Optimal. Leaf size=184 \[ \frac {\left (-b+a x^3\right )^{2/3} \left (-15 b d+18 b c x-20 a d x^3+27 a c x^4\right )}{90 b^2 x^6}+\frac {2 a^2 d \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{7/3}}+\frac {2 a^2 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{27 b^{7/3}}-\frac {a^2 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{27 b^{7/3}} \]
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Rubi [A]
time = 0.16, antiderivative size = 198, normalized size of antiderivative = 1.08, number of steps
used = 11, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1858, 272, 44,
58, 631, 210, 31, 277, 270} \begin {gather*} \frac {2 a^2 d \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{9 \sqrt {3} b^{7/3}}+\frac {a^2 d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{9 b^{7/3}}-\frac {a^2 d \log (x)}{9 b^{7/3}}+\frac {3 a c \left (a x^3-b\right )^{2/3}}{10 b^2 x^2}-\frac {2 a d \left (a x^3-b\right )^{2/3}}{9 b^2 x^3}+\frac {c \left (a x^3-b\right )^{2/3}}{5 b x^5}-\frac {d \left (a x^3-b\right )^{2/3}}{6 b x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 58
Rule 210
Rule 270
Rule 272
Rule 277
Rule 631
Rule 1858
Rubi steps
\begin {align*} \int \frac {-d+c x}{x^7 \sqrt [3]{-b+a x^3}} \, dx &=\int \left (-\frac {d}{x^7 \sqrt [3]{-b+a x^3}}+\frac {c}{x^6 \sqrt [3]{-b+a x^3}}\right ) \, dx\\ &=c \int \frac {1}{x^6 \sqrt [3]{-b+a x^3}} \, dx-d \int \frac {1}{x^7 \sqrt [3]{-b+a x^3}} \, dx\\ &=\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}+\frac {(3 a c) \int \frac {1}{x^3 \sqrt [3]{-b+a x^3}} \, dx}{5 b}-\frac {1}{3} d \text {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{-b+a x}} \, dx,x,x^3\right )\\ &=-\frac {d \left (-b+a x^3\right )^{2/3}}{6 b x^6}+\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}+\frac {3 a c \left (-b+a x^3\right )^{2/3}}{10 b^2 x^2}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{-b+a x}} \, dx,x,x^3\right )}{9 b}\\ &=-\frac {d \left (-b+a x^3\right )^{2/3}}{6 b x^6}+\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}-\frac {2 a d \left (-b+a x^3\right )^{2/3}}{9 b^2 x^3}+\frac {3 a c \left (-b+a x^3\right )^{2/3}}{10 b^2 x^2}-\frac {\left (2 a^2 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{-b+a x}} \, dx,x,x^3\right )}{27 b^2}\\ &=-\frac {d \left (-b+a x^3\right )^{2/3}}{6 b x^6}+\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}-\frac {2 a d \left (-b+a x^3\right )^{2/3}}{9 b^2 x^3}+\frac {3 a c \left (-b+a x^3\right )^{2/3}}{10 b^2 x^2}-\frac {a^2 d \log (x)}{9 b^{7/3}}+\frac {\left (a^2 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{9 b^{7/3}}-\frac {\left (a^2 d\right ) \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{9 b^2}\\ &=-\frac {d \left (-b+a x^3\right )^{2/3}}{6 b x^6}+\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}-\frac {2 a d \left (-b+a x^3\right )^{2/3}}{9 b^2 x^3}+\frac {3 a c \left (-b+a x^3\right )^{2/3}}{10 b^2 x^2}-\frac {a^2 d \log (x)}{9 b^{7/3}}+\frac {a^2 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{9 b^{7/3}}-\frac {\left (2 a^2 d\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}\right )}{9 b^{7/3}}\\ &=-\frac {d \left (-b+a x^3\right )^{2/3}}{6 b x^6}+\frac {c \left (-b+a x^3\right )^{2/3}}{5 b x^5}-\frac {2 a d \left (-b+a x^3\right )^{2/3}}{9 b^2 x^3}+\frac {3 a c \left (-b+a x^3\right )^{2/3}}{10 b^2 x^2}+\frac {2 a^2 d \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3}}-\frac {a^2 d \log (x)}{9 b^{7/3}}+\frac {a^2 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{9 b^{7/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.07, size = 100, normalized size = 0.54 \begin {gather*} \frac {\left (b-a x^3\right ) \left (a x^3 (20 d-27 c x)+3 b (5 d-6 c x)\right )+20 a^2 d \sqrt [3]{1-\frac {b}{a x^3}} x^6 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {b}{a x^3}\right )}{90 b^2 x^6 \sqrt [3]{-b+a x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {c x -d}{x^{7} \left (a \,x^{3}-b \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 219, normalized size = 1.19 \begin {gather*} -\frac {1}{54} \, {\left (\frac {4 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} + \frac {2 \, a^{2} \log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {7}{3}}} - \frac {4 \, a^{2} \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (4 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}} a^{2} + 7 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a^{2} b\right )}}{{\left (a x^{3} - b\right )}^{2} b^{2} + 2 \, {\left (a x^{3} - b\right )} b^{3} + b^{4}}\right )} d + \frac {c {\left (\frac {5 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a}{x^{2}} - \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}}}{x^{5}}\right )}}{10 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 122.98, size = 405, normalized size = 2.20 \begin {gather*} \left [\frac {30 \, \sqrt {\frac {1}{3}} a^{2} b d x^{6} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (\frac {2 \, a x^{3} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} b^{\frac {2}{3}} + {\left (a x^{3} - b\right )}^{\frac {1}{3}} b - b^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} - 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {2}{3}} - 3 \, b}{x^{3}}\right ) + 20 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}}{x}\right ) - 10 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (27 \, a b c x^{4} - 20 \, a b d x^{3} + 18 \, b^{2} c x - 15 \, b^{2} d\right )} {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{270 \, b^{3} x^{6}}, -\frac {60 \, \sqrt {\frac {1}{3}} a^{2} b^{\frac {2}{3}} d x^{6} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}}}\right ) - 20 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}}{x}\right ) + 10 \, a^{2} b^{\frac {2}{3}} d x^{6} \log \left (\frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (27 \, a b c x^{4} - 20 \, a b d x^{3} + 18 \, b^{2} c x - 15 \, b^{2} d\right )} {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{270 \, b^{3} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.52, size = 345, normalized size = 1.88 \begin {gather*} c \left (\begin {cases} - \frac {3 a^{\frac {11}{3}} x^{6} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} + \frac {a^{\frac {8}{3}} b x^{3} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} + \frac {2 a^{\frac {5}{3}} b^{2} \left (-1 + \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a^{2} b^{2} x^{6} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) - 9 a b^{3} x^{3} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\\frac {a^{\frac {5}{3}} \left (1 - \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 b^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {2 a^{\frac {2}{3}} \left (1 - \frac {b}{a x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 b x^{3} \Gamma \left (\frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {d \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x^{7} \Gamma \left (\frac {10}{3}\right )} \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 [B]
time = 2.28, size = 250, normalized size = 1.36 \begin {gather*} \frac {2\,a^2\,d\,\ln \left ({\left (a\,x^3-b\right )}^{1/3}+b^{1/3}\right )}{27\,b^{7/3}}-\frac {\frac {7\,a^2\,d\,{\left (a\,x^3-b\right )}^{2/3}}{18\,b}+\frac {2\,a^2\,d\,{\left (a\,x^3-b\right )}^{5/3}}{9\,b^2}}{{\left (b-a\,x^3\right )}^2-2\,b\,\left (b-a\,x^3\right )+b^2}+\frac {2\,a^2\,d\,\ln \left (\frac {4\,a^4\,d^2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,b^{11/3}}+\frac {4\,a^4\,d^2\,{\left (a\,x^3-b\right )}^{1/3}}{81\,b^4}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{7/3}}-\frac {2\,a^2\,d\,\ln \left (\frac {4\,a^4\,d^2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,b^{11/3}}+\frac {4\,a^4\,d^2\,{\left (a\,x^3-b\right )}^{1/3}}{81\,b^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,b^{7/3}}+\frac {c\,{\left (a\,x^3-b\right )}^{2/3}\,\left (3\,a\,x^3+2\,b\right )}{10\,b^2\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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