Optimal. Leaf size=114 \[ \frac{x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{a A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{b B x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
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Rubi [A] time = 0.0593381, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {770, 76} \[ \frac{x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{a A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{b B x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^3 (A+B x) \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 ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a A b x^3+b (A b+a B) x^4+b^2 B x^5\right ) \, dx}{a b+b^2 x}\\ &=\frac{a A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{(A b+a B) x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b B x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0160524, size = 49, normalized size = 0.43 \[ \frac{x^4 \sqrt{(a+b x)^2} (3 a (5 A+4 B x)+2 b x (6 A+5 B x))}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 0.4 \begin{align*}{\frac{{x}^{4} \left ( 10\,Bb{x}^{2}+12\,Abx+12\,aBx+15\,aA \right ) }{60\,bx+60\,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 = 1.3018, size = 66, normalized size = 0.58 \begin{align*} \frac{1}{6} \, B b x^{6} + \frac{1}{4} \, A a x^{4} + \frac{1}{5} \,{\left (B a + A b\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09913, size = 29, normalized size = 0.25 \begin{align*} \frac{A a x^{4}}{4} + \frac{B b x^{6}}{6} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16, size = 105, normalized size = 0.92 \begin{align*} \frac{1}{6} \, B b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, B a x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, A a x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (2 \, B a^{6} - 3 \, A a^{5} b\right )} \mathrm{sgn}\left (b x + a\right )}{60 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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