Optimal. Leaf size=94 \[ -\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e} \]
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Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2356, 52, 65,
214} \begin {gather*} \frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e}-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 2356
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(2 b n) \int \frac {(d+e x)^{3/2}}{x} \, dx}{3 e}\\ &=-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {(2 b d n) \int \frac {\sqrt {d+e x}}{x} \, dx}{3 e}\\ &=-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {\left (2 b d^2 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx}{3 e}\\ &=-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{3 e^2}\\ &=-\frac {4 b d n \sqrt {d+e x}}{3 e}-\frac {4 b n (d+e x)^{3/2}}{9 e}+\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 77, normalized size = 0.82 \begin {gather*} \frac {2 \left (6 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\sqrt {d+e x} \left (3 a (d+e x)-2 b n (4 d+e x)+3 b (d+e x) \log \left (c x^n\right )\right )\right )}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 96, normalized size = 1.02 \begin {gather*} \frac {2}{3} \, {\left (x e + d\right )}^{\frac {3}{2}} b e^{\left (-1\right )} \log \left (c x^{n}\right ) - \frac {2}{9} \, {\left (3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {x e + d} - \sqrt {d}}{\sqrt {x e + d} + \sqrt {d}}\right ) + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} + 6 \, \sqrt {x e + d} d\right )} b n e^{\left (-1\right )} + \frac {2}{3} \, {\left (x e + d\right )}^{\frac {3}{2}} a e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 191, normalized size = 2.03 \begin {gather*} \left [\frac {2}{9} \, {\left (3 \, b d^{\frac {3}{2}} n \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (8 \, b d n + {\left (2 \, b n - 3 \, a\right )} x e - 3 \, a d - 3 \, {\left (b x e + b d\right )} \log \left (c\right ) - 3 \, {\left (b n x e + b d n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}, -\frac {2}{9} \, {\left (6 \, b \sqrt {-d} d n \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (8 \, b d n + {\left (2 \, b n - 3 \, a\right )} x e - 3 \, a d - 3 \, {\left (b x e + b d\right )} \log \left (c\right ) - 3 \, {\left (b n x e + b d n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.75, size = 102, normalized size = 1.09 \begin {gather*} \frac {2 \left (\frac {a \left (d + e x\right )^{\frac {3}{2}}}{3} + b \left (\frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (\frac {d^{2} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{\sqrt {- d}} + d e \sqrt {d + e x} + \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right )\right )}{e} \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 \left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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