Optimal. Leaf size=113 \[ \frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {5 A b-2 a B}{2 a^3 \sqrt {a+b x^2}}+\frac {2 a B-5 A b}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ -\frac {5 A b-2 a B}{2 a^3 \sqrt {a+b x^2}}-\frac {5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^3 \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}}+\frac {\left (-\frac {5 A b}{2}+a B\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}}-\frac {(5 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac {5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 A b-2 a B}{2 a^3 \sqrt {a+b x^2}}-\frac {(5 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^3}\\ &=-\frac {5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 A b-2 a B}{2 a^3 \sqrt {a+b x^2}}-\frac {(5 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^3 b}\\ &=-\frac {5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{2 a x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 A b-2 a B}{2 a^3 \sqrt {a+b x^2}}+\frac {(5 A b-2 a 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 = 0.04, size = 57, normalized size = 0.50 \[ \frac {x^2 (2 a B-5 A b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b x^2}{a}+1\right )-3 a A}{6 a^2 x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 349, normalized size = 3.09 \[ \left [-\frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} + 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} + 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 101, normalized size = 0.89 \[ \frac {{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{3}} + \frac {3 \, {\left (b x^{2} + a\right )} B a + B a^{2} - 6 \, {\left (b x^{2} + a\right )} A b - A a b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x^{2} + a} A}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 140, normalized size = 1.24 \[ -\frac {5 A b}{6 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}+\frac {B}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a}+\frac {5 A b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}-\frac {B \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {5}{2}}}-\frac {5 A b}{2 \sqrt {b \,x^{2}+a}\, a^{3}}+\frac {B}{\sqrt {b \,x^{2}+a}\, a^{2}}-\frac {A}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 117, normalized size = 1.04 \[ -\frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {5 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{2}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {5 \, A b}{2 \, \sqrt {b x^{2} + a} a^{3}} - \frac {5 \, A b}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {A}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 126, normalized size = 1.12 \[ \frac {\frac {B}{3\,a}+\frac {B\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {10\,A\,b}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {A}{2\,a\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {5\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}-\frac {5\,A\,b^2\,x^2}{2\,a^3\,{\left (b\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 68.64, size = 1608, normalized size = 14.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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