Optimal. Leaf size=76 \[ -a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+a A \sqrt {a+b x^2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 76, 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, 80, 50, 63, 208} \[ -a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+a A \sqrt {a+b x^2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} (a A) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {1}{2} \left (a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}+\frac {\left (a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=a A \sqrt {a+b x^2}+\frac {1}{3} A \left (a+b x^2\right )^{3/2}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b}-a^{3/2} A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 1.00 \[ \frac {1}{3} A \left (a+b x^2\right )^{3/2}+a A \left (\sqrt {a+b x^2}-\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {B \left (a+b x^2\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 170, normalized size = 2.24 \[ \left [\frac {15 \, A a^{\frac {3}{2}} b \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b + {\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{30 \, b}, \frac {15 \, A \sqrt {-a} a b \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, B b^{2} x^{4} + 3 \, B a^{2} + 20 \, A a b + {\left (6 \, B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 79, normalized size = 1.04 \[ \frac {A a^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{4} + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{5} + 15 \, \sqrt {b x^{2} + a} A a b^{5}}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.92 \[ -A \,a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\sqrt {b \,x^{2}+a}\, A a +\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{3}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 58, normalized size = 0.76 \[ -A a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A + \sqrt {b x^{2} + a} A a + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 60, normalized size = 0.79 \[ \frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{3}+\frac {B\,{\left (b\,x^2+a\right )}^{5/2}}{5\,b}-A\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,a\,\sqrt {b\,x^2+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 61.94, size = 71, normalized size = 0.93 \[ \frac {A a^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b x^{2}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + A a \sqrt {a + b x^{2}} + \frac {A \left (a + b x^{2}\right )^{\frac {3}{2}}}{3} + \frac {B \left (a + b x^{2}\right )^{\frac {5}{2}}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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