Optimal. Leaf size=480 \[ -\frac {b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}+\frac {6 b d^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}+\frac {6 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{6 e^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6} \]
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Rubi [A] time = 0.46, antiderivative size = 355, normalized size of antiderivative = 0.74, 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 = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2301
Rule 2334
Rule 2398
Rule 2411
Rule 2454
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b e n) \operatorname {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(b n) \operatorname {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=\frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=\frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6}\\ &=\frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac {60 d^6 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{60 e^6}\\ &=\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 d^6 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}\\ &=\frac {15 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}-\frac {6 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6}{36 e^6}-\frac {6 b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac {1}{60} b n \left (\frac {360 d^5 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt [3]{x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt [3]{x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt [3]{x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt [3]{x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {1}{2} x^2 \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 = 0.39, size = 301, normalized size = 0.63 \[ \frac {e \sqrt [3]{x} \left (1800 a^2 e^5 x^{5/3}+60 a b n \left (60 d^5-30 d^4 e \sqrt [3]{x}+20 d^3 e^2 x^{2/3}-15 d^2 e^3 x+12 d e^4 x^{4/3}-10 e^5 x^{5/3}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt [3]{x}-1140 d^3 e^2 x^{2/3}+555 d^2 e^3 x-264 d e^4 x^{4/3}+100 e^5 x^{5/3}\right )\right )-60 b \left (60 a \left (d^6-e^6 x^2\right )+b n \left (-147 d^6-60 d^5 e \sqrt [3]{x}+30 d^4 e^2 x^{2/3}-20 d^3 e^3 x+15 d^2 e^4 x^{4/3}-12 d e^5 x^{5/3}+10 e^6 x^2\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^2\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3600 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 484, normalized size = 1.01 \[ \frac {1800 \, b^{2} e^{6} x^{2} \log \relax (c)^{2} + 100 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{2} + 1800 \, {\left (b^{2} e^{6} n^{2} x^{2} - b^{2} d^{6} n^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x + 147 \, b^{2} d^{6} n^{2} - 60 \, a b d^{6} n - 10 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{2} + 60 \, {\left (b^{2} e^{6} n x^{2} - b^{2} d^{6} n\right )} \log \relax (c) + 6 \, {\left (2 \, b^{2} d e^{5} n^{2} x - 5 \, b^{2} d^{4} e^{2} n^{2}\right )} x^{\frac {2}{3}} - 15 \, {\left (b^{2} d^{2} e^{4} n^{2} x - 4 \, b^{2} d^{5} e n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x - {\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{2}\right )} \log \relax (c) + 6 \, {\left (435 \, b^{2} d^{4} e^{2} n^{2} - 300 \, a b d^{4} e^{2} n - 4 \, {\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x + 60 \, {\left (2 \, b^{2} d e^{5} n x - 5 \, b^{2} d^{4} e^{2} n\right )} \log \relax (c)\right )} x^{\frac {2}{3}} - 15 \, {\left (588 \, b^{2} d^{5} e n^{2} - 240 \, a b d^{5} e n - {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x + 60 \, {\left (b^{2} d^{2} e^{4} n x - 4 \, b^{2} d^{5} e n\right )} \log \relax (c)\right )} x^{\frac {1}{3}}}{3600 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 956, normalized size = 1.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+a \right )^{2} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 323, normalized size = 0.67 \[ \frac {1}{2} \, b^{2} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {1}{60} \, a b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} + a b x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{3600} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{6}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 431, normalized size = 0.90 \[ \frac {a^2\,x^2}{2}+\frac {b^2\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2}+\frac {b^2\,n^2\,x^2}{36}+a\,b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^6}-\frac {a\,b\,n\,x^2}{6}-\frac {b^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{20\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^{4/3}}{240\,e^2}+\frac {29\,b^2\,d^4\,n^2\,x^{2/3}}{40\,e^4}-\frac {49\,b^2\,d^5\,n^2\,x^{1/3}}{20\,e^5}-\frac {19\,b^2\,d^3\,n^2\,x}{60\,e^3}-\frac {11\,b^2\,d\,n^2\,x^{5/3}}{150\,e}-\frac {b^2\,d^2\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e^2}-\frac {b^2\,d^4\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^4}+\frac {b^2\,d^5\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^5}+\frac {a\,b\,d^3\,n\,x}{3\,e^3}+\frac {a\,b\,d\,n\,x^{5/3}}{5\,e}-\frac {a\,b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^6}+\frac {b^2\,d^3\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^3}+\frac {b^2\,d\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e}-\frac {a\,b\,d^2\,n\,x^{4/3}}{4\,e^2}-\frac {a\,b\,d^4\,n\,x^{2/3}}{2\,e^4}+\frac {a\,b\,d^5\,n\,x^{1/3}}{e^5} \]
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 \[ \int x \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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