Optimal. Leaf size=81 \[ \frac {\sqrt {a+b x^2} (4 A+3 B x)}{2 b^2}-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}-\frac {3 a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 780, 217, 206} \[ \frac {\sqrt {a+b x^2} (4 A+3 B x)}{2 b^2}-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}-\frac {3 a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rule 819
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {\int \frac {x (2 a A+3 a B x)}{\sqrt {a+b x^2}} \, dx}{a b}\\ &=-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {(3 a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^2}\\ &=-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {(3 a B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^2}\\ &=-\frac {x^2 (A+B x)}{b \sqrt {a+b x^2}}+\frac {(4 A+3 B x) \sqrt {a+b x^2}}{2 b^2}-\frac {3 a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 72, normalized size = 0.89 \[ \frac {a (4 A+3 B x)+b x^2 (2 A+B x)}{2 b^2 \sqrt {a+b x^2}}-\frac {3 a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 197, normalized size = 2.43 \[ \left [\frac {3 \, {\left (B a b x^{2} + B a^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b^{2} x^{3} + 2 \, A b^{2} x^{2} + 3 \, B a b x + 4 \, A a b\right )} \sqrt {b x^{2} + a}}{4 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {3 \, {\left (B a b x^{2} + B a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (B b^{2} x^{3} + 2 \, A b^{2} x^{2} + 3 \, B a b x + 4 \, A a b\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 70, normalized size = 0.86 \[ \frac {{\left ({\left (\frac {B x}{b} + \frac {2 \, A}{b}\right )} x + \frac {3 \, B a}{b^{2}}\right )} x + \frac {4 \, A a}{b^{2}}}{2 \, \sqrt {b x^{2} + a}} + \frac {3 \, B a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 93, normalized size = 1.15 \[ \frac {B \,x^{3}}{2 \sqrt {b \,x^{2}+a}\, b}+\frac {A \,x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {3 B a x}{2 \sqrt {b \,x^{2}+a}\, b^{2}}-\frac {3 B a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {5}{2}}}+\frac {2 A a}{\sqrt {b \,x^{2}+a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 85, normalized size = 1.05 \[ \frac {B x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {A x^{2}}{\sqrt {b x^{2} + a} b} + \frac {3 \, B a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {3 \, B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {2 \, A a}{\sqrt {b x^{2} + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.34, size = 117, normalized size = 1.44 \[ A \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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