Optimal. Leaf size=463 \[ \frac{d x \left (a+b x^3\right )^{2/3} \left (-42 a^2 b c d^2+10 a^3 d^3-135 a b^2 c^2 d+27 b^3 c^3\right )}{36 a^2 c^2 \left (c+d x^3\right ) (b c-a d)^4}+\frac{b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{12 a^2 c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)^3}+\frac{d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac{d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{13/3}}+\frac{d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} (b c-a d)^{13/3}}-\frac{d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}+\frac{b x (2 a d+3 b c)}{12 a c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)^2} \]
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Rubi [C] time = 8.66242, antiderivative size = 1990, normalized size of antiderivative = 4.3, 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} \[ \text{result too large to display} \]
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 )^3} \, 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 )^3} \, dx}{a^2 \sqrt [3]{a+b x^3}}\\ &=-\frac{522756 c^6 (b c-a d)^3 x^9 \left (a+b x^3\right )^2+1516320 c^5 d (b c-a d)^3 x^{12} \left (a+b x^3\right )^2+2198664 c^4 d^2 (b c-a d)^3 x^{15} \left (a+b x^3\right )^2+1415232 c^3 d^3 (b c-a d)^3 x^{18} \left (a+b x^3\right )^2+341172 c^2 d^4 (b c-a d)^3 x^{21} \left (a+b x^3\right )^2+28042560 c^7 (b c-a d)^2 x^6 \left (a+b x^3\right )^3+107602560 c^6 d (b c-a d)^2 x^9 \left (a+b x^3\right )^3+157697280 c^5 d^2 (b c-a d)^2 x^{12} \left (a+b x^3\right )^3+101088000 c^4 d^3 (b c-a d)^2 x^{15} \left (a+b x^3\right )^3+24261120 c^3 d^4 (b c-a d)^2 x^{18} \left (a+b x^3\right )^3-265470660 c^8 (b c-a d) x^3 \left (a+b x^3\right )^4-1019636800 c^7 d (b c-a d) x^6 \left (a+b x^3\right )^4-1466086440 c^6 d^2 (b c-a d) x^9 \left (a+b x^3\right )^4-930252960 c^5 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^4-221899860 c^4 d^4 (b c-a d) x^{15} \left (a+b x^3\right )^4+335877360 c^9 \left (a+b x^3\right )^5+1279532800 c^8 d x^3 \left (a+b x^3\right )^5+1823334240 c^7 d^2 x^6 \left (a+b x^3\right )^5+1151579520 c^6 d^3 x^9 \left (a+b x^3\right )^5+273939120 c^5 d^4 x^{12} \left (a+b x^3\right )^5-67420080 c^7 (b c-a d)^2 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 )-259692160 c^6 d (b c-a d)^2 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 )-377700960 c^5 d^2 (b c-a d)^2 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 )-241113600 c^4 d^3 (b c-a d)^2 x^{15} \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 )-57723120 c^3 d^4 (b c-a d)^2 x^{18} \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 )+349440000 c^8 (b c-a 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 )+1339520000 c^7 d (b c-a d) 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 )+1921920000 c^6 d^2 (b c-a d) 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 )+1218147840 c^5 d^3 (b c-a d) x^{12} \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 )+290384640 c^4 d^4 (b c-a d) x^{15} \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 )-335877360 c^9 \left (a+b x^3\right )^5 \, _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 )-1279532800 c^8 d x^3 \left (a+b x^3\right )^5 \, _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 )-1823334240 c^7 d^2 x^6 \left (a+b x^3\right )^5 \, _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 )-1151579520 c^6 d^3 x^9 \left (a+b x^3\right )^5 \, _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 )-273939120 c^5 d^4 x^{12} \left (a+b x^3\right )^5 \, _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 )-57834 c^4 (b c-a d)^5 x^{15} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-224532 c^3 d (b c-a d)^5 x^{18} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-326592 c^2 d^2 (b c-a d)^5 x^{21} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-210924 c d^3 (b c-a d)^5 x^{24} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-51030 d^4 (b c-a d)^5 x^{27} \, _4F_3\left (2,2,2,\frac{10}{3};1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-5103 c^4 (b c-a d)^5 x^{15} \, _5F_4\left (2,2,2,2,\frac{10}{3};1,1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-20412 c^3 d (b c-a d)^5 x^{18} \, _5F_4\left (2,2,2,2,\frac{10}{3};1,1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-30618 c^2 d^2 (b c-a d)^5 x^{21} \, _5F_4\left (2,2,2,2,\frac{10}{3};1,1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-20412 c d^3 (b c-a d)^5 x^{24} \, _5F_4\left (2,2,2,2,\frac{10}{3};1,1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-5103 d^4 (b c-a d)^5 x^{27} \, _5F_4\left (2,2,2,2,\frac{10}{3};1,1,1,\frac{19}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{524160 c^6 (b c-a d)^4 x^{11} \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )^2}\\ \end{align*}
Mathematica [A] time = 5.84791, size = 337, normalized size = 0.73 \[ \frac{1}{36} x \left (a+b x^3\right )^{2/3} \left (\frac{27 b^3 (b c-5 a d)}{a^2 \left (a+b x^3\right ) (b c-a d)^4}-\frac{9 b^3}{a \left (a+b x^3\right )^2 (a d-b c)^3}+\frac{2 d^3 (5 a d-21 b c)}{c^2 \left (c+d x^3\right ) (b c-a d)^4}-\frac{6 d^3}{c \left (c+d x^3\right )^2 (b c-a d)^3}\right )+\frac{d^2 \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \left (\log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{54 c^{8/3} (b c-a d)^{13/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.428, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \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 )}^{3}}\,{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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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