Optimal. Leaf size=165 \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}-\frac {\sqrt [3]{e} q \log \left (d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}\right )}{2 \sqrt [3]{d} g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{\sqrt [3]{d} g}-\frac {\sqrt {3} \sqrt [3]{e} q \tan ^{-1}\left (\frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{\sqrt {3} \sqrt [3]{e}}\right )}{\sqrt [3]{d} g} \]
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Rubi [A] time = 0.17, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2483, 2448, 263, 200, 31, 634, 617, 204, 628} \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}-\frac {\sqrt [3]{e} q \log \left (d^{2/3} (f+g x)^2-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+e^{2/3}\right )}{2 \sqrt [3]{d} g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{d} (f+g x)+\sqrt [3]{e}\right )}{\sqrt [3]{d} g}-\frac {\sqrt {3} \sqrt [3]{e} q \tan ^{-1}\left (\frac {\sqrt [3]{e}-2 \sqrt [3]{d} (f+g x)}{\sqrt {3} \sqrt [3]{e}}\right )}{\sqrt [3]{d} g} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 263
Rule 617
Rule 628
Rule 634
Rule 2448
Rule 2483
Rubi steps
\begin {align*} \int \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x^3}\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {(3 e q) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x^3}\right ) x^3} \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {(3 e q) \operatorname {Subst}\left (\int \frac {1}{e+d x^3} \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {\left (\sqrt [3]{e} q\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{e}+\sqrt [3]{d} x} \, dx,x,f+g x\right )}{g}+\frac {\left (\sqrt [3]{e} q\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{e}-\sqrt [3]{d} x}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac {\left (\sqrt [3]{e} q\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 d^{2/3} x}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{2 \sqrt [3]{d} g}+\frac {\left (3 e^{2/3} q\right ) \operatorname {Subst}\left (\int \frac {1}{e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3} x^2} \, dx,x,f+g x\right )}{2 g}\\ &=\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac {\sqrt [3]{e} q \log \left (e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{d} g}+\frac {\left (3 \sqrt [3]{e} q\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}\right )}{\sqrt [3]{d} g}\\ &=-\frac {\sqrt {3} \sqrt [3]{e} q \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} (f+g x)}{\sqrt [3]{e}}}{\sqrt {3}}\right )}{\sqrt [3]{d} g}+\frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}+\frac {\sqrt [3]{e} q \log \left (\sqrt [3]{e}+\sqrt [3]{d} (f+g x)\right )}{\sqrt [3]{d} g}-\frac {\sqrt [3]{e} q \log \left (e^{2/3}-\sqrt [3]{d} \sqrt [3]{e} (f+g x)+d^{2/3} (f+g x)^2\right )}{2 \sqrt [3]{d} g}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 66, normalized size = 0.40 \[ \frac {(f+g x) \log \left (c \left (d+\frac {e}{(f+g x)^3}\right )^q\right )}{g}-\frac {3 e q \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {e}{d (f+g x)^3}\right )}{2 d g (f+g x)^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 3.90, size = 1392, normalized size = 8.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.48, size = 157, normalized size = 0.95 \[ -\frac {3 f q \ln \left (g x +f \right )}{g}+x \ln \left (c \left (\frac {d \,g^{3} x^{3}+3 d f \,g^{2} x^{2}+3 d \,f^{2} g x +d \,f^{3}+e}{\left (g x +f \right )^{3}}\right )^{q}\right )+\frac {q \left (\RootOf \left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )^{2} d f \,g^{2}+2 \RootOf \left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right ) d \,f^{2} g +d \,f^{3}+e \right ) \ln \left (-\RootOf \left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )+x \right )}{d g \left (g^{2} \RootOf \left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )^{2}+2 f g \RootOf \left (d \,g^{3} \textit {\_Z}^{3}+3 d f \,g^{2} \textit {\_Z}^{2}+3 d \,f^{2} g \textit {\_Z} +d \,f^{3}+e \right )+f^{2}\right )} \]
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 \[ 3 \, q \int \frac {d f g^{2} x^{2} + 2 \, d f^{2} g x + d f^{3} + e}{d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e}\,{d x} - \frac {3 \, f q \log \left (g x + f\right ) - g x \log \left ({\left (d g^{3} x^{3} + 3 \, d f g^{2} x^{2} + 3 \, d f^{2} g x + d f^{3} + e\right )}^{q}\right ) + 3 \, g x \log \left ({\left (g x + f\right )}^{q}\right ) - g x \log \relax (c)}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 499, normalized size = 3.02 \[ x\,\ln \left (c\,{\left (d+\frac {e}{{\left (f+g\,x\right )}^3}\right )}^q\right )-\left (\sum _{k=1}^3\ln \left (-d^2\,e^2\,g^{11}\,\left (3\,e\,q^3\,x+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,e\,q^2+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^3\,d\,f\,g^2\,4+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,d\,f^3\,q^2\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^3\,d\,g^3\,x\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^2\,d\,f^2\,g\,q\,8+\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\,d\,f^2\,g\,q^2\,x\,4+{\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )}^2\,d\,f\,g^2\,q\,x\,8\right )\,9\right )\,\mathrm {root}\left (d\,g^3\,z^3+3\,d\,f\,g^2\,q\,z^2+3\,d\,f^2\,g\,q^2\,z+d\,f^3\,q^3+e\,q^3,z,k\right )\right )-\frac {3\,f\,q\,\ln \left (f+g\,x\right )}{g} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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