Optimal. Leaf size=181 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5} \]
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Rubi [A] time = 0.051446, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {645} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5}-\frac{4 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^5}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{4 b^5}-\frac{4 a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^5}+\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^5} \]
Antiderivative was successfully verified.
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Rule 645
Rubi steps
\begin{align*} \int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^4 \left (a b+b^2 x\right )^5}{b^4}-\frac{4 a^3 \left (a b+b^2 x\right )^6}{b^5}+\frac{6 a^2 \left (a b+b^2 x\right )^7}{b^6}-\frac{4 a \left (a b+b^2 x\right )^8}{b^7}+\frac{\left (a b+b^2 x\right )^9}{b^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{a^4 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^5}-\frac{4 a^3 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^5}+\frac{3 a^2 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^5}-\frac{4 a (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^5}+\frac{(a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{10 b^5}\\ \end{align*}
Mathematica [A] time = 0.0186447, size = 77, normalized size = 0.43 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (1800 a^3 b^2 x^2+1575 a^2 b^3 x^3+1050 a^4 b x+252 a^5+700 a b^4 x^4+126 b^5 x^5\right )}{1260 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 74, normalized size = 0.4 \begin{align*}{\frac{{x}^{5} \left ( 126\,{b}^{5}{x}^{5}+700\,a{b}^{4}{x}^{4}+1575\,{a}^{2}{b}^{3}{x}^{3}+1800\,{a}^{3}{b}^{2}{x}^{2}+1050\,{a}^{4}bx+252\,{a}^{5} \right ) }{1260\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 = 1.65244, size = 132, normalized size = 0.73 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} 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^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3708, size = 144, normalized size = 0.8 \begin{align*} \frac{1}{10} \, b^{5} x^{10} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{9} \, a b^{4} x^{9} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, a^{2} b^{3} x^{8} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, a^{3} b^{2} x^{7} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, a^{4} b x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, a^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{a^{10} \mathrm{sgn}\left (b x + a\right )}{1260 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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