Optimal. Leaf size=84 \[ \frac{b \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \]
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Rubi [A] time = 0.0400933, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1355, 266, 45, 37} \[ \frac{b \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{22}} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{\left (a b+b^2 x^3\right )^5}{x^{22}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^5}{x^8} \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}}-\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^5}{x^7} \, dx,x,x^3\right )}{21 a b^3 \left (a b+b^2 x^3\right )}\\ &=-\frac{\left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 a x^{21}}+\frac{b \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{126 a^2 x^{18}}\\ \end{align*}
Mathematica [A] time = 0.0167755, size = 83, normalized size = 0.99 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (105 a^2 b^3 x^9+84 a^3 b^2 x^6+35 a^4 b x^3+6 a^5+70 a b^4 x^{12}+21 b^5 x^{15}\right )}{126 x^{21} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 80, normalized size = 1. \begin{align*} -{\frac{21\,{b}^{5}{x}^{15}+70\,a{b}^{4}{x}^{12}+105\,{a}^{2}{b}^{3}{x}^{9}+84\,{a}^{3}{b}^{2}{x}^{6}+35\,{a}^{4}b{x}^{3}+6\,{a}^{5}}{126\,{x}^{21} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+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 = 1.73008, size = 136, normalized size = 1.62 \begin{align*} -\frac{21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \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 \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{22}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11332, size = 144, normalized size = 1.71 \begin{align*} -\frac{21 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 70 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 105 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 84 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 35 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 6 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{126 \, x^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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