Optimal. Leaf size=111 \[ \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e} \]
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Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2389, 2296, 2300, 2180, 2204} \[ \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2296
Rule 2300
Rule 2389
Rubi steps
\begin {align*} \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {(b n) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e}\\ &=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\left (b (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 e}\\ &=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}-\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{e}\\ &=-\frac {\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e}+\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 106, normalized size = 0.95 \[ \frac {(d+e x) \left (2 \sqrt {a+b \log \left (c (d+e x)^n\right )}-\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{2 e} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]
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 \[ \int \sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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