Optimal. Leaf size=364 \[ \frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]
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Rubi [A] time = 0.199121, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1343, 199, 200, 31, 634, 617, 204, 628} \[ \frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 1343
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac{\left (2 a b+2 b^2 x^3\right )^5 \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^5} \, dx}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^4} \, dx}{24 a b \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (11 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^3} \, dx}{54 a^2 b^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^2} \, dx}{648 a^3 b^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{2 a b+2 b^2 x^3} \, dx}{1944 a^4 b^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}-\sqrt [3]{2} b^{2/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{5832\ 2^{2/3} a^{14/3} b^{14/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{-2^{2/3} \sqrt [3]{a} b+2\ 2^{2/3} b^{4/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{23328 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \int \frac{1}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{3888 \sqrt [3]{2} a^{13/3} b^{13/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{\left (55 \left (2 a b+2 b^2 x^3\right )^5\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3888 a^{14/3} b^{16/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ &=\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.11405, size = 211, normalized size = 0.58 \[ \frac{\left (a+b x^3\right ) \left (-\frac{220 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+660 a^{2/3} x \left (a+b x^3\right )^3+396 a^{5/3} x \left (a+b x^3\right )^2+297 a^{8/3} x \left (a+b x^3\right )+243 a^{11/3} x+\frac{440 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{440 \sqrt{3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}\right )}{2916 a^{14/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 519, normalized size = 1.4 \begin{align*} \text{result too large to display} \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 = 1.87585, size = 1652, normalized size = 4.54 \begin{align*} \text{result too large to display} \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}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\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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