Integrand size = 15, antiderivative size = 116 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=-\frac {35 b^2}{24 a^3 \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {35 b^2}{8 a^4 \sqrt {a+\frac {b}{x^2}}}-\frac {7 b x^2}{8 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^4}{4 a \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}} \] Output:
-35/24*b^2/a^3/(a+b/x^2)^(3/2)-35/8*b^2/a^4/(a+b/x^2)^(1/2)-7/8*b*x^2/a^2/ (a+b/x^2)^(3/2)+1/4*x^4/a/(a+b/x^2)^(3/2)+35/8*b^2*arctanh((a+b/x^2)^(1/2) /a^(1/2))/a^(9/2)
Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {\sqrt {a} x \left (-105 b^3-140 a b^2 x^2-21 a^2 b x^4+6 a^3 x^6\right )+210 b^2 \left (b+a x^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{-\sqrt {b}+\sqrt {b+a x^2}}\right )}{24 a^{9/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \] Input:
Integrate[x^3/(a + b/x^2)^(5/2),x]
Output:
(Sqrt[a]*x*(-105*b^3 - 140*a*b^2*x^2 - 21*a^2*b*x^4 + 6*a^3*x^6) + 210*b^2 *(b + a*x^2)^(3/2)*ArcTanh[(Sqrt[a]*x)/(-Sqrt[b] + Sqrt[b + a*x^2])])/(24* a^(9/2)*Sqrt[a + b/x^2]*x*(b + a*x^2))
Time = 0.36 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 52, 52, 61, 61, 73, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -\frac {1}{2} \int \frac {x^6}{\left (a+\frac {b}{x^2}\right )^{5/2}}d\frac {1}{x^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \int \frac {x^4}{\left (a+\frac {b}{x^2}\right )^{5/2}}d\frac {1}{x^2}}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \left (-\frac {5 b \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{5/2}}d\frac {1}{x^2}}{2 a}-\frac {x^2}{a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \left (-\frac {5 b \left (\frac {\int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^{3/2}}d\frac {1}{x^2}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{2 a}-\frac {x^2}{a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \left (-\frac {5 b \left (\frac {\frac {\int \frac {x^2}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x^2}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x^2}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{2 a}-\frac {x^2}{a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {1}{b x^4}-\frac {a}{b}}d\sqrt {a+\frac {b}{x^2}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x^2}}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{2 a}-\frac {x^2}{a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+\frac {b}{x^2}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{2 a}-\frac {x^2}{a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )}{4 a}+\frac {x^4}{2 a \left (a+\frac {b}{x^2}\right )^{3/2}}\right )\) |
Input:
Int[x^3/(a + b/x^2)^(5/2),x]
Output:
(x^4/(2*a*(a + b/x^2)^(3/2)) + (7*b*(-(x^2/(a*(a + b/x^2)^(3/2))) - (5*b*( 2/(3*a*(a + b/x^2)^(3/2)) + (2/(a*Sqrt[a + b/x^2]) - (2*ArcTanh[Sqrt[a + b /x^2]/Sqrt[a]])/a^(3/2))/a))/(2*a)))/(4*a))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {\left (a \,x^{2}+b \right ) \left (6 x^{7} a^{\frac {9}{2}}-21 a^{\frac {7}{2}} b \,x^{5}-140 a^{\frac {5}{2}} b^{2} x^{3}+105 \left (a \,x^{2}+b \right )^{\frac {3}{2}} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right ) a \,b^{2}-105 a^{\frac {3}{2}} b^{3} x \right )}{24 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} a^{\frac {11}{2}}}\) | \(98\) |
| risch | \(\frac {\left (2 a \,x^{2}-11 b \right ) \left (a \,x^{2}+b \right )}{8 a^{4} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}+\frac {\left (\frac {35 b^{2} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )}{8 a^{\frac {9}{2}}}+\frac {b^{3} \sqrt {a \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{12 a^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}}-\frac {5 b^{2} \sqrt {a \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{3 a^{5} \left (x -\frac {\sqrt {-a b}}{a}\right )}-\frac {b^{3} \sqrt {a \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{12 a^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}}-\frac {5 b^{2} \sqrt {a \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{3 a^{5} \left (x +\frac {\sqrt {-a b}}{a}\right )}\right ) \sqrt {a \,x^{2}+b}}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\) | \(350\) |
Input:
int(x^3/(a+b/x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/24*(a*x^2+b)*(6*x^7*a^(9/2)-21*a^(7/2)*b*x^5-140*a^(5/2)*b^2*x^3+105*(a* x^2+b)^(3/2)*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*a*b^2-105*a^(3/2)*b^3*x)/((a*x^ 2+b)/x^2)^(5/2)/x^5/a^(11/2)
Time = 0.17 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.47 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\left [\frac {105 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{48 \, {\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}, -\frac {105 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{24 \, {\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}\right ] \] Input:
integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="fricas")
Output:
[1/48*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(a)*log(-2*a*x^2 - 2*sqrt (a)*x^2*sqrt((a*x^2 + b)/x^2) - b) + 2*(6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2 *b^2*x^4 - 105*a*b^3*x^2)*sqrt((a*x^2 + b)/x^2))/(a^7*x^4 + 2*a^6*b*x^2 + a^5*b^2), -1/24*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(-a)*arctan(sqr t(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) - (6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2*b^2*x^4 - 105*a*b^3*x^2)*sqrt((a*x^2 + b)/x^2))/(a^7*x^4 + 2*a^6* b*x^2 + a^5*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (107) = 214\).
Time = 7.42 (sec) , antiderivative size = 432, normalized size of antiderivative = 3.72 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {6 a^{\frac {89}{2}} b^{75} x^{7}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {21 a^{\frac {87}{2}} b^{76} x^{5}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {140 a^{\frac {85}{2}} b^{77} x^{3}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {105 a^{\frac {83}{2}} b^{78} x}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {105 a^{42} b^{\frac {155}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {105 a^{41} b^{\frac {157}{2}} \sqrt {\frac {a x^{2}}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} \] Input:
integrate(x**3/(a+b/x**2)**(5/2),x)
Output:
6*a**(89/2)*b**75*x**7/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) - 21*a**(87/2)*b**76*x**5/(24* a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqr t(a*x**2/b + 1)) - 140*a**(85/2)*b**77*x**3/(24*a**(93/2)*b**(151/2)*x**2* sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) - 105*a** (83/2)*b**78*x/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(9 1/2)*b**(153/2)*sqrt(a*x**2/b + 1)) + 105*a**42*b**(155/2)*x**2*sqrt(a*x** 2/b + 1)*asinh(sqrt(a)*x/sqrt(b))/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x** 2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) + 105*a**41*b**(157 /2)*sqrt(a*x**2/b + 1)*asinh(sqrt(a)*x/sqrt(b))/(24*a**(93/2)*b**(151/2)*x **2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1))
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=-\frac {105 \, {\left (a + \frac {b}{x^{2}}\right )}^{3} b^{2} - 175 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} a b^{2} + 56 \, {\left (a + \frac {b}{x^{2}}\right )} a^{2} b^{2} + 8 \, a^{3} b^{2}}{24 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}} a^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{5} + {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{6}\right )}} - \frac {35 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, a^{\frac {9}{2}}} \] Input:
integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="maxima")
Output:
-1/24*(105*(a + b/x^2)^3*b^2 - 175*(a + b/x^2)^2*a*b^2 + 56*(a + b/x^2)*a^ 2*b^2 + 8*a^3*b^2)/((a + b/x^2)^(7/2)*a^4 - 2*(a + b/x^2)^(5/2)*a^5 + (a + b/x^2)^(3/2)*a^6) - 35/16*b^2*log((sqrt(a + b/x^2) - sqrt(a))/(sqrt(a + b /x^2) + sqrt(a)))/a^(9/2)
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, x^{2} {\left (\frac {2 \, x^{2}}{a \mathrm {sgn}\left (x\right )} - \frac {7 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} - \frac {140 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} x^{2} - \frac {105 \, b^{3}}{a^{4} \mathrm {sgn}\left (x\right )}\right )} x}{24 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}}} + \frac {35 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {9}{2}}} - \frac {35 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{8 \, a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="giac")
Output:
1/24*((3*x^2*(2*x^2/(a*sgn(x)) - 7*b/(a^2*sgn(x))) - 140*b^2/(a^3*sgn(x))) *x^2 - 105*b^3/(a^4*sgn(x)))*x/(a*x^2 + b)^(3/2) + 35/16*b^2*log(abs(b))*s gn(x)/a^(9/2) - 35/8*b^2*log(abs(-sqrt(a)*x + sqrt(a*x^2 + b)))/(a^(9/2)*s gn(x))
Time = 1.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,a^{9/2}}-\frac {35\,b^2}{6\,a^3\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}+\frac {x^4}{4\,a\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}-\frac {7\,b\,x^2}{8\,a^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}-\frac {35\,b^3}{8\,a^4\,x^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \] Input:
int(x^3/(a + b/x^2)^(5/2),x)
Output:
(35*b^2*atanh((a + b/x^2)^(1/2)/a^(1/2)))/(8*a^(9/2)) - (35*b^2)/(6*a^3*(a + b/x^2)^(3/2)) + x^4/(4*a*(a + b/x^2)^(3/2)) - (7*b*x^2)/(8*a^2*(a + b/x ^2)^(3/2)) - (35*b^3)/(8*a^4*x^2*(a + b/x^2)^(3/2))
Time = 0.23 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.83 \[ \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx=\frac {48 \sqrt {a \,x^{2}+b}\, a^{4} x^{7}-168 \sqrt {a \,x^{2}+b}\, a^{3} b \,x^{5}-1120 \sqrt {a \,x^{2}+b}\, a^{2} b^{2} x^{3}-840 \sqrt {a \,x^{2}+b}\, a \,b^{3} x +840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) a^{2} b^{2} x^{4}+1680 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) a \,b^{3} x^{2}+840 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x}{\sqrt {b}}\right ) b^{4}+175 \sqrt {a}\, a^{2} b^{2} x^{4}+350 \sqrt {a}\, a \,b^{3} x^{2}+175 \sqrt {a}\, b^{4}}{192 a^{5} \left (a^{2} x^{4}+2 a b \,x^{2}+b^{2}\right )} \] Input:
int(x^3/(a+b/x^2)^(5/2),x)
Output:
(48*sqrt(a*x**2 + b)*a**4*x**7 - 168*sqrt(a*x**2 + b)*a**3*b*x**5 - 1120*s qrt(a*x**2 + b)*a**2*b**2*x**3 - 840*sqrt(a*x**2 + b)*a*b**3*x + 840*sqrt( a)*log((sqrt(a*x**2 + b) + sqrt(a)*x)/sqrt(b))*a**2*b**2*x**4 + 1680*sqrt( a)*log((sqrt(a*x**2 + b) + sqrt(a)*x)/sqrt(b))*a*b**3*x**2 + 840*sqrt(a)*l og((sqrt(a*x**2 + b) + sqrt(a)*x)/sqrt(b))*b**4 + 175*sqrt(a)*a**2*b**2*x* *4 + 350*sqrt(a)*a*b**3*x**2 + 175*sqrt(a)*b**4)/(192*a**5*(a**2*x**4 + 2* a*b*x**2 + b**2))