Optimal. Leaf size=168 \[ \frac {e^4 (c+d x)^2}{2 b d}+\frac {a^{2/3} e^4 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d} \]
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Rubi [A]
time = 0.09, antiderivative size = 168, 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 = {379, 327, 298,
31, 648, 631, 210, 642} \begin {gather*} \frac {a^{2/3} e^4 \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac {e^4 (c+d x)^2}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 327
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^4}{a+b (c+d x)^3} \, dx &=\frac {e^4 \text {Subst}\left (\int \frac {x^4}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^2}{2 b d}-\frac {\left (a e^4\right ) \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{b d}\\ &=\frac {e^4 (c+d x)^2}{2 b d}+\frac {\left (a^{2/3} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b^{4/3} d}-\frac {\left (a^{2/3} e^4\right ) \text {Subst}\left (\int \frac {\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^{4/3} d}\\ &=\frac {e^4 (c+d x)^2}{2 b d}+\frac {a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {\left (a^{2/3} e^4\right ) \text {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^{5/3} d}-\frac {\left (a e^4\right ) \text {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^{4/3} d}\\ &=\frac {e^4 (c+d x)^2}{2 b d}+\frac {a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}-\frac {\left (a^{2/3} e^4\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{b^{5/3} d}\\ &=\frac {e^4 (c+d x)^2}{2 b d}+\frac {a^{2/3} e^4 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 163, normalized size = 0.97 \begin {gather*} e^4 \left (\frac {(c+d x)^2}{2 b d}-\frac {a^{2/3} \tan ^{-1}\left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.23, size = 96, normalized size = 0.57
method | result | size |
default | \(e^{4} \left (\frac {\frac {1}{2} d \,x^{2}+c x}{b}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\right )\) | \(96\) |
risch | \(\frac {e^{4} d \,x^{2}}{2 b}+\frac {e^{4} c x}{b}+\frac {a \,e^{4} \left (\munderset {\textit {\_R} =\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-\textit {\_R} d -c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{3 b^{2} d}\) | \(105\) |
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 [A]
time = 0.39, size = 179, normalized size = 1.07 \begin {gather*} \frac {3 \, d^{2} e^{4} x^{2} + 6 \, c d e^{4} x - 2 \, \sqrt {3} e^{4} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b d x + b c\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) - e^{4} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a d^{2} x^{2} + 2 \, a c d x + a c^{2} - {\left (b d x + b c\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, e^{4} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a d x + a c + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{6 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 66, normalized size = 0.39 \begin {gather*} \frac {e^{4} \operatorname {RootSum} {\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} b^{3} e^{8} + a c e^{8}}{a d e^{8}} \right )} \right )\right )}}{d} + \frac {c e^{4} x}{b} + \frac {d e^{4} x^{2}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 170, normalized size = 1.01 \begin {gather*} \frac {b d^{7} x^{2} e^{4} + 2 \, b c d^{6} x e^{4}}{2 \, b^{2} d^{6}} - \frac {2 \, \sqrt {3} \left (a^{2} b d^{15}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (a^{2} b\right )^{\frac {2}{3}}}\right ) e^{4} + \left (a^{2} b d^{15}\right )^{\frac {1}{3}} e^{4} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (a^{2} b d^{15}\right )^{\frac {1}{3}} e^{4} \log \left ({\left | a b d x + a b c + \left (a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{6 \, b^{2} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.70, size = 151, normalized size = 0.90 \begin {gather*} \frac {d\,e^4\,x^2}{2\,b}+\frac {c\,e^4\,x}{b}+\frac {a^{2/3}\,e^4\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{3\,b^{5/3}\,d}+\frac {a^{2/3}\,e^4\,\ln \left (a^{1/3}-2\,b^{1/3}\,c-2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{5/3}\,d}-\frac {a^{2/3}\,e^4\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{5/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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