Optimal. Leaf size=210 \[ \frac{b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{a b x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 (a+b x)}+\frac{a^2 x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{a^3 A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{b^3 B x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)} \]
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Rubi [A] time = 0.0882991, antiderivative size = 210, 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} \[ \frac{b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{a b x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 (a+b x)}+\frac{a^2 x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{a^3 A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{b^3 B x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^3 (A+B x) \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 (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 (a^3 A b^3 x^3+a^2 b^3 (3 A b+a B) x^4+3 a b^4 (A b+a B) x^5+b^5 (A b+3 a B) x^6+b^6 B x^7\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{a^3 A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{a^2 (3 A b+a B) x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a b (A b+a B) x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^2 (A b+3 a B) x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{b^3 B x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0319616, size = 87, normalized size = 0.41 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (28 a^2 b x (6 A+5 B x)+14 a^3 (5 A+4 B x)+20 a b^2 x^2 (7 A+6 B x)+5 b^3 x^3 (8 A+7 B x)\right )}{280 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 92, normalized size = 0.4 \begin{align*}{\frac{{x}^{4} \left ( 35\,{b}^{3}B{x}^{4}+40\,A{b}^{3}{x}^{3}+120\,{x}^{3}Ba{b}^{2}+140\,{x}^{2}Aa{b}^{2}+140\,{x}^{2}Bb{a}^{2}+168\,xAb{a}^{2}+56\,{a}^{3}Bx+70\,A{a}^{3} \right ) }{280\, \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 = 1.5447, size = 163, normalized size = 0.78 \begin{align*} \frac{1}{8} \, B b^{3} x^{8} + \frac{1}{4} \, A a^{3} x^{4} + \frac{1}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5} \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 (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 = 1.18826, size = 201, normalized size = 0.96 \begin{align*} \frac{1}{8} \, B b^{3} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, B a b^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{7} \, A b^{3} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a^{2} b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a b^{2} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, B a^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, A a^{2} b x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, A a^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{8} - 2 \, A a^{7} b\right )} \mathrm{sgn}\left (b x + a\right )}{280 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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