Optimal. Leaf size=324 \[ \frac {b (3 b c+4 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (9 b^2 c^2-33 a b c d-4 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d x}{3 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {d^2 (9 b c-2 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{10/3}}+\frac {d^2 (9 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{10/3}}-\frac {d^2 (9 b c-2 a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{10/3}} \]
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Rubi [A]
time = 0.25, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {425, 541, 12,
384} \begin {gather*} \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) \text {ArcTan}\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^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 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 425
Rule 541
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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.31, size = 443, normalized size = 1.37 \begin {gather*} \frac {\frac {3 c^{2/3} x \left (4 a^4 d^3+8 a^3 b d^3 x^3-9 b^4 c^2 x^3 \left (c+d x^3\right )+4 a^2 b^2 d \left (9 c^2+9 c d x^3+d^2 x^6\right )+3 a b^3 c \left (-4 c^2+7 c d x^3+11 d^2 x^6\right )\right )}{a^2 (-b c+a d)^3 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {2 i \left (3 i+\sqrt {3}\right ) d^2 (-9 b c+2 a d) \tanh ^{-1}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )}{(b c-a d)^{10/3}}+\frac {2 \left (1+i \sqrt {3}\right ) d^2 (9 b c-2 a d) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{10/3}}+\frac {\left (1+i \sqrt {3}\right ) d^2 (-9 b c+2 a d) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{10/3}}}{36 c^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {7}{3}} \left (d \,x^{3}+c \right )^{2}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{3}\right )^{\frac {7}{3}} \left (c + d x^{3}\right )^{2}}\, dx \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*} \text {could not integrate} \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 {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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