Optimal. Leaf size=290 \[ -\frac {b c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{9 a^{4/3}}-\frac {b c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{a}}+\frac {x}{\sqrt {3}}}{x}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{18 a^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}+b^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {c x \left (a x^3-b\right )^{2/3}}{3 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{3 \sqrt [3]{b}}+\frac {d \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.17, antiderivative size = 190, normalized size of antiderivative = 0.66, number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1844, 266, 56, 617, 204, 31, 321, 239} \begin {gather*} -\frac {b c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{6 a^{4/3}}+\frac {b c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {c x \left (a x^3-b\right )^{2/3}}{3 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {d \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 239
Rule 266
Rule 321
Rule 617
Rule 1844
Rubi steps
\begin {align*} \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx &=\int \left (-\frac {d}{x \sqrt [3]{-b+a x^3}}+\frac {c x^3}{\sqrt [3]{-b+a x^3}}\right ) \, dx\\ &=c \int \frac {x^3}{\sqrt [3]{-b+a x^3}} \, dx-d \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx\\ &=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {(b c) \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx}{3 a}-\frac {1}{3} d \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-b+a x}} \, dx,x,x^3\right )\\ &=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}}-\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{-b+a x^3}\right )+\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}\\ &=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}}\\ &=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}+\frac {b c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {d \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{6 a^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 184, normalized size = 0.63 \begin {gather*} \frac {1}{18} \left (\frac {c \left (b \log \left (\frac {a^{2/3} x^2}{\left (a x^3-b\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1\right )+6 \sqrt [3]{a} x \left (a x^3-b\right )^{2/3}-2 b \log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}\right )+2 \sqrt {3} b \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )\right )}{a^{4/3}}-\frac {9 d \left (a x^3-b\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-\frac {a x^3}{b}\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.88, size = 290, normalized size = 1.00 \begin {gather*} -\frac {b c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{9 a^{4/3}}-\frac {b c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{a}}+\frac {x}{\sqrt {3}}}{x}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{18 a^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}+b^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {c x \left (a x^3-b\right )^{2/3}}{3 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{3 \sqrt [3]{b}}+\frac {d \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}} \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 {c x^{4} - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c \,x^{4}-d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 250, normalized size = 0.86 \begin {gather*} -\frac {1}{18} \, {\left (\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b \log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {4}{3}}} - \frac {6 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} b}{{\left (a^{2} - \frac {{\left (a x^{3} - b\right )} a}{x^{3}}\right )} x^{2}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \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 {1}{3}}} + \frac {\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 {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \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 {d-c\,x^4}{x\,{\left (a\,x^3-b\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.09, size = 83, normalized size = 0.29 \begin {gather*} - \frac {c x^{4} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {7}{3}\right )} + \frac {d \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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