Optimal. Leaf size=479 \[ \frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6} \]
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Rubi [A] time = 0.48, antiderivative size = 355, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6} \]
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 \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx &=-\left (3 \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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,\frac {1}{\sqrt [3]{x}}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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+\frac {e}{\sqrt [3]{x}}\right )\\ &=-\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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+\frac {e}{\sqrt [3]{x}}\right )\\ &=-\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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+\frac {e}{\sqrt [3]{x}}\right )}{60 e^6}\\ &=-\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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+\frac {e}{\sqrt [3]{x}}\right )}{60 e^6}\\ &=-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 d^6 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\\ &=-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {1}{60} b n \left (\frac {360 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {450 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{e^6}+\frac {400 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{e^6}-\frac {225 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{e^6}+\frac {72 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [C] time = 0.35, size = 698, normalized size = 1.46 \[ \frac {-1800 a^2 e^6-3600 a b e^6 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3600 a b d^6 n x^2 \log \left (d \sqrt [3]{x}+e\right )+3600 a b d^6 n x^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )-3600 a b d^5 e n x^{5/3}+1800 a b d^4 e^2 n x^{4/3}-1200 a b d^3 e^3 n x+900 a b d^2 e^4 n x^{2/3}-720 a b d e^5 n \sqrt [3]{x}+600 a b e^6 n-3600 b^2 d^6 n x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3600 b^2 d^6 n x^2 \log \left (d \sqrt [3]{x}+e\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3600 b^2 d^6 n x^2 \log \left (-\frac {e}{d \sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-3600 b^2 d^5 e n x^{5/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1800 b^2 d^4 e^2 n x^{4/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-1200 b^2 d^3 e^3 n x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+900 b^2 d^2 e^4 n x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-1800 b^2 e^6 \log ^2\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+600 b^2 e^6 n \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-720 b^2 d e^5 n \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3600 b^2 d^6 n^2 x^2 \text {Li}_2\left (\frac {e}{d \sqrt [3]{x}}+1\right )+3600 b^2 d^6 n^2 x^2 \text {Li}_2\left (\frac {\sqrt [3]{x} d}{e}+1\right )-1800 b^2 d^6 n^2 x^2 \log ^2\left (d \sqrt [3]{x}+e\right )-5220 b^2 d^6 n^2 x^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+3600 b^2 d^6 n^2 x^2 \log \left (d \sqrt [3]{x}+e\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )+8820 b^2 d^5 e n^2 x^{5/3}-2610 b^2 d^4 e^2 n^2 x^{4/3}+1140 b^2 d^3 e^3 n^2 x-555 b^2 d^2 e^4 n^2 x^{2/3}+264 b^2 d e^5 n^2 \sqrt [3]{x}-100 b^2 e^6 n^2}{3600 e^6 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 597, normalized size = 1.25 \[ -\frac {100 \, b^{2} e^{6} n^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} - 20 \, {\left (90 \, a^{2} e^{6} - {\left (57 \, b^{2} d^{3} e^{3} - 5 \, b^{2} e^{6}\right )} n^{2} + 30 \, {\left (2 \, a b d^{3} e^{3} - a b e^{6}\right )} n\right )} x^{2} - 1800 \, {\left (b^{2} e^{6} x^{2} - b^{2} e^{6}\right )} \log \relax (c)^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{2} - b^{2} e^{6} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{2} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x - b^{2} e^{6} n + 6 \, a b e^{6} - {\left (6 \, a b e^{6} + {\left (2 \, b^{2} d^{3} e^{3} - b^{2} e^{6}\right )} n\right )} x^{2}\right )} \log \relax (c) + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x - 10 \, b^{2} e^{6} n^{2} + 60 \, a b e^{6} n + 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{2} - 60 \, {\left (b^{2} d^{6} n x^{2} - b^{2} e^{6} n\right )} \log \relax (c) + 15 \, {\left (4 \, b^{2} d^{5} e n^{2} x - b^{2} d^{2} e^{4} n^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (5 \, b^{2} d^{4} e^{2} n^{2} x - 2 \, b^{2} d e^{5} n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right ) + 15 \, {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n - 12 \, {\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x + 60 \, {\left (4 \, b^{2} d^{5} e n x - b^{2} d^{2} e^{4} n\right )} \log \relax (c)\right )} x^{\frac {2}{3}} - 6 \, {\left (44 \, b^{2} d e^{5} n^{2} - 120 \, a b d e^{5} n - 15 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x + 60 \, {\left (5 \, b^{2} d^{4} e^{2} n x - 2 \, b^{2} d e^{5} n\right )} \log \relax (c)\right )} x^{\frac {1}{3}}}{3600 \, e^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 1639, normalized size = 3.42 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )+a \right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 387, normalized size = 0.81 \[ \frac {1}{60} \, a b e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{7}} - \frac {20 \, d^{6} \log \relax (x)}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} e x^{\frac {4}{3}} + 20 \, d^{3} e^{2} x - 15 \, d^{2} e^{3} x^{\frac {2}{3}} + 12 \, d e^{4} x^{\frac {1}{3}} - 10 \, e^{5}}{e^{6} x^{2}}\right )} + \frac {1}{3600} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{7}} - \frac {20 \, d^{6} \log \relax (x)}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} e x^{\frac {4}{3}} + 20 \, d^{3} e^{2} x - 15 \, d^{2} e^{3} x^{\frac {2}{3}} + 12 \, d e^{4} x^{\frac {1}{3}} - 10 \, e^{5}}{e^{6} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{2} \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 200 \, d^{6} x^{2} \log \relax (x)^{2} - 2940 \, d^{6} x^{2} \log \relax (x) - 8820 \, d^{5} e x^{\frac {5}{3}} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 555 \, d^{2} e^{4} x^{\frac {2}{3}} - 264 \, d e^{5} x^{\frac {1}{3}} + 100 \, e^{6} - 60 \, {\left (20 \, d^{6} x^{2} \log \relax (x) - 147 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right )\right )} n^{2}}{e^{6} x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 439, normalized size = 0.92 \[ \frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,x^2}-\frac {b^2\,n^2}{36\,x^2}-\frac {a\,b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x^2}-\frac {a^2}{2\,x^2}+\frac {a\,b\,n}{6\,x^2}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{6\,x^2}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{20\,e^6}+\frac {19\,b^2\,d^3\,n^2}{60\,e^3\,x}-\frac {37\,b^2\,d^2\,n^2}{240\,e^2\,x^{4/3}}-\frac {29\,b^2\,d^4\,n^2}{40\,e^4\,x^{2/3}}+\frac {49\,b^2\,d^5\,n^2}{20\,e^5\,x^{1/3}}+\frac {11\,b^2\,d\,n^2}{150\,e\,x^{5/3}}-\frac {b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{3\,e^3\,x}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{4\,e^2\,x^{4/3}}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2\,e^4\,x^{2/3}}-\frac {b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{e^5\,x^{1/3}}-\frac {a\,b\,d\,n}{5\,e\,x^{5/3}}+\frac {a\,b\,d^6\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{e^6}-\frac {b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{5\,e\,x^{5/3}}-\frac {a\,b\,d^3\,n}{3\,e^3\,x}+\frac {a\,b\,d^2\,n}{4\,e^2\,x^{4/3}}+\frac {a\,b\,d^4\,n}{2\,e^4\,x^{2/3}}-\frac {a\,b\,d^5\,n}{e^5\,x^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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