Optimal. Leaf size=284 \[ -\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}+\frac {12 b^3 d n^3 \sqrt {x}}{e}-\frac {12 b^3 d n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2448,
2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2501
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx &=2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )\\ &=2 \text {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 \text {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) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e}\\ &=\frac {2 \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.14, size = 241, normalized size = 0.85 \begin {gather*} \frac {-8 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+4 \left (d+e \sqrt {x}\right )^2 \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 (a-b n) \sqrt {x}+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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 394, normalized size = 1.39 \begin {gather*} -\frac {3}{2} \, {\left ({\left (2 \, d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + {\left (x e - 2 \, d \sqrt {x}\right )} e^{\left (-2\right )}\right )} n e - 2 \, x \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} a^{2} b + \frac {3}{2} \, {\left ({\left (2 \, d^{2} \log \left (\sqrt {x} e + d\right )^{2} + 6 \, d^{2} \log \left (\sqrt {x} e + d\right ) - 6 \, d \sqrt {x} e + x e^{2}\right )} n^{2} e^{\left (-2\right )} - 2 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + {\left (x e - 2 \, d \sqrt {x}\right )} e^{\left (-2\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + 2 \, x \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2}\right )} a b^{2} - \frac {1}{4} \, {\left (6 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + {\left (x e - 2 \, d \sqrt {x}\right )} e^{\left (-2\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} - 4 \, x \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{3} + {\left ({\left (4 \, d^{2} \log \left (\sqrt {x} e + d\right )^{3} + 18 \, d^{2} \log \left (\sqrt {x} e + d\right )^{2} + 42 \, d^{2} \log \left (\sqrt {x} e + d\right ) - 42 \, d \sqrt {x} e + 3 \, x e^{2}\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (2 \, d^{2} \log \left (\sqrt {x} e + d\right )^{2} + 6 \, d^{2} \log \left (\sqrt {x} e + d\right ) - 6 \, d \sqrt {x} e + x e^{2}\right )} n e^{\left (-3\right )} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 496, normalized size = 1.75 \begin {gather*} \frac {1}{4} \, {\left (4 \, b^{3} x e^{2} \log \left (c\right )^{3} - 6 \, {\left (b^{3} n - 2 \, a b^{2}\right )} x e^{2} \log \left (c\right )^{2} - 4 \, {\left (b^{3} d^{2} n^{3} - b^{3} n^{3} x e^{2}\right )} \log \left (\sqrt {x} e + d\right )^{3} + 6 \, {\left (b^{3} n^{2} - 2 \, a b^{2} n + 2 \, a^{2} b\right )} x e^{2} \log \left (c\right ) - {\left (3 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 6 \, a^{2} b n - 4 \, a^{3}\right )} x e^{2} + 6 \, {\left (2 \, b^{3} d n^{3} \sqrt {x} e + 3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2} - {\left (b^{3} n^{3} - 2 \, a b^{2} n^{2}\right )} x e^{2} - 2 \, {\left (b^{3} d^{2} n^{2} - b^{3} n^{2} x e^{2}\right )} \log \left (c\right )\right )} \log \left (\sqrt {x} e + d\right )^{2} - 6 \, {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n - {\left (b^{3} n^{3} - 2 \, a b^{2} n^{2} + 2 \, a^{2} b n\right )} x e^{2} + 2 \, {\left (b^{3} d^{2} n - b^{3} n x e^{2}\right )} \log \left (c\right )^{2} - 2 \, {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n - {\left (b^{3} n^{2} - 2 \, a b^{2} n\right )} x e^{2}\right )} \log \left (c\right ) - 2 \, {\left (2 \, b^{3} d n^{2} e \log \left (c\right ) - {\left (3 \, b^{3} d n^{3} - 2 \, a b^{2} d n^{2}\right )} e\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right ) + 6 \, {\left (2 \, b^{3} d n e \log \left (c\right )^{2} - 2 \, {\left (3 \, b^{3} d n^{2} - 2 \, a b^{2} d n\right )} e \log \left (c\right ) + {\left (7 \, b^{3} d n^{3} - 6 \, a b^{2} d n^{2} + 2 \, a^{2} b d n\right )} e\right )} \sqrt {x}\right )} e^{\left (-2\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 {x}\right )^{n} \right )}\right )^{3}\, 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. 763 vs.
\(2 (252) = 504\).
time = 2.59, size = 763, normalized size = 2.69 \begin {gather*} \frac {1}{4} \, {\left ({\left (4 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right )^{3} - 8 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right )^{3} - 6 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right )^{2} + 24 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right )^{2} + 6 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - 3 \, {\left (\sqrt {x} e + d\right )}^{2} + 48 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{3} n^{3} e^{\left (-1\right )} + 6 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right )^{2} - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right )^{2} - 2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) + 8 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) + {\left (\sqrt {x} e + d\right )}^{2} - 8 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 6 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} + 4 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 6 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right )^{2} - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right )^{2} - 2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) + 8 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) + {\left (\sqrt {x} e + d\right )}^{2} - 8 \, {\left (\sqrt {x} e + d\right )} d\right )} a b^{2} n^{2} e^{\left (-1\right )} + 12 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} a b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 12 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} a b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} + 6 \, {\left (2 \, {\left (\sqrt {x} e + d\right )}^{2} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (\sqrt {x} e + d\right )} d \log \left (\sqrt {x} e + d\right ) - {\left (\sqrt {x} e + d\right )}^{2} + 4 \, {\left (\sqrt {x} e + d\right )} d\right )} a^{2} b n e^{\left (-1\right )} + 12 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} a^{2} b e^{\left (-1\right )} \log \left (c\right ) + 4 \, {\left ({\left (\sqrt {x} e + d\right )}^{2} - 2 \, {\left (\sqrt {x} e + d\right )} d\right )} a^{3} e^{\left (-1\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 350, normalized size = 1.23 \begin {gather*} x\,\left (a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (b^3\,x-\frac {b^3\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )-\frac {3\,b\,x\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\sqrt {x}\,\left (\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}\right )+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}-\frac {3\,b^2\,x\,\left (2\,a-b\,n\right )}{2}\right )-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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