Optimal. Leaf size=72 \[ -\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}+\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \begin {gather*} \frac {A}{a^2 \sqrt {a+b x^2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}
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 \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a^2 b}\\ &=\frac {A b-a B}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {A}{a^2 \sqrt {a+b x^2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.85 \begin {gather*} \frac {a (A b-a B)+3 A b \left (a+b x^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^2}{a}+1\right )}{3 a^2 b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 69, normalized size = 0.96 \begin {gather*} \frac {a^2 (-B)+4 a A b+3 A b^2 x^2}{3 a^2 b \left (a+b x^2\right )^{3/2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 241, normalized size = 3.35 \begin {gather*} \left [\frac {3 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\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 \, A a b^{2} x^{2} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, \frac {3 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, A a b^{2} x^{2} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 66, normalized size = 0.92 \begin {gather*} \frac {A \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {B a^{2} - 3 \, {\left (b x^{2} + a\right )} A b - A a b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 1.04 \begin {gather*} \frac {A}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a}-\frac {B}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}-\frac {A \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {5}{2}}}+\frac {A}{\sqrt {b \,x^{2}+a}\, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 63, normalized size = 0.88 \begin {gather*} -\frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a^{2}} + \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 65, normalized size = 0.90 \begin {gather*} \frac {\frac {A}{3\,a}+\frac {A\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {B}{3\,b\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 42.17, size = 66, normalized size = 0.92 \begin {gather*} \frac {A}{a^{2} \sqrt {a + b x^{2}}} + \frac {A \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{a^{2} \sqrt {- a}} - \frac {- A b + B a}{3 a b \left (a + b x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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