Optimal. Leaf size=255 \[ \frac {b^5 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 270} \begin {gather*} \frac {b^5 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^7 \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x^7+5 a^4 b^6 x^{10}+10 a^3 b^7 x^{13}+10 a^2 b^8 x^{16}+5 a b^9 x^{19}+b^{10} x^{22}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {a^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {5 a^4 b x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^3 b^2 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {a b^4 x^{20} \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^5 x^{23} \sqrt {a^2+2 a b x^3+b^2 x^6}}{23 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} \frac {x^8 \sqrt {\left (a+b x^3\right )^2} \left (30107 a^5+109480 a^4 b x^3+172040 a^3 b^2 x^6+141680 a^2 b^3 x^9+60214 a b^4 x^{12}+10472 b^5 x^{15}\right )}{240856 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 13.72, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (30107 a^5 x^8+109480 a^4 b x^{11}+172040 a^3 b^2 x^{14}+141680 a^2 b^3 x^{17}+60214 a b^4 x^{20}+10472 b^5 x^{23}\right )}{240856 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{23} \, b^{5} x^{23} + \frac {1}{4} \, a b^{4} x^{20} + \frac {10}{17} \, a^{2} b^{3} x^{17} + \frac {5}{7} \, a^{3} b^{2} x^{14} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{8} \, a^{5} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 105, normalized size = 0.41 \begin {gather*} \frac {1}{23} \, b^{5} x^{23} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{4} \, a b^{4} x^{20} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{17} \, a^{2} b^{3} x^{17} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{7} \, a^{3} b^{2} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{11} \, a^{4} b x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{8} \, a^{5} x^{8} \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 = 0.01, size = 80, normalized size = 0.31 \begin {gather*} \frac {\left (10472 b^{5} x^{15}+60214 a \,b^{4} x^{12}+141680 a^{2} b^{3} x^{9}+172040 a^{3} b^{2} x^{6}+109480 a^{4} b \,x^{3}+30107 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x^{8}}{240856 \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 = 0.77, size = 57, normalized size = 0.22 \begin {gather*} \frac {1}{23} \, b^{5} x^{23} + \frac {1}{4} \, a b^{4} x^{20} + \frac {10}{17} \, a^{2} b^{3} x^{17} + \frac {5}{7} \, a^{3} b^{2} x^{14} + \frac {5}{11} \, a^{4} b x^{11} + \frac {1}{8} \, a^{5} x^{8} \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 x^7\,{\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 x^{7} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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