Optimal. Leaf size=104 \[ \frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}-\frac {a \sqrt {a+b x^2} (16 A+9 B x)}{24 b^2}+\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {833, 780, 217, 206} \begin {gather*} \frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}-\frac {a \sqrt {a+b x^2} (16 A+9 B x)}{24 b^2}+\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rule 833
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\sqrt {a+b x^2}} \, dx &=\frac {B x^3 \sqrt {a+b x^2}}{4 b}+\frac {\int \frac {x^2 (-3 a B+4 A b x)}{\sqrt {a+b x^2}} \, dx}{4 b}\\ &=\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}+\frac {\int \frac {x (-8 a A b-9 a b B x)}{\sqrt {a+b x^2}} \, dx}{12 b^2}\\ &=\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {a (16 A+9 B x) \sqrt {a+b x^2}}{24 b^2}+\frac {\left (3 a^2 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^2}\\ &=\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {a (16 A+9 B x) \sqrt {a+b x^2}}{24 b^2}+\frac {\left (3 a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^2}\\ &=\frac {A x^2 \sqrt {a+b x^2}}{3 b}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {a (16 A+9 B x) \sqrt {a+b x^2}}{24 b^2}+\frac {3 a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 0.73 \begin {gather*} \frac {9 a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\sqrt {b} \sqrt {a+b x^2} \left (-16 a A-9 a B x+8 A b x^2+6 b B x^3\right )}{24 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 77, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-16 a A-9 a B x+8 A b x^2+6 b B x^3\right )}{24 b^2}-\frac {3 a^2 B \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{8 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 158, normalized size = 1.52 \begin {gather*} \left [\frac {9 \, B a^{2} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} - 9 \, B a b x - 16 \, A a b\right )} \sqrt {b x^{2} + a}}{48 \, b^{3}}, -\frac {9 \, B a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} - 9 \, B a b x - 16 \, A a b\right )} \sqrt {b x^{2} + a}}{24 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 74, normalized size = 0.71 \begin {gather*} \frac {1}{24} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (\frac {3 \, B x}{b} + \frac {4 \, A}{b}\right )} x - \frac {9 \, B a}{b^{2}}\right )} x - \frac {16 \, A a}{b^{2}}\right )} - \frac {3 \, B a^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 0.92 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, B \,x^{3}}{4 b}+\frac {\sqrt {b \,x^{2}+a}\, A \,x^{2}}{3 b}+\frac {3 B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}-\frac {3 \sqrt {b \,x^{2}+a}\, B a x}{8 b^{2}}-\frac {2 \sqrt {b \,x^{2}+a}\, A a}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 88, normalized size = 0.85 \begin {gather*} \frac {\sqrt {b x^{2} + a} B x^{3}}{4 \, b} + \frac {\sqrt {b x^{2} + a} A x^{2}}{3 \, b} - \frac {3 \, \sqrt {b x^{2} + a} B a x}{8 \, b^{2}} + \frac {3 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {2 \, \sqrt {b x^{2} + a} A a}{3 \, b^{2}} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (A+B\,x\right )}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.91, size = 150, normalized size = 1.44 \begin {gather*} A \left (\begin {cases} - \frac {2 a \sqrt {a + b x^{2}}}{3 b^{2}} + \frac {x^{2} \sqrt {a + b x^{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {3 B a^{\frac {3}{2}} x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B \sqrt {a} x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {B x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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