Optimal. Leaf size=248 \[ \frac {b^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {a b^4 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
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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 {b^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {a b^4 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \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 \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^8} \, 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^8} \, 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 (\frac {a^5 b^5}{x^8}+\frac {5 a^4 b^6}{x^5}+\frac {10 a^3 b^7}{x^2}+10 a^2 b^8 x+5 a b^9 x^4+b^{10} x^7\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}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {a b^4 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {b^5 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \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 {\sqrt {\left (a+b x^3\right )^2} \left (-8 a^5-70 a^4 b x^3-560 a^3 b^2 x^6+280 a^2 b^3 x^9+56 a b^4 x^{12}+7 b^5 x^{15}\right )}{56 x^7 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 18.57, size = 83, normalized size = 0.33 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (-8 a^5-70 a^4 b x^3-560 a^3 b^2 x^6+280 a^2 b^3 x^9+56 a b^4 x^{12}+7 b^5 x^{15}\right )}{56 x^7 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 59, normalized size = 0.24 \begin {gather*} \frac {7 \, b^{5} x^{15} + 56 \, a b^{4} x^{12} + 280 \, a^{2} b^{3} x^{9} - 560 \, a^{3} b^{2} x^{6} - 70 \, a^{4} b x^{3} - 8 \, a^{5}}{56 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 107, normalized size = 0.43 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + a b^{4} x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {280 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 35 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{28 \, x^{7}} \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 (-7 b^{5} x^{15}-56 a \,b^{4} x^{12}-280 a^{2} b^{3} x^{9}+560 a^{3} b^{2} x^{6}+70 a^{4} b \,x^{3}+8 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{56 \left (b \,x^{3}+a \right )^{5} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 59, normalized size = 0.24 \begin {gather*} \frac {7 \, b^{5} x^{15} + 56 \, a b^{4} x^{12} + 280 \, a^{2} b^{3} x^{9} - 560 \, a^{3} b^{2} x^{6} - 70 \, a^{4} b x^{3} - 8 \, a^{5}}{56 \, x^{7}} \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^8} \,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^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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