Optimal. Leaf size=78 \[ \frac {a}{12 b^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {1}{9 b^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 43} \[ \frac {a}{12 b^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {1}{9 b^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac {x^5}{\left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a b+b^2 x\right )^5} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \left (-\frac {a}{b^6 (a+b x)^5}+\frac {1}{b^6 (a+b x)^4}\right ) \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {a}{12 b^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {1}{9 b^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.50 \[ \frac {-a-4 b x^3}{36 b^2 \left (a+b x^3\right )^3 \sqrt {\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 58, normalized size = 0.74 \[ -\frac {4 \, b x^{3} + a}{36 \, {\left (b^{6} x^{12} + 4 \, a b^{5} x^{9} + 6 \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.41 \[ -\frac {\left (b \,x^{3}+a \right ) \left (4 b \,x^{3}+a \right )}{36 \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 43, normalized size = 0.55 \[ -\frac {1}{9 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {a}{12 \, {\left (x^{3} + \frac {a}{b}\right )}^{4} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 42, normalized size = 0.54 \[ -\frac {\left (4\,b\,x^3+a\right )\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{36\,b^2\,{\left (b\,x^3+a\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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