Optimal. Leaf size=261 \[ -\frac{d (3 b c-a d) \log \left (c+d x^3\right )}{9 c^{5/3} (b c-a d)^{7/3}}+\frac{d (3 b c-a d) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} (b c-a d)^{7/3}}-\frac{2 d (3 b c-a d) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^{7/3}}+\frac{b x (a d+3 b c)}{3 a c \sqrt [3]{a+b x^3} (b c-a d)^2}-\frac{d x}{3 c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)} \]
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Rubi [C] time = 1.93369, antiderivative size = 625, normalized size of antiderivative = 2.39, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ -\frac{c \left (a+b x^3\right )^{2/3} \left (-\frac{54 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^5 \left (a+b x^3\right )^3}-\frac{108 d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac{54 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac{2520 d^2 x^9 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}-\frac{945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac{5320 d x^6 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}-\frac{1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}-\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}+\frac{2240 x^3 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\frac{525 x^3 (b c-a d)}{c \left (a+b x^3\right )}+\frac{6300 d^2 x^6}{c^2}+\frac{13720 d x^3}{c}+6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )^2} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac{c \left (a+b x^3\right )^{2/3} \left (6860+\frac{13720 d x^3}{c}+\frac{6300 d^2 x^6}{c^2}-\frac{525 (b c-a d) x^3}{c \left (a+b x^3\right )}-\frac{1890 d (b c-a d) x^6}{c^2 \left (a+b x^3\right )}-\frac{945 d^2 (b c-a d) x^9}{c^3 \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c}-\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2}+\frac{2240 (b c-a d) x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c \left (a+b x^3\right )}+\frac{5320 d (b c-a d) x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac{2520 d^2 (b c-a d) x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac{54 (b c-a d)^3 x^9 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )^3}-\frac{108 d (b c-a d)^3 x^{12} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac{54 d^2 (b c-a d)^3 x^{15} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^5 \left (a+b x^3\right )^3}\right )}{420 (b c-a d)^2 x^5 \left (c+d x^3\right )}\\ \end{align*}
Mathematica [C] time = 1.45104, size = 625, normalized size = 2.39 \[ \frac{c \left (a+b x^3\right )^{2/3} \left (\frac{54 d^2 x^{15} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^5 \left (a+b x^3\right )^3}+\frac{108 d x^{12} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^4 \left (a+b x^3\right )^3}+\frac{54 x^9 (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac{2520 d^2 x^9 (a d-b c) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}+\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}+\frac{945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac{5320 d x^6 (a d-b c) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac{1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}+\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}-\frac{2240 x^3 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}+6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\frac{525 x^3 (b c-a d)}{c \left (a+b x^3\right )}-\frac{6300 d^2 x^6}{c^2}-\frac{13720 d x^3}{c}-6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, 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{4}{3}}{\left (d x^{3} + c\right )}^{2}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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