Optimal. Leaf size=392 \[ -\frac {\sqrt [3]{a+b x^3} \left (130 a^2 d^2-143 a b c d+4 b^2 c^2\right )}{910 a c^3 x^7}+\frac {\sqrt [3]{a+b x^3} \left (455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right )}{1820 a^2 c^4 x^4}-\frac {\sqrt [3]{a+b x^3} \left (1820 a^4 d^4-2275 a^3 b c d^3+260 a^2 b^2 c^2 d^2+78 a b^3 c^3 d+36 b^4 c^4\right )}{1820 a^3 c^5 x}-\frac {d^3 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{16/3}}+\frac {d^3 (b c-a d)^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{16/3}}+\frac {d^3 (b c-a d)^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{16/3}}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{130 c^2 x^{10}}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}} \]
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Rubi [C] time = 4.39, antiderivative size = 1446, normalized size of antiderivative = 3.69, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510}
result too large to display
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx &=\frac {\left (a \sqrt [3]{a+b x^3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=-\frac {140 a^2 c^5+840 a b c^5 x^3-686 a^2 c^4 d x^3+700 b^2 c^5 x^6-1316 a b c^4 d x^6+612 a^2 c^3 d^2 x^6-630 b^2 c^4 d x^9+1152 a b c^3 d^2 x^9-513 a^2 c^2 d^3 x^9+540 b^2 c^3 d^2 x^{12}-918 a b c^2 d^3 x^{12}+324 a^2 c d^4 x^{12}-405 b^2 c^2 d^3 x^{15}+324 a b c d^4 x^{15}-828 a b c^5 x^3 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+828 a^2 c^4 d x^3 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-828 b^2 c^5 x^6 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+918 a b c^4 d x^6 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-90 a^2 c^3 d^2 x^6 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+90 b^2 c^4 d x^9 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+234 a b c^3 d^2 x^9 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-324 a^2 c^2 d^3 x^9 \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+324 b^2 c^3 d^2 x^{12} \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-918 a b c^2 d^3 x^{12} \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+594 a^2 c d^4 x^{12} \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-594 b^2 c^2 d^3 x^{15} \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+594 a b c d^4 x^{15} \, _2F_1\left (-\frac {1}{3},2;\frac {2}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-280 b^2 c^5 x^6 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+560 a b c^4 d x^6 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-280 a^2 c^3 d^2 x^6 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+252 b^2 c^4 d x^9 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-504 a b c^3 d^2 x^9 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+252 a^2 c^2 d^3 x^9 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-216 b^2 c^3 d^2 x^{12} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+432 a b c^2 d^3 x^{12} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-216 a^2 c d^4 x^{12} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+162 b^2 c^2 d^3 x^{15} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-324 a b c d^4 x^{15} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+162 a^2 d^5 x^{15} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+54 c (b c-a d) x^3 \left (a+b x^3\right ) \left (7 c-6 d x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (-\frac {1}{3},2,2;\frac {2}{3},1;\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-54 c (b c-a d) x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^3 \, _4F_3\left (-\frac {1}{3},2,2,2;\frac {2}{3},1,1;\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{1820 c^6 x^{13} \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 5.44, size = 1446, normalized size = 3.69
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Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 4.75, size = 577, normalized size = 1.47 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-140 a^4 c^4+182 a^4 c^3 d x^3-260 a^4 c^2 d^2 x^6+455 a^4 c d^3 x^9-1820 a^4 d^4 x^{12}-196 a^3 b c^4 x^3+286 a^3 b c^3 d x^6-520 a^3 b c^2 d^2 x^9+2275 a^3 b c d^3 x^{12}-8 a^2 b^2 c^4 x^6+26 a^2 b^2 c^3 d x^9-260 a^2 b^2 c^2 d^2 x^{12}+12 a b^3 c^4 x^9-78 a b^3 c^3 d x^{12}-36 b^4 c^4 x^{12}\right )}{1820 a^3 c^5 x^{13}}+\frac {i \left (\sqrt {3} d^3 (b c-a d)^{4/3}+i d^3 (b c-a d)^{4/3}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{16/3}}-\frac {\sqrt {\frac {1}{6} \left (-1-i \sqrt {3}\right )} d^3 (b c-a d)^{4/3} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{16/3}}+\frac {\left (d^3 (b c-a d)^{4/3}-i \sqrt {3} d^3 (b c-a d)^{4/3}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{16/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{14}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}}}{\left (d \,x^{3}+c \right ) x^{14}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{14}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{4/3}}{x^{14}\,\left (d\,x^3+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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