Optimal. Leaf size=267 \[ -\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \]
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Rubi [A]
time = 0.20, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2445,
2458, 45, 2372, 12, 14, 2338} \begin {gather*} \frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2501
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac {1}{3} b n \left (\frac {18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac {2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac {1}{3} b n \left (\frac {18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac {2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3}\\ &=-\frac {1}{3} b n \left (\frac {18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac {2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3}\\ &=-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {1}{3} b n \left (\frac {18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac {2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {\left (2 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}\\ &=-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {1}{3} b n \left (\frac {18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac {2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac {6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 249, normalized size = 0.93 \begin {gather*} \frac {-36 a b d^2 e n \sqrt [3]{x}+66 b^2 d^2 e n^2 \sqrt [3]{x}+18 a b d e^2 n x^{2/3}-15 b^2 d e^2 n^2 x^{2/3}+18 a^2 e^3 x-12 a b e^3 n x+4 b^2 e^3 n^2 x-18 b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )-6 b \left (-6 a e^3 x+b \left (6 d^2 e n \sqrt [3]{x}-3 d e^2 n x^{2/3}+2 e^3 n x\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+18 b^2 e^3 x \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+6 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{18 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 217, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, {\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 )} a b - \frac {1}{18} \, {\left ({\left (18 \, d^{3} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 66 \, d^{3} \log \left (x^{\frac {1}{3}} e + d\right ) - 66 \, d^{2} x^{\frac {1}{3}} e + 15 \, d x^{\frac {2}{3}} e^{2} - 4 \, x e^{3}\right )} n^{2} e^{\left (-3\right )} - 6 \, {\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 \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) - 18 \, x \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{2}\right )} b^{2} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 272, normalized size = 1.02 \begin {gather*} \frac {1}{18} \, {\left (18 \, b^{2} x e^{3} \log \left (c\right )^{2} - 12 \, {\left (b^{2} n - 3 \, a b\right )} x e^{3} \log \left (c\right ) + 2 \, {\left (2 \, b^{2} n^{2} - 6 \, a b n + 9 \, a^{2}\right )} x e^{3} + 18 \, {\left (b^{2} d^{3} n^{2} + b^{2} n^{2} x e^{3}\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 6 \, {\left (6 \, b^{2} d^{2} n^{2} x^{\frac {1}{3}} e + 11 \, b^{2} d^{3} n^{2} - 3 \, b^{2} d n^{2} x^{\frac {2}{3}} e^{2} - 6 \, a b d^{3} n + 2 \, {\left (b^{2} n^{2} - 3 \, a b n\right )} x e^{3} - 6 \, {\left (b^{2} d^{3} n + b^{2} n x e^{3}\right )} \log \left (c\right )\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 3 \, {\left (6 \, b^{2} d n e^{2} \log \left (c\right ) - {\left (5 \, b^{2} d n^{2} - 6 \, a b d n\right )} e^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (6 \, b^{2} d^{2} n e \log \left (c\right ) - {\left (11 \, b^{2} d^{2} n^{2} - 6 \, a b d^{2} n\right )} e\right )} x^{\frac {1}{3}}\right )} e^{\left (-3\right )} \end {gather*}
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 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \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. 479 vs.
\(2 (239) = 478\).
time = 4.15, size = 479, normalized size = 1.79 \begin {gather*} \frac {1}{18} \, {\left (18 \, b^{2} x e \log \left (c\right )^{2} + {\left (18 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 54 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 54 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 12 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 54 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 108 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 4 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} e^{\left (-2\right )} - 27 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d e^{\left (-2\right )} + 108 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{2} e^{\left (-2\right )}\right )} b^{2} n^{2} + 6 \, {\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 )} b^{2} n \log \left (c\right ) + 36 \, a b x e \log \left (c\right ) + 6 \, {\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 )} a b n + 18 \, a^{2} x e\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.51, size = 290, normalized size = 1.09 \begin {gather*} \ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {2\,b\,x\,\left (3\,a-b\,n\right )}{3}-x^{2/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )+\frac {d\,x^{1/3}\,\left (\frac {2\,b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {6\,a\,b\,d}{e}\right )}{e}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{e}\right )}{e}+\frac {2\,b^2\,d^2\,n^2}{e^2}\right )+x\,\left (a^2-\frac {2\,a\,b\,n}{3}+\frac {2\,b^2\,n^2}{9}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d^3}{e^3}\right )-\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{3\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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