Optimal. Leaf size=156 \[ \frac {\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac {\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} e^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} d}+\frac {e^3 x}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {372, 321, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac {\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} e^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} d}+\frac {e^3 x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 321
Rule 372
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{a+b (c+d x)^3} \, dx &=\frac {e^3 \operatorname {Subst}\left (\int \frac {x^3}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 x}{b}-\frac {\left (a e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{b d}\\ &=\frac {e^3 x}{b}-\frac {\left (\sqrt [3]{a} e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b d}-\frac {\left (\sqrt [3]{a} e^3\right ) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 b d}\\ &=\frac {e^3 x}{b}-\frac {\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\left (\sqrt [3]{a} e^3\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{6 b^{4/3} d}-\frac {\left (a^{2/3} e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{2 b d}\\ &=\frac {e^3 x}{b}-\frac {\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac {\left (\sqrt [3]{a} e^3\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{b^{4/3} d}\\ &=\frac {e^3 x}{b}+\frac {\sqrt [3]{a} e^3 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 145, normalized size = 0.93 \begin {gather*} \frac {e^3 \left (\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )+6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x\right )}{6 b^{4/3} d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c e+d e x)^3}{a+b (c+d x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.70, size = 147, normalized size = 0.94 \begin {gather*} \frac {6 \, d e^{3} x + 2 \, \sqrt {3} e^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b d x + b c\right )} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - e^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (d x + c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, e^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d x + c - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 188, normalized size = 1.21 \begin {gather*} \frac {x e^{3}}{b} + \frac {2 \, \sqrt {3} \left (-\frac {a d^{6} e^{9}}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {a d^{6} e^{9}}{b}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {a d^{6} e^{9}}{b}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 84, normalized size = 0.54 \begin {gather*} \frac {e^{3} x}{b}-\frac {a \,e^{3} \ln \left (-\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+x \right )}{3 b^{2} d \left (d^{2} \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )^{2}+2 c d \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+c^{2}\right )} \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*} -\frac {\frac {1}{6} \, a e^{3} {\left (\frac {2 \, \sqrt {3} \left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right )}{d} - \frac {\left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right )}{d} + \frac {2 \, \left (\frac {1}{a^{2} b}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{d}\right )}}{b} + \frac {e^{3} x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 162, normalized size = 1.04 \begin {gather*} \frac {e^3\,x}{b}+\frac {{\left (-a\right )}^{1/3}\,e^3\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,c+a\,b^{1/3}\,d\,x\right )}{3\,b^{4/3}\,d}-\frac {{\left (-a\right )}^{1/3}\,e^3\,\ln \left (2\,a\,b^{1/3}\,c-{\left (-a\right )}^{4/3}+2\,a\,b^{1/3}\,d\,x-\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{4/3}\,d}+\frac {{\left (-a\right )}^{1/3}\,e^3\,\ln \left (2\,a\,b^{1/3}\,c-{\left (-a\right )}^{4/3}+2\,a\,b^{1/3}\,d\,x+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{4/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 44, normalized size = 0.28 \begin {gather*} \frac {e^{3} \operatorname {RootSum} {\left (27 t^{3} b^{4} + a, \left (t \mapsto t \log {\left (x + \frac {- 3 t b e^{3} + c e^{3}}{d e^{3}} \right )} \right )\right )}}{d} + \frac {e^{3} x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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