Optimal. Leaf size=252 \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{12 x^{12} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b^5 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3} \]
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Rubi [A]
time = 0.04, 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 = {1369, 272, 45}
\begin {gather*} \frac {b^5 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 \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {10 a^2 b^3 \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}}{12 x^{12} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1369
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{13}} \, 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^{13}} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^5}{x^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} \text {Subst}\left (\int \left (b^{10}+\frac {a^5 b^5}{x^5}+\frac {5 a^4 b^6}{x^4}+\frac {10 a^3 b^7}{x^3}+\frac {10 a^2 b^8}{x^2}+\frac {5 a b^9}{x}\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}}{12 x^{12} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^6 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b^5 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 \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 = 0.01, size = 85, normalized size = 0.34 \begin {gather*} -\frac {\sqrt {\left (a+b x^3\right )^2} \left (3 a^5+20 a^4 b x^3+60 a^3 b^2 x^6+120 a^2 b^3 x^9-12 b^5 x^{15}-180 a b^4 x^{12} \log (x)\right )}{36 x^{12} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 82, normalized size = 0.33
method | result | size |
default | \(\frac {\left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} \left (12 b^{5} x^{15}+180 b^{4} a \ln \left (x \right ) x^{12}-120 a^{2} b^{3} x^{9}-60 b^{2} a^{3} x^{6}-20 a^{4} b \,x^{3}-3 a^{5}\right )}{36 \left (b \,x^{3}+a \right )^{5} x^{12}}\) | \(82\) |
risch | \(\frac {b^{5} x^{3} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{3 b \,x^{3}+3 a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {10}{3} a^{2} b^{3} x^{9}-\frac {5}{3} b^{2} a^{3} x^{6}-\frac {5}{9} a^{4} b \,x^{3}-\frac {1}{12} a^{5}\right )}{\left (b \,x^{3}+a \right ) x^{12}}+\frac {5 a \,b^{4} \ln \left (x \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 342, normalized size = 1.36 \begin {gather*} \frac {5 \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{5} x^{3}}{6 \, a} + \frac {5}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a b^{4} \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {5}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a b^{4} \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^{5} x^{3}}{12 \, a^{3}} + \frac {5}{2} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{4} + \frac {35 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b^{4}}{36 \, a^{2}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{4}}{9 \, a^{4}} - \frac {2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {5}{2}} b^{3}}{9 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b^{2}}{9 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}} b}{36 \, a^{3} x^{9}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {7}{2}}}{12 \, a^{2} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 61, normalized size = 0.24 \begin {gather*} \frac {12 \, b^{5} x^{15} + 180 \, a b^{4} x^{12} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{9} - 60 \, a^{3} b^{2} x^{6} - 20 \, a^{4} b x^{3} - 3 \, a^{5}}{36 \, x^{12}} \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^{13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.02, size = 125, normalized size = 0.50 \begin {gather*} \frac {1}{3} \, b^{5} x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {125 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 120 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 60 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 20 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 3 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{36 \, x^{12}} \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^{13}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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