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 = 0.25, 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 = {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 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2389
Rule 2390
Rule 2401
Rule 2451
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*}
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Mathematica [A] time = 0.21, 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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fricas [B] time = 0.46, size = 527, normalized size = 1.86 \[ \frac {4 \, b^{3} e^{2} x \log \relax (c)^{3} + 4 \, {\left (b^{3} e^{2} n^{3} x - b^{3} d^{2} n^{3}\right )} \log \left (e \sqrt {x} + d\right )^{3} - 6 \, {\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} x \log \relax (c)^{2} + 6 \, {\left (2 \, b^{3} d e n^{3} \sqrt {x} + 3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2}\right )} x + 2 \, {\left (b^{3} e^{2} n^{2} x - b^{3} d^{2} n^{2}\right )} \log \relax (c)\right )} \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} x \log \relax (c) - {\left (3 \, b^{3} e^{2} n^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2}\right )} x - 6 \, {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n - 2 \, {\left (b^{3} e^{2} n x - b^{3} d^{2} n\right )} \log \relax (c)^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n\right )} x - 2 \, {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n - {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n\right )} x\right )} \log \relax (c) + 2 \, {\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \relax (c) - 2 \, a b^{2} d e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) + 6 \, {\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \relax (c)^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \, {\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \relax (c)\right )} \sqrt {x}}{4 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 763, normalized size = 2.69 \[ \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 \relax (c) + 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 \relax (c)^{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 \relax (c)^{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 \relax (c) + 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 \relax (c)^{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 \relax (c) + 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 )} \]
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 \[ \int \left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 381, normalized size = 1.34 \[ -\frac {3}{2} \, {\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a^{2} b - \frac {3}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} a b^{2} - \frac {1}{4} \, {\left (6 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - 4 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + e n {\left (\frac {{\left (4 \, d^{2} \log \left (e \sqrt {x} + d\right )^{3} + 18 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + 3 \, e^{2} x + 42 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 42 \, d e \sqrt {x}\right )} n^{2}}{e^{3}} - \frac {6 \, {\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{3}}\right )}\right )} b^{3} + a^{3} x \]
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 \[ x\,\left (a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (b^3\,x-\frac {b^3\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )-\frac {3\,b\,x\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\sqrt {x}\,\left (\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}\right )+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}-\frac {3\,b^2\,x\,\left (2\,a-b\,n\right )}{2}\right )-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \]
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 \[ \int \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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