Optimal. Leaf size=285 \[ \frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448,
2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} -\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}-\frac {12 b^3 d n^3}{e \sqrt {x}} \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 2504
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx &=-\left (2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\left (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,\frac {1}{\sqrt {x}}\right )\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,\frac {1}{\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,\frac {1}{\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+\frac {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+\frac {e}{\sqrt {x}}\right )}{e^2}\\ &=\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{e^2}\\ &=-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {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+\frac {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+\frac {e}{\sqrt {x}}\right )}{e^2}\\ &=\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {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+\frac {e}{\sqrt {x}}\right )}{e^2}\\ &=\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 558, normalized size = 1.96 \begin {gather*} \frac {-4 a^3 e^2+6 a^2 b e^2 n-6 a b^2 e^2 n^2+3 b^3 e^2 n^3-12 a^2 b d e n \sqrt {x}+36 a b^2 d e n^2 \sqrt {x}-42 b^3 d e n^3 \sqrt {x}-8 b^3 d^2 n^3 x \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-4 b^3 e^2 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+12 a^2 b d^2 n x \log \left (e+d \sqrt {x}\right )-36 a b^2 d^2 n^2 x \log \left (e+d \sqrt {x}\right )+42 b^3 d^2 n^3 x \log \left (e+d \sqrt {x}\right )+6 b^2 d^2 n^2 x \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-2 a+3 b n-2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (2 \log \left (e+d \sqrt {x}\right )-\log (x)\right )-6 a^2 b d^2 n x \log (x)+18 a b^2 d^2 n^2 x \log (x)-21 b^3 d^2 n^3 x \log (x)+6 b^2 d^2 n^2 x \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (2 a-3 b n+2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+2 b n \log \left (e+d \sqrt {x}\right )-b n \log (x)\right )+6 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-2 a e+b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n x \log \left (e+d \sqrt {x}\right )-b d^2 n x \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (2 a^2 e+b^2 n^2 \left (e-6 d \sqrt {x}\right )-2 a b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n (-2 a+3 b n) x \log \left (e+d \sqrt {x}\right )+b d^2 n (2 a-3 b n) x \log (x)\right )}{4 e^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )^{3}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 579 vs.
\(2 (253) = 506\).
time = 0.33, size = 579, normalized size = 2.03 \begin {gather*} \frac {3}{2} \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (d \sqrt {x} + e\right ) - d^{2} e^{\left (-3\right )} \log \left (x\right ) - \frac {{\left (2 \, d \sqrt {x} - e\right )} e^{\left (-2\right )}}{x}\right )} a^{2} b n e - \frac {b^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{3}}{x} + \frac {3}{4} \, {\left (4 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (d \sqrt {x} + e\right ) - d^{2} e^{\left (-3\right )} \log \left (x\right ) - \frac {{\left (2 \, d \sqrt {x} - e\right )} e^{\left (-2\right )}}{x}\right )} n e \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d \sqrt {x} e - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right ) + 2 \, e^{2}\right )} n^{2} e^{\left (-2\right )}}{x}\right )} a b^{2} + \frac {1}{8} \, {\left (12 \, {\left (2 \, d^{2} e^{\left (-3\right )} \log \left (d \sqrt {x} + e\right ) - d^{2} e^{\left (-3\right )} \log \left (x\right ) - \frac {{\left (2 \, d \sqrt {x} - e\right )} e^{\left (-2\right )}}{x}\right )} n e \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2} + {\left (\frac {{\left (8 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{3} - d^{2} x \log \left (x\right )^{3} + 9 \, d^{2} x \log \left (x\right )^{2} - 42 \, d^{2} x \log \left (x\right ) - 12 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )^{2} - 84 \, d \sqrt {x} e + 6 \, {\left (d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) + 14 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right ) + 6 \, e^{2}\right )} n^{2} e^{\left (-3\right )}}{x} - \frac {6 \, {\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d \sqrt {x} e - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right ) + 2 \, e^{2}\right )} n e^{\left (-3\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x}\right )} n e\right )} b^{3} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x} - \frac {a^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 519 vs.
\(2 (253) = 506\).
time = 0.40, size = 519, normalized size = 1.82 \begin {gather*} -\frac {{\left (4 \, b^{3} e^{2} \log \left (c\right )^{3} - 6 \, {\left (b^{3} n - 2 \, a b^{2}\right )} e^{2} \log \left (c\right )^{2} - 4 \, {\left (b^{3} d^{2} n^{3} x - b^{3} n^{3} e^{2}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right )^{3} + 6 \, {\left (b^{3} n^{2} - 2 \, a b^{2} n + 2 \, a^{2} b\right )} e^{2} \log \left (c\right ) + 6 \, {\left (2 \, b^{3} d n^{3} \sqrt {x} e + {\left (3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2}\right )} x - {\left (b^{3} n^{3} - 2 \, a b^{2} n^{2}\right )} e^{2} - 2 \, {\left (b^{3} d^{2} n^{2} x - b^{3} n^{2} e^{2}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right )^{2} - {\left (3 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 6 \, a^{2} b n - 4 \, a^{3}\right )} e^{2} - 6 \, {\left (2 \, {\left (b^{3} d^{2} n x - b^{3} n e^{2}\right )} \log \left (c\right )^{2} + {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n\right )} x - {\left (b^{3} n^{3} - 2 \, a b^{2} n^{2} + 2 \, a^{2} b n\right )} e^{2} - 2 \, {\left ({\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n\right )} x - {\left (b^{3} n^{2} - 2 \, a b^{2} n\right )} 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 (\frac {d x + \sqrt {x} e}{x}\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 )}}{4 \, x} \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 \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}}{x^{2}}\, 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. 1127 vs.
\(2 (253) = 506\).
time = 4.68, size = 1127, normalized size = 3.95 \begin {gather*} \frac {1}{4} \, {\left (\frac {8 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{3}}{\sqrt {x}} - \frac {4 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{3}}{x} - \frac {24 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{2} \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{\sqrt {x}} + \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{2} \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x} + \frac {48 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {48 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{2} \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} b^{3} d n \log \left (c\right )^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} a b^{2} d n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{\sqrt {x}} - \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} + \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{2} \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n \log \left (c\right )^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x} - \frac {48 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3}}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{2} \log \left (c\right )}{\sqrt {x}} - \frac {24 \, {\left (d \sqrt {x} + e\right )} b^{3} d n \log \left (c\right )^{2}}{\sqrt {x}} + \frac {8 \, {\left (d \sqrt {x} + e\right )} b^{3} d \log \left (c\right )^{3}}{\sqrt {x}} - \frac {48 \, {\left (d \sqrt {x} + e\right )} a b^{2} d n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {48 \, {\left (d \sqrt {x} + e\right )} a b^{2} d n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {3 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3}}{x} - \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{2} \log \left (c\right )}{x} + \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} n \log \left (c\right )^{2}}{x} - \frac {4 \, {\left (d \sqrt {x} + e\right )}^{2} b^{3} \log \left (c\right )^{3}}{x} + \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {24 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} + \frac {48 \, {\left (d \sqrt {x} + e\right )} a b^{2} d n^{2}}{\sqrt {x}} - \frac {48 \, {\left (d \sqrt {x} + e\right )} a b^{2} d n \log \left (c\right )}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} a b^{2} d \log \left (c\right )^{2}}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} a^{2} b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} - \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} n^{2}}{x} + \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} n \log \left (c\right )}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a b^{2} \log \left (c\right )^{2}}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a^{2} b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {24 \, {\left (d \sqrt {x} + e\right )} a^{2} b d n}{\sqrt {x}} + \frac {24 \, {\left (d \sqrt {x} + e\right )} a^{2} b d \log \left (c\right )}{\sqrt {x}} + \frac {6 \, {\left (d \sqrt {x} + e\right )}^{2} a^{2} b n}{x} - \frac {12 \, {\left (d \sqrt {x} + e\right )}^{2} a^{2} b \log \left (c\right )}{x} + \frac {8 \, {\left (d \sqrt {x} + e\right )} a^{3} d}{\sqrt {x}} - \frac {4 \, {\left (d \sqrt {x} + e\right )}^{2} a^{3}}{x}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 357, normalized size = 1.25 \begin {gather*} \frac {\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}}{\sqrt {x}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^3\,\left (\frac {b^3}{x}-\frac {b^3\,d^2}{e^2}\right )+\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {\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}}{\sqrt {x}}-\frac {3\,b\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2\,x}\right )+{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}}{\sqrt {x}}-\frac {3\,b^2\,\left (2\,a-b\,n\right )}{2\,x}+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}\right )-\frac {a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}}{x}+\frac {\ln \left (d+\frac {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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