Optimal. Leaf size=47 \[ \frac {4 \sqrt {a x^2+b x^3}}{b^2 x}-\frac {2 x^2}{b \sqrt {a x^2+b x^3}} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 1588} \begin {gather*} \frac {4 \sqrt {a x^2+b x^3}}{b^2 x}-\frac {2 x^2}{b \sqrt {a x^2+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac {2 x^2}{b \sqrt {a x^2+b x^3}}+\frac {2 \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx}{b}\\ &=-\frac {2 x^2}{b \sqrt {a x^2+b x^3}}+\frac {4 \sqrt {a x^2+b x^3}}{b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.55 \begin {gather*} \frac {2 x (2 a+b x)}{b^2 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.41, size = 37, normalized size = 0.79 \begin {gather*} \frac {2 (2 a+b x) \sqrt {a x^2+b x^3}}{b^2 x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 38, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )}}{b^{3} x^{2} + a b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 28, normalized size = 0.60 \begin {gather*} \frac {2 \, {\left (\frac {1}{b} + \frac {2 \, a}{b^{2} x}\right )}}{\sqrt {\frac {b}{x} + \frac {a}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 0.72 \begin {gather*} \frac {2 \left (b x +a \right ) \left (b x +2 a \right ) x^{3}}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.54, size = 19, normalized size = 0.40 \begin {gather*} \frac {2 \, {\left (b x + 2 \, a\right )}}{\sqrt {b x + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 35, normalized size = 0.74 \begin {gather*} \frac {2\,\left (2\,a+b\,x\right )\,\sqrt {b\,x^3+a\,x^2}}{b^2\,x\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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