Optimal. Leaf size=269 \[ -\frac {b}{12 a^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {4 b}{3 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^5 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b}{2 a^4 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.14, antiderivative size = 269, 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, 44} \[ -\frac {b}{2 a^4 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b}{12 a^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {4 b}{3 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^5 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a b+b^2 x\right )^5} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^5 b^5 x^2}-\frac {5}{a^6 b^4 x}+\frac {1}{a^2 b^3 (a+b x)^5}+\frac {2}{a^3 b^3 (a+b x)^4}+\frac {3}{a^4 b^3 (a+b x)^3}+\frac {4}{a^5 b^3 (a+b x)^2}+\frac {5}{a^6 b^3 (a+b x)}\right ) \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {4 b}{3 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b}{12 a^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b}{2 a^4 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^5 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 b \left (a+b x^3\right ) \log (x)}{a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 119, normalized size = 0.44 \[ \frac {-a \left (12 a^4+125 a^3 b x^3+260 a^2 b^2 x^6+210 a b^3 x^9+60 b^4 x^{12}\right )-180 b x^3 \log (x) \left (a+b x^3\right )^4+60 b x^3 \left (a+b x^3\right )^4 \log \left (a+b x^3\right )}{36 a^6 x^3 \left (a+b x^3\right )^3 \sqrt {\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 207, normalized size = 0.77 \[ -\frac {60 \, a b^{4} x^{12} + 210 \, a^{2} b^{3} x^{9} + 260 \, a^{3} b^{2} x^{6} + 125 \, a^{4} b x^{3} + 12 \, a^{5} - 60 \, {\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 180 \, {\left (b^{5} x^{15} + 4 \, a b^{4} x^{12} + 6 \, a^{2} b^{3} x^{9} + 4 \, a^{3} b^{2} x^{6} + a^{4} b x^{3}\right )} \log \relax (x)}{36 \, {\left (a^{6} b^{4} x^{15} + 4 \, a^{7} b^{3} x^{12} + 6 \, a^{8} b^{2} x^{9} + 4 \, a^{9} b x^{6} + a^{10} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 219, normalized size = 0.81 \[ -\frac {\left (180 b^{5} x^{15} \ln \relax (x )-60 b^{5} x^{15} \ln \left (b \,x^{3}+a \right )+720 a \,b^{4} x^{12} \ln \relax (x )-240 a \,b^{4} x^{12} \ln \left (b \,x^{3}+a \right )+60 a \,b^{4} x^{12}+1080 a^{2} b^{3} x^{9} \ln \relax (x )-360 a^{2} b^{3} x^{9} \ln \left (b \,x^{3}+a \right )+210 a^{2} b^{3} x^{9}+720 a^{3} b^{2} x^{6} \ln \relax (x )-240 a^{3} b^{2} x^{6} \ln \left (b \,x^{3}+a \right )+260 a^{3} b^{2} x^{6}+180 a^{4} b \,x^{3} \ln \relax (x )-60 a^{4} b \,x^{3} \ln \left (b \,x^{3}+a \right )+125 a^{4} b \,x^{3}+12 a^{5}\right ) \left (b \,x^{3}+a \right )}{36 \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}} a^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 163, normalized size = 0.61 \[ \frac {5 \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right )}{3 \, a^{6}} - \frac {5 \, b}{9 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{3}} - \frac {5 \, b}{3 \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{5}} - \frac {1}{3 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{3}} - \frac {5}{6 \, {\left (x^{3} + \frac {a}{b}\right )}^{2} a^{4} b} - \frac {1}{12 \, {\left (x^{3} + \frac {a}{b}\right )}^{4} a^{2} b^{3}} \]
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 \[ \int \frac {1}{x^4\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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