Optimal. Leaf size=114 \[ \frac{x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac{a A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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Rubi [A] time = 0.0670668, 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} \[ \frac{x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac{a A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^4 (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^4 \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^4+b (A b+a B) x^5+b^2 B x^6\right ) \, dx}{a b+b^2 x}\\ &=\frac{a A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{(A b+a B) x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{b B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0216679, size = 49, normalized size = 0.43 \[ \frac{x^5 \sqrt{(a+b x)^2} (7 a (6 A+5 B x)+5 b x (7 A+6 B x))}{210 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 0.4 \begin{align*}{\frac{{x}^{5} \left ( 30\,Bb{x}^{2}+35\,Abx+35\,aBx+42\,aA \right ) }{210\,bx+210\,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 = 1.34436, size = 66, normalized size = 0.58 \begin{align*} \frac{1}{7} \, B b x^{7} + \frac{1}{5} \, A a x^{5} + \frac{1}{6} \,{\left (B a + A b\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.116205, size = 29, normalized size = 0.25 \begin{align*} \frac{A a x^{5}}{5} + \frac{B b x^{7}}{7} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23265, size = 105, normalized size = 0.92 \begin{align*} \frac{1}{7} \, B b x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, B a x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A a x^{5} \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (5 \, B a^{7} - 7 \, A a^{6} b\right )} \mathrm{sgn}\left (b x + a\right )}{210 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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