Optimal. Leaf size=341 \[ -\frac {3 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^4}+\frac {4 b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {b^2 d^4 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {4 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}+\frac {3 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}-\frac {4 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{4 e^4}+\frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2} \]
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Rubi [A]
time = 0.25, antiderivative size = 341, normalized size of antiderivative = 1.00, 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 = {2504, 2445,
2458, 45, 2372, 12, 14, 2338} \begin {gather*} \frac {b d^4 n \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}-\frac {4 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^4}+\frac {3 b d^2 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 )}{e^4}-\frac {4 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{4 e^4}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 d^4 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}+\frac {4 b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {3 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2504
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^3} \, dx &=-\left (2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}+(b e n) \text {Subst}\left (\int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}+(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )\\ &=-\frac {1}{12} b n \left (\frac {48 d^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^4}+\frac {16 d \left (d+\frac {e}{\sqrt {x}}\right )^3}{e^4}-\frac {3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^4}-\frac {12 d^4 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}-\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{12 e^4 x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )\\ &=-\frac {1}{12} b n \left (\frac {48 d^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^4}+\frac {16 d \left (d+\frac {e}{\sqrt {x}}\right )^3}{e^4}-\frac {3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^4}-\frac {12 d^4 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{12 e^4}\\ &=-\frac {1}{12} b n \left (\frac {48 d^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^4}+\frac {16 d \left (d+\frac {e}{\sqrt {x}}\right )^3}{e^4}-\frac {3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^4}-\frac {12 d^4 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3+\frac {12 d^4 \log (x)}{x}\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{12 e^4}\\ &=-\frac {3 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^4}+\frac {4 b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {1}{12} b n \left (\frac {48 d^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^4}+\frac {16 d \left (d+\frac {e}{\sqrt {x}}\right )^3}{e^4}-\frac {3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^4}-\frac {12 d^4 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}\\ &=-\frac {3 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{16 e^4}+\frac {4 b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {b^2 d^4 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {1}{12} b n \left (\frac {48 d^3 \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {36 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{e^4}+\frac {16 d \left (d+\frac {e}{\sqrt {x}}\right )^3}{e^4}-\frac {3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{e^4}-\frac {12 d^4 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.25, size = 473, normalized size = 1.39 \begin {gather*} -\frac {72 e^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+b n \left (-36 a e^4+9 b e^4 n+48 a d e^3 \sqrt {x}-28 b d e^3 n \sqrt {x}-72 a d^2 e^2 x+78 b d^2 e^2 n x+144 a d^3 e x^{3/2}-300 b d^3 e n x^{3/2}+156 b d^4 n x^2 \log \left (d+\frac {e}{\sqrt {x}}\right )-36 b e^4 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+48 b d e^3 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-72 b d^2 e^2 x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+144 b d^3 e x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+144 b d^4 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-144 a d^4 x^2 \log \left (e+d \sqrt {x}\right )-144 b d^4 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )+72 b d^4 n x^2 \log ^2\left (e+d \sqrt {x}\right )-144 a d^4 x^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-144 b d^4 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )-144 b d^4 n x^2 \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )-144 b d^4 n x^2 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )-144 b d^4 n x^2 \text {Li}_2\left (1+\frac {d \sqrt {x}}{e}\right )\right )}{144 e^4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )^{2}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 319, normalized size = 0.94 \begin {gather*} \frac {1}{12} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (d \sqrt {x} + e\right ) - 6 \, d^{4} e^{\left (-5\right )} \log \left (x\right ) - \frac {{\left (12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} x e + 4 \, d \sqrt {x} e^{2} - 3 \, e^{3}\right )} e^{\left (-4\right )}}{x^{2}}\right )} a b n e + \frac {1}{144} \, {\left (12 \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (d \sqrt {x} + e\right ) - 6 \, d^{4} e^{\left (-5\right )} \log \left (x\right ) - \frac {{\left (12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} x e + 4 \, d \sqrt {x} e^{2} - 3 \, e^{3}\right )} e^{\left (-4\right )}}{x^{2}}\right )} n e \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} x^{\frac {3}{2}} e + 78 \, d^{2} x e^{2} - 28 \, d \sqrt {x} e^{3} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right ) + 9 \, e^{4}\right )} n^{2} e^{\left (-4\right )}}{x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 342, normalized size = 1.00 \begin {gather*} -\frac {{\left (72 \, b^{2} e^{4} \log \left (c\right )^{2} + 6 \, {\left (13 \, b^{2} d^{2} n^{2} - 12 \, a b d^{2} n\right )} x e^{2} - 72 \, {\left (b^{2} d^{4} n^{2} x^{2} - b^{2} n^{2} e^{4}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right )^{2} + 9 \, {\left (b^{2} n^{2} - 4 \, a b n + 8 \, a^{2}\right )} e^{4} - 36 \, {\left (2 \, b^{2} d^{2} n x e^{2} + {\left (b^{2} n - 4 \, a b\right )} e^{4}\right )} \log \left (c\right ) - 12 \, {\left (6 \, b^{2} d^{2} n^{2} x e^{2} - {\left (25 \, b^{2} d^{4} n^{2} - 12 \, a b d^{4} n\right )} x^{2} + 3 \, {\left (b^{2} n^{2} - 4 \, a b n\right )} e^{4} + 12 \, {\left (b^{2} d^{4} n x^{2} - b^{2} n e^{4}\right )} \log \left (c\right ) - 4 \, {\left (3 \, b^{2} d^{3} n^{2} x e + b^{2} d n^{2} e^{3}\right )} \sqrt {x}\right )} \log \left (\frac {d x + \sqrt {x} e}{x}\right ) - 4 \, {\left (3 \, {\left (25 \, b^{2} d^{3} n^{2} - 12 \, a b d^{3} n\right )} x e + {\left (7 \, b^{2} d n^{2} - 12 \, a b d n\right )} e^{3} - 12 \, {\left (3 \, b^{2} d^{3} n x e + b^{2} d n e^{3}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} e^{\left (-4\right )}}{144 \, x^{2}} \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 )^{2}}{x^{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. 1071 vs.
\(2 (300) = 600\).
time = 4.36, size = 1071, normalized size = 3.14 \begin {gather*} \frac {1}{144} \, {\left (\frac {288 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{\sqrt {x}} - \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x} - \frac {576 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {576 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {288 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x^{\frac {3}{2}}} + \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {864 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2}}{x^{2}} + \frac {576 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n^{2}}{\sqrt {x}} - \frac {576 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} n \log \left (c\right )}{\sqrt {x}} + \frac {288 \, {\left (d \sqrt {x} + e\right )} b^{2} d^{3} \log \left (c\right )^{2}}{\sqrt {x}} - \frac {192 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} + \frac {576 \, {\left (d \sqrt {x} + e\right )} a b d^{3} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{\sqrt {x}} + \frac {576 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} - \frac {216 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n^{2}}{x} + \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} n \log \left (c\right )}{x} - \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} b^{2} d^{2} \log \left (c\right )^{2}}{x} + \frac {36 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} n^{2} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {864 \, {\left (d \sqrt {x} + e\right )}^{2} a b d^{2} n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x} - \frac {144 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} n \log \left (c\right ) \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} + \frac {64 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n^{2}}{x^{\frac {3}{2}}} - \frac {576 \, {\left (d \sqrt {x} + e\right )} a b d^{3} n}{\sqrt {x}} - \frac {192 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d n \log \left (c\right )}{x^{\frac {3}{2}}} + \frac {576 \, {\left (d \sqrt {x} + e\right )} a b d^{3} \log \left (c\right )}{\sqrt {x}} + \frac {288 \, {\left (d \sqrt {x} + e\right )}^{3} b^{2} d \log \left (c\right )^{2}}{x^{\frac {3}{2}}} + \frac {576 \, {\left (d \sqrt {x} + e\right )}^{3} a b d n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{\frac {3}{2}}} - \frac {9 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} n^{2}}{x^{2}} + \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} a b d^{2} n}{x} + \frac {36 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} n \log \left (c\right )}{x^{2}} - \frac {864 \, {\left (d \sqrt {x} + e\right )}^{2} a b d^{2} \log \left (c\right )}{x} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{4} b^{2} \log \left (c\right )^{2}}{x^{2}} - \frac {144 \, {\left (d \sqrt {x} + e\right )}^{4} a b n \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )}{x^{2}} - \frac {192 \, {\left (d \sqrt {x} + e\right )}^{3} a b d n}{x^{\frac {3}{2}}} + \frac {288 \, {\left (d \sqrt {x} + e\right )} a^{2} d^{3}}{\sqrt {x}} + \frac {576 \, {\left (d \sqrt {x} + e\right )}^{3} a b d \log \left (c\right )}{x^{\frac {3}{2}}} + \frac {36 \, {\left (d \sqrt {x} + e\right )}^{4} a b n}{x^{2}} - \frac {432 \, {\left (d \sqrt {x} + e\right )}^{2} a^{2} d^{2}}{x} - \frac {144 \, {\left (d \sqrt {x} + e\right )}^{4} a b \log \left (c\right )}{x^{2}} + \frac {288 \, {\left (d \sqrt {x} + e\right )}^{3} a^{2} d}{x^{\frac {3}{2}}} - \frac {72 \, {\left (d \sqrt {x} + e\right )}^{4} a^{2}}{x^{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 424, normalized size = 1.24 \begin {gather*} \ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {\frac {b\,d\,\left (4\,a-b\,n\right )}{3\,e}-\frac {4\,a\,b\,d}{3\,e}}{x^{3/2}}-\frac {b\,\left (4\,a-b\,n\right )}{4\,x^2}-\frac {d\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{2\,e\,x}+\frac {d^2\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{e^2\,\sqrt {x}}\right )+\frac {\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{3\,e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{9\,e}}{x^{3/2}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {b^2}{2\,x^2}-\frac {b^2\,d^4}{2\,e^4}\right )-\frac {\frac {a^2}{2}-\frac {a\,b\,n}{4}+\frac {b^2\,n^2}{16}}{x^2}-\frac {\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2}{4\,e^2}}{x}+\frac {\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{2\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2}{e^3}}{\sqrt {x}}-\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (25\,b^2\,d^4\,n^2-12\,a\,b\,d^4\,n\right )}{12\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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