Optimal. Leaf size=70 \[ -\frac {b e^2 n \sqrt [3]{x}}{d^2}+\frac {b e n x^{2/3}}{2 d}+a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {b e^3 n \log \left (e+d \sqrt [3]{x}\right )}{d^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2498, 269, 196,
45} \begin {gather*} a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {b e^3 n \log \left (d \sqrt [3]{x}+e\right )}{d^3}-\frac {b e^2 n \sqrt [3]{x}}{d^2}+\frac {b e n x^{2/3}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 269
Rule 2498
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {1}{3} (b e n) \int \frac {1}{\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {1}{3} (b e n) \int \frac {1}{e+d \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \text {Subst}\left (\int \frac {x^2}{e+d x} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \text {Subst}\left (\int \left (-\frac {e}{d^2}+\frac {x}{d}+\frac {e^2}{d^2 (e+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {b e^2 n \sqrt [3]{x}}{d^2}+\frac {b e n x^{2/3}}{2 d}+a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+\frac {b e^3 n \log \left (e+d \sqrt [3]{x}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 79, normalized size = 1.13 \begin {gather*} a x+b x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-b e n \left (\frac {e \sqrt [3]{x}}{d^2}-\frac {x^{2/3}}{2 d}-\frac {e^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {e^2 \log (x)}{3 d^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 115, normalized size = 1.64
method | result | size |
default | \(a x +x b \ln \left (c \left (\frac {e +d \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right )^{n}\right )+\frac {b \,e^{3} n \ln \left (d^{3} x +e^{3}\right )}{3 d^{3}}+\frac {b e n \,x^{\frac {2}{3}}}{2 d}-\frac {b \,e^{3} n \ln \left (d^{2} x^{\frac {2}{3}}-e d \,x^{\frac {1}{3}}+e^{2}\right )}{3 d^{3}}+\frac {2 b \,e^{3} n \ln \left (e +d \,x^{\frac {1}{3}}\right )}{3 d^{3}}-\frac {b \,e^{2} n \,x^{\frac {1}{3}}}{d^{2}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 62, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, {\left (n {\left (\frac {d x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} e}{d^{2}} + \frac {2 \, e^{2} \log \left (d x^{\frac {1}{3}} + e\right )}{d^{3}}\right )} e + 2 \, x \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 109, normalized size = 1.56 \begin {gather*} \frac {2 \, b d^{3} x \log \left (c\right ) - 2 \, b d^{3} n \log \left (x^{\frac {1}{3}}\right ) + b d^{2} n x^{\frac {2}{3}} e + 2 \, a d^{3} x - 2 \, b d n x^{\frac {1}{3}} e^{2} + 2 \, {\left (b d^{3} n + b n e^{3}\right )} \log \left (d x^{\frac {1}{3}} + e\right ) + 2 \, {\left (b d^{3} n x - b d^{3} n\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right )}{2 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.67, size = 92, normalized size = 1.31 \begin {gather*} a x + b \left (\frac {e n \left (\frac {3 x^{\frac {2}{3}}}{2 d} - \frac {3 e \sqrt [3]{x}}{d^{2}} + \frac {3 e^{3} \left (\begin {cases} \frac {1}{d \sqrt [3]{x}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{\sqrt [3]{x}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} - \frac {3 e^{2} \log {\left (\frac {1}{\sqrt [3]{x}} \right )}}{d^{3}}\right )}{3} + x \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.64, size = 66, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, {\left ({\left ({\left (\frac {d x^{\frac {2}{3}} - 2 \, x^{\frac {1}{3}} e}{d^{2}} + \frac {2 \, e^{2} \log \left ({\left | d x^{\frac {1}{3}} + e \right |}\right )}{d^{3}}\right )} e + 2 \, x \log \left (d + \frac {e}{x^{\frac {1}{3}}}\right )\right )} n + 2 \, x \log \left (c\right )\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 59, normalized size = 0.84 \begin {gather*} a\,x+b\,x\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )+\frac {b\,\left (2\,e^3\,n\,\ln \left (e+d\,x^{1/3}\right )-2\,d\,e^2\,n\,x^{1/3}+d^2\,e\,n\,x^{2/3}\right )}{2\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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