Optimal. Leaf size=248 \[ \frac {b^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 248, 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 {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{15}} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^{15}} \, 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 (b^{10}+\frac {a^5 b^5}{x^{15}}+\frac {5 a^4 b^6}{x^{12}}+\frac {10 a^3 b^7}{x^9}+\frac {10 a^2 b^8}{x^6}+\frac {5 a b^9}{x^3}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac {2 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {b^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (22 a^5+140 a^4 b x^3+385 a^3 b^2 x^6+616 a^2 b^3 x^9+770 a b^4 x^{12}-308 b^5 x^{15}\right )}{308 x^{14} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 19.62, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-22 a^5-140 a^4 b x^3-385 a^3 b^2 x^6-616 a^2 b^3 x^9-770 a b^4 x^{12}+308 b^5 x^{15}\right )}{308 x^{14} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 59, normalized size = 0.24 \begin {gather*} \frac {308 \, b^{5} x^{15} - 770 \, a b^{4} x^{12} - 616 \, a^{2} b^{3} x^{9} - 385 \, a^{3} b^{2} x^{6} - 140 \, a^{4} b x^{3} - 22 \, a^{5}}{308 \, x^{14}} \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 = 0.42 \begin {gather*} b^{5} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {770 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 616 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 385 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 140 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 22 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{308 \, x^{14}} \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 = 0.32 \begin {gather*} -\frac {\left (-308 b^{5} x^{15}+770 a \,b^{4} x^{12}+616 a^{2} b^{3} x^{9}+385 a^{3} b^{2} x^{6}+140 a^{4} b \,x^{3}+22 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{308 \left (b \,x^{3}+a \right )^{5} x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 59, normalized size = 0.24 \begin {gather*} \frac {308 \, b^{5} x^{15} - 770 \, a b^{4} x^{12} - 616 \, a^{2} b^{3} x^{9} - 385 \, a^{3} b^{2} x^{6} - 140 \, a^{4} b x^{3} - 22 \, a^{5}}{308 \, x^{14}} \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 \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^{15}} \,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 \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{15}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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