Optimal. Leaf size=269 \[ -\frac{b}{2 a^4 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{9 a^3 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{b}{12 a^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{4 b}{3 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^5 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 b \log (x) \left (a+b x^3\right )}{a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^6 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.142528, antiderivative size = 269, normalized size of antiderivative = 1., 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 1355
Rule 266
Rule 44
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*}
Mathematica [A] time = 0.0522613, size = 119, normalized size = 0.44 \[ \frac{-a \left (260 a^2 b^2 x^6+125 a^3 b x^3+12 a^4+210 a b^3 x^9+60 b^4 x^{12}\right )-180 b x^3 \log (x) \left (a+b x^3\right )^4+60 b x^3 \left (a+b x^3\right )^4 \log \left (a+b x^3\right )}{36 a^6 x^3 \left (a+b x^3\right )^3 \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 219, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 180\,{b}^{5}\ln \left ( x \right ){x}^{15}-60\,\ln \left ( b{x}^{3}+a \right ){x}^{15}{b}^{5}+720\,a{b}^{4}\ln \left ( x \right ){x}^{12}-240\,\ln \left ( b{x}^{3}+a \right ){x}^{12}a{b}^{4}+60\,a{b}^{4}{x}^{12}+1080\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{9}-360\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{a}^{2}{b}^{3}+210\,{a}^{2}{b}^{3}{x}^{9}+720\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{6}-240\,\ln \left ( b{x}^{3}+a \right ){x}^{6}{a}^{3}{b}^{2}+260\,{a}^{3}{b}^{2}{x}^{6}+180\,{a}^{4}b\ln \left ( x \right ){x}^{3}-60\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{4}b+125\,{a}^{4}b{x}^{3}+12\,{a}^{5} \right ) \left ( b{x}^{3}+a \right ) }{36\,{x}^{3}{a}^{6}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61363, size = 446, normalized size = 1.66 \begin{align*} -\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 \left (x\right )}{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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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