Optimal. Leaf size=252 \[ \frac {b^5 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {10 a^3 b^2 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 252, 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} \[ \frac {b^5 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}+\frac {10 a^3 b^2 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^7} \, 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^7} \, 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 \frac {\left (a b+b^2 x\right )^5}{x^3} \, 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 (10 a^2 b^8+\frac {a^5 b^5}{x^3}+\frac {5 a^4 b^6}{x^2}+\frac {10 a^3 b^7}{x}+5 a b^9 x+b^{10} x^2\right ) \, dx,x,x^3\right )}{3 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}}{6 x^6 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 \left (a+b x^3\right )}+\frac {b^5 x^9 \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 \left (a+b x^3\right )}+\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 85, normalized size = 0.34 \[ \frac {\sqrt {\left (a+b x^3\right )^2} \left (-3 a^5-30 a^4 b x^3+180 a^3 b^2 x^6 \log (x)+60 a^2 b^3 x^9+15 a b^4 x^{12}+2 b^5 x^{15}\right )}{18 x^6 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 61, normalized size = 0.24 \[ \frac {2 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 180 \, a^{3} b^{2} x^{6} \log \relax (x) - 30 \, a^{4} b x^{3} - 3 \, a^{5}}{18 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 126, normalized size = 0.50 \[ \frac {1}{9} \, b^{5} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{6} \, a b^{4} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{3} \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {30 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 10 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 82, normalized size = 0.33 \[ \frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{5} x^{15}+15 a \,b^{4} x^{12}+60 a^{2} b^{3} x^{9}+180 a^{3} b^{2} x^{6} \ln \relax (x )-30 a^{4} b \,x^{3}-3 a^{5}\right )}{18 \left (b \,x^{3}+a \right )^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 282, normalized size = 1.12 \[ \frac {5}{3} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a b^{3} x^{3} + \frac {10}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a^{3} b^{2} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {10}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a^{3} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{3} x^{3}}{6 \, a} + 5 \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{2} b^{2} + \frac {35}{18} \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{2} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{2}}{6 \, a^{2}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b}{2 \, a x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{6 \, a^{2} x^{6}} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^7} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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