Optimal. Leaf size=324 \[ \frac{b x \left (-4 a^2 d^2-33 a b c d+9 b^2 c^2\right )}{12 a^2 c \sqrt [3]{a+b x^3} (b c-a d)^3}+\frac{d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac{d^2 (9 b c-2 a d) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}}+\frac{d^2 (9 b c-2 a d) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^{10/3}}-\frac{d x}{3 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}+\frac{b x (4 a d+3 b c)}{12 a c \left (a+b x^3\right )^{4/3} (b c-a d)^2} \]
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Rubi [C] time = 5.6921, antiderivative size = 1214, normalized size of antiderivative = 3.75, 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{6804 d^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}+1134 d^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}+21546 c d^2 (b c-a d)^4 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}+3402 c d^2 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}+26325 c^2 d^3 (b c-a d)^2 \left (b x^3+a\right )^2 x^{15}+22680 c^2 d (b c-a d)^4 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+3402 c^2 d (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+589680 c^3 d^3 (b c-a d) \left (b x^3+a\right )^3 x^{12}+84240 c^3 d^2 (b c-a d)^2 \left (b x^3+a\right )^2 x^{12}-966420 c^3 d^3 (b c-a d) \left (b x^3+a\right )^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}+7938 c^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+1134 c^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-1506960 c^4 d^3 \left (b x^3+a\right )^4 x^9+1916460 c^4 d^2 (b c-a d) \left (b x^3+a\right )^3 x^9+89505 c^4 d (b c-a d)^2 \left (b x^3+a\right )^2 x^9+1506960 c^4 d^3 \left (b x^3+a\right )^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^9-3144960 c^4 d^2 (b c-a d) \left (b x^3+a\right )^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^9-4914000 c^5 d^2 \left (b x^3+a\right )^4 x^6+2113020 c^5 d (b c-a d) \left (b x^3+a\right )^3 x^6+26130 c^5 (b c-a d)^2 \left (b x^3+a\right )^2 x^6+4914000 c^5 d^2 \left (b x^3+a\right )^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^6-3478020 c^5 d (b c-a d) \left (b x^3+a\right )^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^6-5460000 c^6 d \left (b x^3+a\right )^4 x^3+748020 c^6 (b c-a d) \left (b x^3+a\right )^3 x^3+5460000 c^6 d \left (b x^3+a\right )^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^3-1248520 c^6 (b c-a d) \left (b x^3+a\right )^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^3-2002000 c^7 \left (b x^3+a\right )^4+2002000 c^7 \left (b x^3+a\right )^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 )}{21840 c^5 (b c-a d)^3 x^8 \left (b x^3+a\right )^{10/3} \left (d x^3+c\right )} \]
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 )^{7/3} \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 )^{7/3} \left (c+d x^3\right )^2} \, dx}{a^2 \sqrt [3]{a+b x^3}}\\ &=\frac{26130 c^5 (b c-a d)^2 x^6 \left (a+b x^3\right )^2+89505 c^4 d (b c-a d)^2 x^9 \left (a+b x^3\right )^2+84240 c^3 d^2 (b c-a d)^2 x^{12} \left (a+b x^3\right )^2+26325 c^2 d^3 (b c-a d)^2 x^{15} \left (a+b x^3\right )^2+748020 c^6 (b c-a d) x^3 \left (a+b x^3\right )^3+2113020 c^5 d (b c-a d) x^6 \left (a+b x^3\right )^3+1916460 c^4 d^2 (b c-a d) x^9 \left (a+b x^3\right )^3+589680 c^3 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^3-2002000 c^7 \left (a+b x^3\right )^4-5460000 c^6 d x^3 \left (a+b x^3\right )^4-4914000 c^5 d^2 x^6 \left (a+b x^3\right )^4-1506960 c^4 d^3 x^9 \left (a+b x^3\right )^4-1248520 c^6 (b c-a d) x^3 \left (a+b x^3\right )^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 )-3478020 c^5 d (b c-a d) x^6 \left (a+b x^3\right )^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 )-3144960 c^4 d^2 (b c-a d) x^9 \left (a+b x^3\right )^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 )-966420 c^3 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^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 )+2002000 c^7 \left (a+b x^3\right )^4 \, _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 )+5460000 c^6 d x^3 \left (a+b x^3\right )^4 \, _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 )+4914000 c^5 d^2 x^6 \left (a+b x^3\right )^4 \, _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 )+1506960 c^4 d^3 x^9 \left (a+b x^3\right )^4 \, _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 )+7938 c^3 (b c-a d)^4 x^{12} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+22680 c^2 d (b c-a d)^4 x^{15} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+21546 c d^2 (b c-a d)^4 x^{18} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+6804 d^3 (b c-a d)^4 x^{21} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1134 c^3 (b c-a d)^4 x^{12} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+3402 c^2 d (b c-a d)^4 x^{15} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+3402 c d^2 (b c-a d)^4 x^{18} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1134 d^3 (b c-a d)^4 x^{21} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{21840 c^5 (b c-a d)^3 x^8 \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}\\ \end{align*}
Mathematica [C] time = 4.10342, size = 1216, normalized size = 3.75 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.423, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{7}{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{7}{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{7}{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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