Optimal. Leaf size=402 \[ -\frac {1}{2} \sqrt {b} d^3 e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{16} \sqrt {b} e^3 e^{-\frac {4 a}{b n}} \sqrt {n} \sqrt {\pi } x^4 \left (c x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {3}{4} \sqrt {b} d^2 e e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{2} \sqrt {b} d e^2 e^{-\frac {3 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d^3 x \sqrt {a+b \log \left (c x^n\right )}+\frac {3}{2} d^2 e x^2 \sqrt {a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt {a+b \log \left (c x^n\right )}+\frac {1}{4} e^3 x^4 \sqrt {a+b \log \left (c x^n\right )} \]
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Rubi [A]
time = 0.45, antiderivative size = 402, normalized size of antiderivative = 1.00, 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 = {2367, 2333,
2337, 2211, 2235, 2342, 2347} \begin {gather*} -\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 {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 {\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2342
Rule 2347
Rule 2367
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.34, size = 366, normalized size = 0.91 \begin {gather*} \frac {1}{48} e^{-\frac {4 a}{b n}} x \left (c x^n\right )^{-4/n} \left (-24 \sqrt {b} d^3 e^{\frac {3 a}{b n}} \sqrt {n} \sqrt {\pi } \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 {b} e^3 \sqrt {n} \sqrt {\pi } x^3 \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (-9 \sqrt {b} d^2 e e^{\frac {a}{b n}} \sqrt {n} \sqrt {2 \pi } x \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 {b} d e^2 \sqrt {n} \sqrt {3 \pi } x^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+6 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \sqrt {a+b \log \left (c x^n\right )}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{3} \sqrt {a +b \ln \left (c \,x^{n}\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \sqrt {a + b \log {\left (c x^{n} \right )}} \left (d + e x\right )^{3}\, dx \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 = -0.00 \begin {gather*} \int \sqrt {a+b\,\ln \left (c\,x^n\right )}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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