Optimal. Leaf size=284 \[ \frac{3 b^2 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^2}-\frac{12 a b^2 d n^2 \sqrt{x}}{e}-\frac{3 b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^2}+\frac{6 b d n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{12 b^3 d n^2 \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^2}-\frac{3 b^3 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^2}+\frac{12 b^3 d n^3 \sqrt{x}}{e} \]
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Rubi [A] time = 0.251127, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2451, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{3 b^2 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^2}-\frac{12 a b^2 d n^2 \sqrt{x}}{e}-\frac{3 b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^2}+\frac{6 b d n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{12 b^3 d n^2 \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^2}-\frac{3 b^3 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^2}+\frac{12 b^3 d n^3 \sqrt{x}}{e} \]
Antiderivative was successfully verified.
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Rule 2451
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \, dx &=2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )}{e}-\frac{(2 d) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^2}-\frac{(2 d) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^2}\\ &=-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{(3 b n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt{x}\right )}{e^2}+\frac{(6 b d n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt{x}\right )}{e^2}\\ &=\frac{6 b d n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}-\frac{3 b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}+\frac{\left (3 b^2 n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt{x}\right )}{e^2}-\frac{\left (12 b^2 d n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt{x}\right )}{e^2}\\ &=-\frac{3 b^3 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^2}-\frac{12 a b^2 d n^2 \sqrt{x}}{e}+\frac{3 b^2 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^2}+\frac{6 b d n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}-\frac{3 b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}-\frac{\left (12 b^3 d n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt{x}\right )}{e^2}\\ &=-\frac{3 b^3 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^2}-\frac{12 a b^2 d n^2 \sqrt{x}}{e}+\frac{12 b^3 d n^3 \sqrt{x}}{e}-\frac{12 b^3 d n^2 \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^2}+\frac{3 b^2 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^2}+\frac{6 b d n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^2}-\frac{3 b n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^2}-\frac{2 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}+\frac{\left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^2}\\ \end{align*}
Mathematica [A] time = 0.202161, size = 241, normalized size = 0.85 \[ \frac{4 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3-8 d \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3+24 b d n \left (\left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-2 b n \left (e \sqrt{x} (a-b n)+b \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )\right )-3 b n \left (2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+b n \left (b e n \left (2 d \sqrt{x}+e x\right )-2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )\right )\right )}{4 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12399, size = 514, normalized size = 1.81 \begin{align*} -\frac{3}{2} \,{\left (e n{\left (\frac{2 \, d^{2} \log \left (e \sqrt{x} + d\right )}{e^{3}} + \frac{e x - 2 \, d \sqrt{x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )\right )} a^{2} b - \frac{3}{2} \,{\left (2 \, e n{\left (\frac{2 \, d^{2} \log \left (e \sqrt{x} + d\right )}{e^{3}} + \frac{e x - 2 \, d \sqrt{x}}{e^{2}}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} - \frac{{\left (2 \, d^{2} \log \left (e \sqrt{x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt{x} + d\right ) - 6 \, d e \sqrt{x}\right )} n^{2}}{e^{2}}\right )} a b^{2} - \frac{1}{4} \,{\left (6 \, e n{\left (\frac{2 \, d^{2} \log \left (e \sqrt{x} + d\right )}{e^{3}} + \frac{e x - 2 \, d \sqrt{x}}{e^{2}}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} - 4 \, x \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{3} + e n{\left (\frac{{\left (4 \, d^{2} \log \left (e \sqrt{x} + d\right )^{3} + 18 \, d^{2} \log \left (e \sqrt{x} + d\right )^{2} + 3 \, e^{2} x + 42 \, d^{2} \log \left (e \sqrt{x} + d\right ) - 42 \, d e \sqrt{x}\right )} n^{2}}{e^{3}} - \frac{6 \,{\left (2 \, d^{2} \log \left (e \sqrt{x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt{x} + d\right ) - 6 \, d e \sqrt{x}\right )} n \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )}{e^{3}}\right )}\right )} b^{3} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91796, size = 1146, normalized size = 4.04 \begin{align*} \frac{4 \, b^{3} e^{2} x \log \left (c\right )^{3} + 4 \,{\left (b^{3} e^{2} n^{3} x - b^{3} d^{2} n^{3}\right )} \log \left (e \sqrt{x} + d\right )^{3} - 6 \,{\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} x \log \left (c\right )^{2} + 6 \,{\left (2 \, b^{3} d e n^{3} \sqrt{x} + 3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2} -{\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2}\right )} x + 2 \,{\left (b^{3} e^{2} n^{2} x - b^{3} d^{2} n^{2}\right )} \log \left (c\right )\right )} \log \left (e \sqrt{x} + d\right )^{2} + 6 \,{\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} x \log \left (c\right ) -{\left (3 \, b^{3} e^{2} n^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2}\right )} x - 6 \,{\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n - 2 \,{\left (b^{3} e^{2} n x - b^{3} d^{2} n\right )} \log \left (c\right )^{2} -{\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n\right )} x - 2 \,{\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n -{\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n\right )} x\right )} \log \left (c\right ) + 2 \,{\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \left (c\right ) - 2 \, a b^{2} d e n^{2}\right )} \sqrt{x}\right )} \log \left (e \sqrt{x} + d\right ) + 6 \,{\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \left (c\right )^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \,{\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \left (c\right )\right )} \sqrt{x}}{4 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17528, size = 1030, normalized size = 3.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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