Optimal. Leaf size=160 \[ \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^8}{27 b^4}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{8 b^4}+\frac {a^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{7 b^4}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^4} \]
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Rubi [A] time = 0.12, antiderivative size = 160, 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} \begin {gather*} \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^8}{27 b^4}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{8 b^4}+\frac {a^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{7 b^4}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int x^{11} \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^{11} \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} \operatorname {Subst}\left (\int x^3 \left (a b+b^2 x\right )^5 \, dx,x,x^3\right )}{3 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 \left (-\frac {a^3 \left (a b+b^2 x\right )^5}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^6}{b^4}-\frac {3 a \left (a b+b^2 x\right )^7}{b^5}+\frac {\left (a b+b^2 x\right )^8}{b^6}\right ) \, dx,x,x^3\right )}{3 b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^3 \left (a+b x^3\right )^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{18 b^4}+\frac {a^2 \left (a+b x^3\right )^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 b^4}-\frac {a \left (a+b x^3\right )^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 b^4}+\frac {\left (a+b x^3\right )^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{27 b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.52 \begin {gather*} \frac {x^{12} \sqrt {\left (a+b x^3\right )^2} \left (126 a^5+504 a^4 b x^3+840 a^3 b^2 x^6+720 a^2 b^3 x^9+315 a b^4 x^{12}+56 b^5 x^{15}\right )}{1512 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 21.67, size = 83, normalized size = 0.52 \begin {gather*} \frac {x^{12} \sqrt {\left (a+b x^3\right )^2} \left (126 a^5+504 a^4 b x^3+840 a^3 b^2 x^6+720 a^2 b^3 x^9+315 a b^4 x^{12}+56 b^5 x^{15}\right )}{1512 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 57, normalized size = 0.36 \begin {gather*} \frac {1}{27} \, b^{5} x^{27} + \frac {5}{24} \, a b^{4} x^{24} + \frac {10}{21} \, a^{2} b^{3} x^{21} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {1}{3} \, a^{4} b x^{15} + \frac {1}{12} \, a^{5} x^{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 105, normalized size = 0.66 \begin {gather*} \frac {1}{27} \, b^{5} x^{27} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{24} \, a b^{4} x^{24} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{21} \, a^{2} b^{3} x^{21} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{9} \, a^{3} b^{2} x^{18} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{3} \, a^{4} b x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{12} \, a^{5} x^{12} \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 = 0.50 \begin {gather*} \frac {\left (56 b^{5} x^{15}+315 a \,b^{4} x^{12}+720 a^{2} b^{3} x^{9}+840 a^{3} b^{2} x^{6}+504 a^{4} b \,x^{3}+126 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} x^{12}}{1512 \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 = 0.72, size = 145, normalized size = 0.91 \begin {gather*} \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} x^{6}}{27 \, b^{2}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a^{3} x^{3}}{18 \, b^{3}} - \frac {11 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} a x^{3}}{216 \, b^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} a^{4}}{18 \, b^{4}} + \frac {83 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} a^{2}}{1512 \, b^{4}} \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 = -0.01 \begin {gather*} \int x^{11}\,{\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^{11} \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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