Optimal. Leaf size=132 \[ \frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {4 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}-\frac {4 b n \sqrt {f+g x} (e f-d g)}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g} \]
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Rubi [A] time = 0.09, 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 = {2395, 50, 63, 208} \[ \frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {4 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}-\frac {4 b n \sqrt {f+g x} (e f-d g)}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 2395
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 ) \operatorname {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 = 0.14, size = 118, normalized size = 0.89 \[ \frac {2 \left (\sqrt {e} \sqrt {f+g x} \left (3 a e (f+g x)+3 b e (f+g x) \log \left (c (d+e x)^n\right )-2 b n (-3 d g+4 e f+e g x)\right )+6 b n (e f-d g)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )}{9 e^{3/2} g} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 311, normalized size = 2.36 \[ \left [-\frac {2 \, {\left (3 \, {\left (b e f - b d g\right )} n \sqrt {\frac {e f - d g}{e}} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {g x + f} e \sqrt {\frac {e f - d g}{e}}}{e x + d}\right ) - {\left (3 \, a e f - 2 \, {\left (4 \, b e f - 3 \, b d g\right )} n - {\left (2 \, b e g n - 3 \, a e g\right )} x + 3 \, {\left (b e g n x + b e f n\right )} \log \left (e x + d\right ) + 3 \, {\left (b e g x + b e f\right )} \log \relax (c)\right )} \sqrt {g x + f}\right )}}{9 \, e g}, \frac {2 \, {\left (6 \, {\left (b e f - b d g\right )} n \sqrt {-\frac {e f - d g}{e}} \arctan \left (-\frac {\sqrt {g x + f} e \sqrt {-\frac {e f - d g}{e}}}{e f - d g}\right ) + {\left (3 \, a e f - 2 \, {\left (4 \, b e f - 3 \, b d g\right )} n - {\left (2 \, b e g n - 3 \, a e g\right )} x + 3 \, {\left (b e g n x + b e f n\right )} \log \left (e x + d\right ) + 3 \, {\left (b e g x + b e f\right )} \log \relax (c)\right )} \sqrt {g x + f}\right )}}{9 \, e g}\right ] \]
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 \[ \int \sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \sqrt {g x +f}\, \left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )\, dx \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \sqrt {f+g\,x}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.72, size = 139, normalized size = 1.05 \[ \frac {2 \left (\frac {a \left (f + g x\right )^{\frac {3}{2}}}{3} + b \left (- \frac {2 e n \left (\frac {g \left (f + g x\right )^{\frac {3}{2}}}{3 e} + \frac {\sqrt {f + g x} \left (- d g^{2} + e f g\right )}{e^{2}} + \frac {g \left (d g - e f\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{3} \sqrt {\frac {d g - e f}{e}}}\right )}{3 g} + \frac {\left (f + g x\right )^{\frac {3}{2}} \log {\left (c \left (d - \frac {e f}{g} + \frac {e \left (f + g x\right )}{g}\right )^{n} \right )}}{3}\right )\right )}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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