Optimal. Leaf size=404 \[ -\frac {\sqrt {b} e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^3}-\frac {\sqrt {b} e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {n} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b n}} g^2 \sqrt {n} \sqrt {\frac {\pi }{3}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{6 e^3}+\frac {(e f-d g)^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3} \]
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Rubi [A]
time = 0.51, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2448, 2436,
2333, 2337, 2211, 2235, 2437, 2342, 2347} \begin {gather*} -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} g \sqrt {n} e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^3}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) (e f-d g)^2 \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^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} g^2 \sqrt {n} e^{-\frac {3 a}{b n}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{6 e^3}+\frac {g (d+e x)^2 (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {(d+e x) (e f-d g)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2342
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rubi steps
\begin {align*} \int (f+g x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac {(e f-d g)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {2 g (e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}\right ) \, dx\\ &=\frac {g^2 \int (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}+\frac {(2 g (e f-d g)) \int (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}+\frac {(e f-d g)^2 \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx}{e^2}\\ &=\frac {g^2 \text {Subst}\left (\int x^2 \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}+\frac {(2 g (e f-d g)) \text {Subst}\left (\int x \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}+\frac {(e f-d g)^2 \text {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^3}\\ &=\frac {(e f-d g)^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac {\left (b g^2 n\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{6 e^3}-\frac {(b g (e f-d g) n) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^3}-\frac {\left (b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^3}\\ &=\frac {(e f-d g)^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac {\left (b g^2 (d+e x)^3 \left (c (d+e 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 (d+e x)^n\right )\right )}{6 e^3}-\frac {\left (b g (e f-d g) (d+e x)^2 \left (c (d+e 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 (d+e x)^n\right )\right )}{2 e^3}-\frac {\left (b (e f-d g)^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {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^3}\\ &=\frac {(e f-d g)^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3}-\frac {\left (g^2 (d+e x)^3 \left (c (d+e 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 (d+e x)^n\right )}\right )}{3 e^3}-\frac {\left (g (e f-d g) (d+e x)^2 \left (c (d+e 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 (d+e x)^n\right )}\right )}{e^3}-\frac {\left ((e f-d g)^2 (d+e x) \left (c (d+e 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 (d+e x)^n\right )}\right )}{e^3}\\ &=-\frac {\sqrt {b} e^{-\frac {a}{b n}} (e f-d g)^2 \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^3}-\frac {\sqrt {b} e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {n} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b n}} g^2 \sqrt {n} \sqrt {\frac {\pi }{3}} (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{6 e^3}+\frac {(e f-d g)^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g (e f-d g) (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^3}+\frac {g^2 (d+e x)^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 374, normalized size = 0.93 \begin {gather*} \frac {(d+e x) \left (-18 \sqrt {b} e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {n} \sqrt {\pi } \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 )+9 \sqrt {b} e^{-\frac {2 a}{b n}} g (-e f+d g) \sqrt {n} \sqrt {2 \pi } (d+e x) \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-2 \sqrt {b} e^{-\frac {3 a}{b n}} g^2 \sqrt {n} \sqrt {3 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+36 (e f-d g)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}+36 g (e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}+12 g^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{36 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (g x +f \right )^{2} \sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}\, dx\]
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 \left (d + e x\right )^{n} \right )}} \left (f + g x\right )^{2}\, 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 {\left (f+g\,x\right )}^2\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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