Optimal. Leaf size=132 \[ -\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g} \]
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Rubi [A]
time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2442, 52, 65,
214} \begin {gather*} \frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {4 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}-\frac {4 b n \sqrt {f+g x} (e f-d g)}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 2442
Rubi steps
\begin {align*} \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {(2 b e n) \int \frac {(f+g x)^{3/2}}{d+e x} \, dx}{3 g}\\ &=-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {(2 b (e f-d g) n) \int \frac {\sqrt {f+g x}}{d+e x} \, dx}{3 g}\\ &=-\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {\left (2 b (e f-d g)^2 n\right ) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{3 e g}\\ &=-\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {\left (4 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 e g^2}\\ &=-\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 118, normalized size = 0.89 \begin {gather*} \frac {2 \left (6 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+\sqrt {e} \sqrt {f+g x} \left (3 a e (f+g x)-2 b n (4 e f-3 d g+e g x)+3 b e (f+g x) \log \left (c (d+e x)^n\right )\right )\right )}{9 e^{3/2} g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \sqrt {g x +f}\, \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 303, normalized size = 2.30 \begin {gather*} \left [-\frac {2 \, {\left (3 \, {\left (b d g n - b f n e\right )} \sqrt {-{\left (d g - f e\right )} e^{\left (-1\right )}} \log \left (-\frac {d g - {\left (g x + 2 \, f\right )} e - 2 \, \sqrt {g x + f} \sqrt {-{\left (d g - f e\right )} e^{\left (-1\right )}} e}{x e + d}\right ) - {\left (6 \, b d g n + 3 \, {\left (b g n x + b f n\right )} e \log \left (x e + d\right ) + 3 \, {\left (b g x + b f\right )} e \log \left (c\right ) - {\left (8 \, b f n - 3 \, a f + {\left (2 \, b g n - 3 \, a g\right )} x\right )} e\right )} \sqrt {g x + f}\right )} e^{\left (-1\right )}}{9 \, g}, \frac {2 \, {\left (6 \, {\left (b d g n - b f n e\right )} \sqrt {d g - f e} \arctan \left (-\frac {\sqrt {g x + f} e^{\frac {1}{2}}}{\sqrt {d g - f e}}\right ) e^{\left (-\frac {1}{2}\right )} + {\left (6 \, b d g n + 3 \, {\left (b g n x + b f n\right )} e \log \left (x e + d\right ) + 3 \, {\left (b g x + b f\right )} e \log \left (c\right ) - {\left (8 \, b f n - 3 \, a f + {\left (2 \, b g n - 3 \, a g\right )} x\right )} e\right )} \sqrt {g x + f}\right )} e^{\left (-1\right )}}{9 \, g}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.30, size = 139, normalized size = 1.05 \begin {gather*} \frac {2 \left (\frac {a \left (f + g x\right )^{\frac {3}{2}}}{3} + b \left (- \frac {2 e n \left (\frac {g \left (f + g x\right )^{\frac {3}{2}}}{3 e} + \frac {\sqrt {f + g x} \left (- d g^{2} + e f g\right )}{e^{2}} + \frac {g \left (d g - e f\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{3} \sqrt {\frac {d g - e f}{e}}}\right )}{3 g} + \frac {\left (f + g x\right )^{\frac {3}{2}} \log {\left (c \left (d - \frac {e f}{g} + \frac {e \left (f + g x\right )}{g}\right )^{n} \right )}}{3}\right )\right )}{g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {f+g\,x}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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