Optimal. Leaf size=262 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (20 a^2 d^2+12 a b c d+9 b^2 c^2\right )}{40 a^3 c^3 x^2}+\frac{\left (a+b x^3\right )^{2/3} (4 a d+3 b c)}{20 a^2 c^2 x^5}-\frac{d^3 \log \left (c+d x^3\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}+\frac{d^3 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{11/3} \sqrt [3]{b c-a d}}-\frac{\left (a+b x^3\right )^{2/3}}{8 a c x^8} \]
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Rubi [A] time = 0.324543, antiderivative size = 317, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {494, 461, 200, 31, 634, 617, 204, 628} \[ -\frac{\left (a+b x^3\right )^{2/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 a^3 c^3 x^2}+\frac{\left (a+b x^3\right )^{5/3} (a d+2 b c)}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac{d^3 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{11/3} \sqrt [3]{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 494
Rule 461
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^9 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-b x^3\right )^3}{x^9 \left (c-(b c-a d) x^3\right )} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{c x^9}+\frac{-2 b c-a d}{c^2 x^6}+\frac{b^2 c^2+a b c d+a^2 d^2}{c^3 x^3}+\frac{a^3 d^3}{c^3 \left (-c+(b c-a d) x^3\right )}\right ) \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{a^3}\\ &=-\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac{(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{-c+(b c-a d) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{c^3}\\ &=-\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac{(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{-\sqrt [3]{c}+\sqrt [3]{b c-a d} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3}}+\frac{d^3 \operatorname{Subst}\left (\int \frac{-2 \sqrt [3]{c}-\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3}}\\ &=-\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac{(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac{d^3 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{2 c^{10/3}}-\frac{d^3 \operatorname{Subst}\left (\int \frac{\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}\\ &=-\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac{(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac{d^3 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{11/3} \sqrt [3]{b c-a d}}\\ &=-\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac{(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac{\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}-\frac{d^3 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c^{11/3} \sqrt [3]{b c-a d}}+\frac{d^3 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac{d^3 \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}\\ \end{align*}
Mathematica [C] time = 2.29045, size = 821, normalized size = 3.13 \[ -\frac{648 b c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+648 a c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 b c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-243 a c d^3 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+243 b c^2 d^2 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 a c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-72 b c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+45 a c^2 d^2 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-45 b c^3 d \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-72 a c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+9 b c^4 \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-9 a c^3 d \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-54 (b c-a d) \left (c-3 d x^3\right ) \left (d x^3+c\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},2,2\right \},\left \{1,\frac{7}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},2,2,2\right \},\left \{1,1,\frac{7}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{320 c^5 x^8 \left (b x^3+a\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9} \left ( d{x}^{3}+c \right ) }{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )} x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{9} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, 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*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )} x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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