Optimal. Leaf size=151 \[ \frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rubi [A] time = 0.0413578, antiderivative size = 151, 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} \[ \frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^3 \left (a b+b^2 x\right )^3 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^3 b^3 x^3+3 a^2 b^4 x^4+3 a b^5 x^5+b^6 x^6\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.013957, size = 55, normalized size = 0.36 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (84 a^2 b x+35 a^3+70 a b^2 x^2+20 b^3 x^3\right )}{140 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.176, size = 52, normalized size = 0.3 \begin{align*}{\frac{{x}^{4} \left ( 20\,{b}^{3}{x}^{3}+70\,a{b}^{2}{x}^{2}+84\,{a}^{2}bx+35\,{a}^{3} \right ) }{140\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 2.02851, size = 80, normalized size = 0.53 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25165, size = 99, normalized size = 0.66 \begin{align*} \frac{1}{7} \, b^{3} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a b^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, a^{2} b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{7} \mathrm{sgn}\left (b x + a\right )}{140 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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