Optimal. Leaf size=71 \[ \frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rubi [A] time = 0.0235502, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 43} \[ \frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^3 \left (a b+b^2 x\right ) \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b x^3+b^2 x^4\right ) \, dx}{a b+b^2 x}\\ &=\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0087176, size = 33, normalized size = 0.46 \[ \frac{x^4 \sqrt{(a+b x)^2} (5 a+4 b x)}{20 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 30, normalized size = 0.4 \begin{align*}{\frac{{x}^{4} \left ( 4\,bx+5\,a \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86497, size = 31, normalized size = 0.44 \begin{align*} \frac{1}{5} \, b x^{5} + \frac{1}{4} \, a x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.208986, size = 12, normalized size = 0.17 \begin{align*} \frac{a x^{4}}{4} + \frac{b x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25429, size = 53, normalized size = 0.75 \begin{align*} \frac{1}{5} \, b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x + a\right )}{20 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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