Optimal. Leaf size=247 \[ \frac {b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac {5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac {a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac {a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac {5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac {10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5} \]
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Rubi [A] time = 0.05, antiderivative size = 247, 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} \begin {gather*} \frac {b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac {5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac {a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac {10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5}+\frac {5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac {a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5} \end {gather*}
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 )^{5/2} \, dx &=\frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \int \left (2 a b+2 b^2 x^3\right )^5 \, dx}{\left (2 a b+2 b^2 x^3\right )^5}\\ &=\frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \int \left (32 a^5 b^5+160 a^4 b^6 x^3+320 a^3 b^7 x^6+320 a^2 b^8 x^9+160 a b^9 x^{12}+32 b^{10} x^{15}\right ) \, dx}{\left (2 a b+2 b^2 x^3\right )^5}\\ &=\frac {a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac {5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac {10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5}+\frac {a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac {5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac {b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 0.33 \begin {gather*} \frac {x \sqrt {\left (a+b x^3\right )^2} \left (1456 a^5+1820 a^4 b x^3+2080 a^3 b^2 x^6+1456 a^2 b^3 x^9+560 a b^4 x^{12}+91 b^5 x^{15}\right )}{1456 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 12.92, size = 81, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (1456 a^5 x+1820 a^4 b x^4+2080 a^3 b^2 x^7+1456 a^2 b^3 x^{10}+560 a b^4 x^{13}+91 b^5 x^{16}\right )}{1456 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 53, normalized size = 0.21 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{4} \, a^{4} b x^{4} + a^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 101, normalized size = 0.41 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{13} \, a b^{4} x^{13} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{2} b^{3} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{7} \, a^{3} b^{2} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{4} \, a^{4} b x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{5} x \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 78, normalized size = 0.32 \begin {gather*} \frac {\left (91 b^{5} x^{15}+560 a \,b^{4} x^{12}+1456 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1820 a^{4} b \,x^{3}+1456 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x}{1456 \left (b \,x^{3}+a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 53, normalized size = 0.21 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac {10}{7} \, a^{3} b^{2} x^{7} + \frac {5}{4} \, a^{4} b x^{4} + a^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \end {gather*}
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 \begin {gather*} \int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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