Optimal. Leaf size=210 \[ \frac{b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac{3 a b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{a^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac{a^3 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^3 B x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
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Rubi [A] time = 0.092834, 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} \[ \frac{b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac{3 a b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{a^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac{a^3 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^3 B x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int x^4 (A+B x) \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^4 \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^4+a^2 b^3 (3 A b+a B) x^5+3 a b^4 (A b+a B) x^6+b^5 (A b+3 a B) x^7+b^6 B x^8\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{a^3 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a^2 (3 A b+a B) x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac{3 a b (A b+a B) x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{b^2 (A b+3 a B) x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{b^3 B x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0325856, size = 87, normalized size = 0.41 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (180 a^2 b x (7 A+6 B x)+84 a^3 (6 A+5 B x)+135 a b^2 x^2 (8 A+7 B x)+35 b^3 x^3 (9 A+8 B x)\right )}{2520 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 92, normalized size = 0.4 \begin{align*}{\frac{{x}^{5} \left ( 280\,{b}^{3}B{x}^{4}+315\,A{b}^{3}{x}^{3}+945\,{x}^{3}Ba{b}^{2}+1080\,{x}^{2}Aa{b}^{2}+1080\,{x}^{2}B{a}^{2}b+1260\,xA{a}^{2}b+420\,{a}^{3}Bx+504\,A{a}^{3} \right ) }{2520\, \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 = 1.52969, size = 163, normalized size = 0.78 \begin{align*} \frac{1}{9} \, B b^{3} x^{9} + \frac{1}{5} \, A a^{3} x^{5} + \frac{1}{8} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + \frac{3}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \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^{4} \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 = 1.2644, size = 203, normalized size = 0.97 \begin{align*} \frac{1}{9} \, B b^{3} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{8} \, B a b^{2} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{8} \, A b^{3} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, B a^{2} b x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{7} \, A a b^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, B a^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a^{2} b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, A a^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (5 \, B a^{9} - 9 \, A a^{8} b\right )} \mathrm{sgn}\left (b x + a\right )}{2520 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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