Optimal. Leaf size=176 \[ -\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac {e \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3} d}+\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {372, 290, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}-\frac {e \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3} d}+\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 290
Rule 292
Rule 372
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac {e \operatorname {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}+\frac {e \operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {e \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 a^{4/3} \sqrt [3]{b} d}+\frac {e \operatorname {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 )}{9 a^{4/3} \sqrt [3]{b} d}\\ &=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \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 )}{18 a^{4/3} b^{2/3} d}+\frac {e \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 )}{6 a \sqrt [3]{b} d}\\ &=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}+\frac {e \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{4/3} b^{2/3} d}\\ &=\frac {e (c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {e \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3} d}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 153, normalized size = 0.87 \begin {gather*} \frac {e \left (\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac {6 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}\right )}{18 a^{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}{\left (a+b (c+d x)^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.02, size = 894, normalized size = 5.08 \begin {gather*} \left [\frac {6 \, a b^{2} d^{2} e x^{2} + 12 \, a b^{2} c d e x + 6 \, a b^{2} c^{2} e + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d^{3} e x^{3} + 3 \, a b^{2} c d^{2} e x^{2} + 3 \, a b^{2} c^{2} d e x + {\left (a b^{2} c^{3} + a^{2} b\right )} e\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} d^{3} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 6 \, b^{2} c^{2} d x + 2 \, b^{2} c^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b d x + a b c + 2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} {\left (d x + c\right )}}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\right ) + {\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x + {\left (b c^{3} + a\right )} e\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (b d x + b c\right )} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x + {\left (b c^{3} + a\right )} e\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{2} b^{3} d^{4} x^{3} + 3 \, a^{2} b^{3} c d^{3} x^{2} + 3 \, a^{2} b^{3} c^{2} d^{2} x + {\left (a^{2} b^{3} c^{3} + a^{3} b^{2}\right )} d\right )}}, \frac {6 \, a b^{2} d^{2} e x^{2} + 12 \, a b^{2} c d e x + 6 \, a b^{2} c^{2} e + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d^{3} e x^{3} + 3 \, a b^{2} c d^{2} e x^{2} + 3 \, a b^{2} c^{2} d e x + {\left (a b^{2} c^{3} + a^{2} b\right )} e\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b d x + 2 \, b c + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x + {\left (b c^{3} + a\right )} e\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} {\left (b d x + b c\right )} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b d^{3} e x^{3} + 3 \, b c d^{2} e x^{2} + 3 \, b c^{2} d e x + {\left (b c^{3} + a\right )} e\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{2} b^{3} d^{4} x^{3} + 3 \, a^{2} b^{3} c d^{3} x^{2} + 3 \, a^{2} b^{3} c^{2} d^{2} x + {\left (a^{2} b^{3} c^{3} + a^{3} b^{2}\right )} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 218, normalized size = 1.24 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-\frac {e^{3}}{a b^{2} d^{3}}\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 ) + \left (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \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 (-\frac {e^{3}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{18 \, a} + \frac {d^{2} x^{2} e + 2 \, c d x e + c^{2} e}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 216, normalized size = 1.23 \begin {gather*} \frac {d e \,x^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a}+\frac {2 c e x}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a}+\frac {c^{2} e}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a d}+\frac {e \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 ) d +c \right ) \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 )}{9 a b 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} \, {\left (2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\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 ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \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 (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )\right )} e}{3 \, a} + \frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{3 \, {\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x + {\left (a b c^{3} + a^{2}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 192, normalized size = 1.09 \begin {gather*} \frac {\frac {c^2\,e}{3\,a\,d}+\frac {2\,c\,e\,x}{3\,a}+\frac {d\,e\,x^2}{3\,a}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}-\frac {e\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{4/3}\,b^{2/3}\,d}+\frac {\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 (e-\sqrt {3}\,e\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d}+\frac {\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 (e+\sqrt {3}\,e\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.37, size = 122, normalized size = 0.69 \begin {gather*} \frac {c^{2} e + 2 c d e x + d^{2} e x^{2}}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {e \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{2} + 1, \left (t \mapsto t \log {\left (x + \frac {81 t^{2} a^{3} b e^{2} + c e^{2}}{d e^{2}} \right )} \right )\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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