Optimal. Leaf size=138 \[ -\frac {105 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}}+\frac {105 b^3}{8 a^5 \sqrt {a+\frac {b}{x}}}+\frac {35 b^3}{8 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {21 b^2 x}{8 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {3 b x^2}{4 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 134, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac {105 b^2 x \sqrt {a+\frac {b}{x}}}{8 a^5}-\frac {105 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}}-\frac {35 b x^2 \sqrt {a+\frac {b}{x}}}{4 a^4}+\frac {7 x^3 \sqrt {a+\frac {b}{x}}}{a^3}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {21 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{a^3}+\frac {(35 b) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3}\\ &=-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{4 a^4}-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{a^3}-\frac {\left (105 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{8 a^4}\\ &=\frac {105 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^5}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{4 a^4}-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{a^3}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{16 a^5}\\ &=\frac {105 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^5}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{4 a^4}-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{a^3}+\frac {\left (105 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{8 a^5}\\ &=\frac {105 b^2 \sqrt {a+\frac {b}{x}} x}{8 a^5}-\frac {35 b \sqrt {a+\frac {b}{x}} x^2}{4 a^4}-\frac {2 x^3}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {6 x^3}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {7 \sqrt {a+\frac {b}{x}} x^3}{a^3}-\frac {105 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 39, normalized size = 0.28 \[ \frac {2 b^3 \, _2F_1\left (-\frac {3}{2},4;-\frac {1}{2};\frac {b}{a x}+1\right )}{3 a^4 \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 273, normalized size = 1.98 \[ \left [\frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 134, normalized size = 0.97 \[ \frac {1}{24} \, b^{3} {\left (\frac {315 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{5}} + \frac {16 \, a^{4} + \frac {144 \, {\left (a x + b\right )} a^{3}}{x} - \frac {693 \, {\left (a x + b\right )}^{2} a^{2}}{x^{2}} + \frac {840 \, {\left (a x + b\right )}^{3} a}{x^{3}} - \frac {315 \, {\left (a x + b\right )}^{4}}{x^{4}}}{{\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}^{3} a^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 616, normalized size = 4.46 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-336 a^{4} b^{3} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+21 a^{4} b^{3} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-84 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b \,x^{4}-1008 a^{3} b^{4} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{3} b^{4} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-294 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{2} x^{3}+672 \sqrt {\left (a x +b \right ) x}\, a^{\frac {9}{2}} b^{2} x^{3}-1008 a^{2} b^{5} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{2} b^{5} x \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{3}-378 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{3} x^{2}+2016 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b^{3} x^{2}-336 a \,b^{6} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+21 a \,b^{6} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{2}-210 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{4} x +2016 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{4} x +48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x -384 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x -42 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{5}+672 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{5}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}-352 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}\right ) x}{48 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{3} a^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 171, normalized size = 1.24 \[ \frac {315 \, {\left (a + \frac {b}{x}\right )}^{4} b^{3} - 840 \, {\left (a + \frac {b}{x}\right )}^{3} a b^{3} + 693 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b^{3} - 144 \, {\left (a + \frac {b}{x}\right )} a^{3} b^{3} - 16 \, a^{4} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{5} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{6} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{7} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{8}\right )}} + \frac {105 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 113, normalized size = 0.82 \[ \frac {35\,b^3}{2\,a^4\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{11/2}}+\frac {x^3}{3\,a\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {3\,b\,x^2}{4\,a^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {21\,b^2\,x}{8\,a^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {105\,b^4}{8\,a^5\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.64, size = 532, normalized size = 3.86 \[ \frac {8 a^{\frac {133}{2}} b^{128} x^{72}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {18 a^{\frac {131}{2}} b^{129} x^{71}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {63 a^{\frac {129}{2}} b^{130} x^{70}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {420 a^{\frac {127}{2}} b^{131} x^{69}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {315 a^{\frac {125}{2}} b^{132} x^{68}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{63} b^{\frac {263}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{62} b^{\frac {265}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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