Optimal. Leaf size=227 \[ -\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {2 b e^2 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2} \]
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Rubi [A] time = 0.53, antiderivative size = 248, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {2451, 2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac {2 b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {e}{d \sqrt [3]{x}}+1\right )}{d^3}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}-\frac {2 b e^3 n \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {2 b e^2 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2451
Rule 2454
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx &=3 \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {(2 b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (2 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {\left (2 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (2 b e^3 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (b^2 e n^2\right ) \operatorname {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\left (2 b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {\left (2 b^2 e^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {2 b^2 e^3 n^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 237, normalized size = 1.04 \[ x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b e n \left (-3 d^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-6 e^2 \log \left (d \sqrt [3]{x}+e\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )+6 a d e \sqrt [3]{x}+6 b d e \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3 b e^2 n \left (\log \left (d \sqrt [3]{x}+e\right ) \left (\log \left (d \sqrt [3]{x}+e\right )-2 \log \left (-\frac {d \sqrt [3]{x}}{e}\right )\right )-2 \text {Li}_2\left (\frac {\sqrt [3]{x} d}{e}+1\right )\right )+2 b e^2 n \left (3 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+\log (x)\right )+3 b e n \left (e \log \left (d \sqrt [3]{x}+e\right )-d \sqrt [3]{x}\right )\right )}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \log \left (c \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{n}\right ) + a^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )+a \right )^{2}\, dx \]
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 \[ {\left (e n {\left (\frac {2 \, e^{2} \log \left (d x^{\frac {1}{3}} + e\right )}{d^{3}} + \frac {d x^{\frac {2}{3}} - 2 \, e x^{\frac {1}{3}}}{d^{2}}\right )} + 2 \, x \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )\right )} a b + {\left (x \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right )^{2} - \int -\frac {3 \, d x \log \relax (c)^{2} + 3 \, e x^{\frac {2}{3}} \log \relax (c)^{2} + 3 \, {\left (d x + e x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 2 \, {\left (d n x - 3 \, d x \log \relax (c) - 3 \, e x^{\frac {2}{3}} \log \relax (c) + 3 \, {\left (d x + e x^{\frac {2}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac {1}{3}} + e\right )}^{n}\right ) - 6 \, {\left (d x \log \relax (c) + e x^{\frac {2}{3}} \log \relax (c)\right )} \log \left (x^{\frac {1}{3} \, n}\right )}{3 \, {\left (d x + e x^{\frac {2}{3}}\right )}}\,{d x}\right )} b^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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