Optimal. Leaf size=85 \[ x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {n} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2296, 2300, 2180, 2204} \[ x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {n} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2296
Rule 2300
Rubi steps
\begin {align*} \int \sqrt {a+b \log \left (c x^n\right )} \, dx &=x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} (b n) \int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx\\ &=x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \left (b x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c x^n\right )\right )\\ &=x \sqrt {a+b \log \left (c x^n\right )}-\left (x \left (c 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 x^n\right )}\right )\\ &=-\frac {1}{2} \sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+x \sqrt {a+b \log \left (c x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 1.00 \[ x \sqrt {a+b \log \left (c x^n\right )}-\frac {1}{2} \sqrt {\pi } \sqrt {b} \sqrt {n} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \]
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 \[ \int \sqrt {b \log \left (c x^{n}\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.73, size = 0, normalized size = 0.00 \[ \int \sqrt {b \ln \left (c \,x^{n}\right )+a}\, dx \]
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 \[ \int \sqrt {b \log \left (c x^{n}\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a+b\,\ln \left (c\,x^n\right )} \,d x \]
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 \[ \int \sqrt {a + b \log {\left (c x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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