Optimal. Leaf size=88 \[ \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {5 b}{2 a^3 \sqrt {a+b x^2}}-\frac {5 b}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {1}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac {5 \sqrt {a+b x^2}}{2 a^3 x^2}+\frac {5}{3 a^2 x^2 \sqrt {a+b x^2}}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}+\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{6 a}\\ &=\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {5}{3 a^2 x^2 \sqrt {a+b x^2}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {5}{3 a^2 x^2 \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 x^2}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^3}\\ &=\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {5}{3 a^2 x^2 \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 x^2}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^3}\\ &=\frac {1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {5}{3 a^2 x^2 \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 x^2}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.42 \[ -\frac {b \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b x^2}{a}+1\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 241, normalized size = 2.74 \[ \left [\frac {15 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (15 \, a b^{2} x^{4} + 20 \, a^{2} b x^{2} + 3 \, a^{3}\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}, -\frac {15 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, a b^{2} x^{4} + 20 \, a^{2} b x^{2} + 3 \, a^{3}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 73, normalized size = 0.83 \[ -\frac {5 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{3}} - \frac {6 \, {\left (b x^{2} + a\right )} b + a b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x^{2} + a}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 0.89 \[ -\frac {5 b}{6 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}+\frac {5 b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}-\frac {5 b}{2 \sqrt {b \,x^{2}+a}\, a^{3}}-\frac {1}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 66, normalized size = 0.75 \[ \frac {5 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} - \frac {5 \, b}{2 \, \sqrt {b x^{2} + a} a^{3}} - \frac {5 \, b}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {1}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 73, normalized size = 0.83 \[ \frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}-\frac {1}{2\,a\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {10\,b}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {5\,b^2\,x^2}{2\,a^3\,{\left (b\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.17, size = 864, normalized size = 9.82 \[ - \frac {6 a^{17} \sqrt {1 + \frac {b x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {46 a^{16} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {15 a^{16} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} + \frac {30 a^{16} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {70 a^{15} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {45 a^{15} b^{2} x^{4} \log {\left (\frac {b x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} + \frac {90 a^{15} b^{2} x^{4} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {30 a^{14} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {45 a^{14} b^{3} x^{6} \log {\left (\frac {b x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} + \frac {90 a^{14} b^{3} x^{6} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} - \frac {15 a^{13} b^{4} x^{8} \log {\left (\frac {b x^{2}}{a} \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} + \frac {30 a^{13} b^{4} x^{8} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{12 a^{\frac {39}{2}} x^{2} + 36 a^{\frac {37}{2}} b x^{4} + 36 a^{\frac {35}{2}} b^{2} x^{6} + 12 a^{\frac {33}{2}} b^{3} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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