Optimal. Leaf size=402 \[ -\frac{3}{4} \sqrt{\frac{\pi }{2}} \sqrt{b} d^2 e \sqrt{n} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} d^3 \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d^3 x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\frac{\pi }{3}} \sqrt{b} d e^2 \sqrt{n} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{16} \sqrt{\pi } \sqrt{b} e^3 \sqrt{n} x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )} \]
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Rubi [A] time = 0.615256, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {2330, 2296, 2300, 2180, 2204, 2305, 2310} \[ -\frac{3}{4} \sqrt{\frac{\pi }{2}} \sqrt{b} d^2 e \sqrt{n} x^2 e^{-\frac{2 a}{b n}} \left (c x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\pi } \sqrt{b} d^3 \sqrt{n} x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d^3 x \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \sqrt{\frac{\pi }{3}} \sqrt{b} d e^2 \sqrt{n} x^3 e^{-\frac{3 a}{b n}} \left (c x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{16} \sqrt{\pi } \sqrt{b} e^3 \sqrt{n} x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2330
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rule 2305
Rule 2310
Rubi steps
\begin{align*} \int (d+e x)^3 \sqrt{a+b \log \left (c x^n\right )} \, dx &=\int \left (d^3 \sqrt{a+b \log \left (c x^n\right )}+3 d^2 e x \sqrt{a+b \log \left (c x^n\right )}+3 d e^2 x^2 \sqrt{a+b \log \left (c x^n\right )}+e^3 x^3 \sqrt{a+b \log \left (c x^n\right )}\right ) \, dx\\ &=d^3 \int \sqrt{a+b \log \left (c x^n\right )} \, dx+\left (3 d^2 e\right ) \int x \sqrt{a+b \log \left (c x^n\right )} \, dx+\left (3 d e^2\right ) \int x^2 \sqrt{a+b \log \left (c x^n\right )} \, dx+e^3 \int x^3 \sqrt{a+b \log \left (c x^n\right )} \, dx\\ &=d^3 x \sqrt{a+b \log \left (c x^n\right )}+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{2} \left (b d^3 n\right ) \int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx-\frac{1}{4} \left (3 b d^2 e n\right ) \int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx-\frac{1}{2} \left (b d e^2 n\right ) \int \frac{x^2}{\sqrt{a+b \log \left (c x^n\right )}} \, dx-\frac{1}{8} \left (b e^3 n\right ) \int \frac{x^3}{\sqrt{a+b \log \left (c x^n\right )}} \, dx\\ &=d^3 x \sqrt{a+b \log \left (c x^n\right )}+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{8} \left (b e^3 x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{2} \left (b d e^2 x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{4} \left (3 b d^2 e x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c x^n\right )\right )-\frac{1}{2} \left (b d^3 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 )\\ &=d^3 x \sqrt{a+b \log \left (c x^n\right )}+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )}-\frac{1}{4} \left (e^3 x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{4 a}{b n}+\frac{4 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c x^n\right )}\right )-\left (d e^2 x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b n}+\frac{3 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c x^n\right )}\right )-\frac{1}{2} \left (3 d^2 e x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c x^n\right )}\right )-\left (d^3 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} d^3 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 )-\frac{1}{16} \sqrt{b} e^3 e^{-\frac{4 a}{b n}} \sqrt{n} \sqrt{\pi } x^4 \left (c x^n\right )^{-4/n} \text{erfi}\left (\frac{2 \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )-\frac{3}{4} \sqrt{b} d^2 e e^{-\frac{2 a}{b n}} \sqrt{n} \sqrt{\frac{\pi }{2}} x^2 \left (c x^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )-\frac{1}{2} \sqrt{b} d e^2 e^{-\frac{3 a}{b n}} \sqrt{n} \sqrt{\frac{\pi }{3}} x^3 \left (c x^n\right )^{-3/n} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )+d^3 x \sqrt{a+b \log \left (c x^n\right )}+\frac{3}{2} d^2 e x^2 \sqrt{a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt{a+b \log \left (c x^n\right )}+\frac{1}{4} e^3 x^4 \sqrt{a+b \log \left (c x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.490493, size = 366, normalized size = 0.91 \[ \frac{1}{48} x e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \left (2 e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (6 e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right ) \sqrt{a+b \log \left (c x^n\right )}-9 \sqrt{2 \pi } \sqrt{b} d^2 e \sqrt{n} x e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )-4 \sqrt{3 \pi } \sqrt{b} d e^2 \sqrt{n} x^2 \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right )-24 \sqrt{\pi } \sqrt{b} d^3 \sqrt{n} e^{\frac{3 a}{b n}} \left (c x^n\right )^{3/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )-3 \sqrt{\pi } \sqrt{b} e^3 \sqrt{n} x^3 \text{Erfi}\left (\frac{2 \sqrt{a+b \log \left (c x^n\right )}}{\sqrt{b} \sqrt{n}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.389, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3}\sqrt{a+b\ln \left ( c{x}^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{3} \sqrt{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \log{\left (c x^{n} \right )}} \left (d + e x\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{3} \sqrt{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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