Optimal. Leaf size=121 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (A b-2 a B)}{5 b^3}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B)}{4 b^3}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]
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Rubi [A] time = 0.068253, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {770, 76} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (A b-2 a B)}{5 b^3}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B)}{4 b^3}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x (A+B x) \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 \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a (-A b+a B) \left (a b+b^2 x\right )^3}{b^2}+\frac{(A b-2 a B) \left (a b+b^2 x\right )^4}{b^3}+\frac{B \left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{a (A b-a B) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^3}+\frac{(A b-2 a B) (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0291283, size = 87, normalized size = 0.72 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (15 a^2 b x (4 A+3 B x)+10 a^3 (3 A+2 B x)+9 a b^2 x^2 (5 A+4 B x)+2 b^3 x^3 (6 A+5 B x)\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 92, normalized size = 0.8 \begin{align*}{\frac{{x}^{2} \left ( 10\,{b}^{3}B{x}^{4}+12\,A{b}^{3}{x}^{3}+36\,{x}^{3}Ba{b}^{2}+45\,{x}^{2}Aa{b}^{2}+45\,{x}^{2}B{a}^{2}b+60\,xA{a}^{2}b+20\,{a}^{3}Bx+30\,A{a}^{3} \right ) }{60\, \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 = 1.49734, size = 163, normalized size = 1.35 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} + \frac{1}{2} \, A a^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \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 \left (A + B x\right ) \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 = 1.16915, size = 200, normalized size = 1.65 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, B a b^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A b^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, B a^{2} b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, A a b^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + A a^{2} b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{6} - 3 \, A a^{5} b\right )} \mathrm{sgn}\left (b x + a\right )}{60 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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