Optimal. Leaf size=77 \[ -\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}+a x-\frac {b n x}{3}+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2498, 272, 45}
\begin {gather*} a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}-\frac {b n x}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2498
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac {1}{3} (b e n) \int \frac {\sqrt [3]{x}}{d+e \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-(b e n) \text {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-(b e n) \text {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}+a x-\frac {b n x}{3}+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 77, normalized size = 1.00 \begin {gather*} -\frac {b d^2 n \sqrt [3]{x}}{e^2}+\frac {b d n x^{2/3}}{2 e}+a x-\frac {b n x}{3}+\frac {b d^3 n \log \left (d+e \sqrt [3]{x}\right )}{e^3}+b x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 66, normalized size = 0.86
method | result | size |
default | \(-\frac {b \,d^{2} n \,x^{\frac {1}{3}}}{e^{2}}+\frac {b d n \,x^{\frac {2}{3}}}{2 e}+a x -\frac {b n x}{3}+\frac {b \,d^{3} n \ln \left (d +e \,x^{\frac {1}{3}}\right )}{e^{3}}+b x \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 70, normalized size = 0.91 \begin {gather*} \frac {1}{6} \, {\left ({\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (3 \, d x^{\frac {2}{3}} e - 6 \, d^{2} x^{\frac {1}{3}} - 2 \, x e^{2}\right )} e^{\left (-3\right )}\right )} n e + 6 \, x \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 71, normalized size = 0.92 \begin {gather*} -\frac {1}{6} \, {\left (6 \, b d^{2} n x^{\frac {1}{3}} e - 3 \, b d n x^{\frac {2}{3}} e^{2} - 6 \, b x e^{3} \log \left (c\right ) + 2 \, {\left (b n - 3 \, a\right )} x e^{3} - 6 \, {\left (b d^{3} n + b n x e^{3}\right )} \log \left (x^{\frac {1}{3}} e + d\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 82, normalized size = 1.06 \begin {gather*} a x + b \left (- \frac {e n \left (- \frac {3 d^{3} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {3 d^{2} \sqrt [3]{x}}{e^{3}} - \frac {3 d x^{\frac {2}{3}}}{2 e^{2}} + \frac {x}{e}\right )}{3} + x \log {\left (c \left (d + 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs.
\(2 (64) = 128\).
time = 4.38, size = 135, normalized size = 1.75 \begin {gather*} \frac {1}{6} \, {\left (6 \, x e \log \left (c\right ) + {\left (6 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 18 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 18 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 2 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} + 9 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} - 18 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )}\right )} n\right )} b e^{\left (-1\right )} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 65, normalized size = 0.84 \begin {gather*} a\,x+b\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {b\,n\,x}{3}+\frac {b\,d\,n\,x^{2/3}}{2\,e}+\frac {b\,d^3\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^3}-\frac {b\,d^2\,n\,x^{1/3}}{e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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