Optimal. Leaf size=162 \[ \frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}+\frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3} \]
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Rubi [A] time = 0.03, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1343, 194} \[ \frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}+\frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3} \]
Antiderivative was successfully verified.
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Rule 194
Rule 1343
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx &=\frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \int \left (2 a b+2 b^2 x^3\right )^3 \, dx}{\left (2 a b+2 b^2 x^3\right )^3}\\ &=\frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^3+24 a b^5 x^6+8 b^6 x^9\right ) \, dx}{\left (2 a b+2 b^2 x^3\right )^3}\\ &=\frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 0.36 \[ \frac {x \sqrt {\left (a+b x^3\right )^2} \left (140 a^3+105 a^2 b x^3+60 a b^2 x^6+14 b^3 x^9\right )}{140 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 32, normalized size = 0.20 \[ \frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 64, normalized size = 0.40 \[ \frac {1}{10} \, b^{3} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{7} \, a b^{2} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{4} \, a^{2} b x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 56, normalized size = 0.35 \[ \frac {\left (14 b^{3} x^{9}+60 a \,b^{2} x^{6}+105 a^{2} b \,x^{3}+140 a^{3}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} x}{140 \left (b \,x^{3}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 32, normalized size = 0.20 \[ \frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \]
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 \[ \int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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