**6 + 15*sqrt(c)*sqrt(a*c + 
 b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2 
)*a*c - sqrt(c)*sqrt(c + d*x**2)*b)*b**2*c*d**2*x**4 + 15*sqrt(c)*sqrt(a*c 
 + b)*log(sqrt(a*c + b)*sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x* 
*2)*a*c - sqrt(c)*sqrt(c + d*x**2)*b)*b**2*d**3*x**6 - 12*sqrt(c)*sqrt(a*c 
 + b)*log(x)*a*b*c**2*d**2*x**4 - 12*sqrt(c)*sqrt(a*c + b)*log(x)*a*b*c*d* 
*3*x**6 - 15*sqrt(c)*sqrt(a*c + b)*log(x)*b**2*c*d**2*x**4 - 15*sqrt(c)*sq 
rt(a*c + b)*log(x)*b**2*d**3*x**6)/(8*c**4*x**4*(a*c**2 + a*c*d*x**2 + b*c 
 + b*d*x**2))