Optimal. Leaf size=383 \[ \frac {\sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {3 \sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}} \]
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Rubi [A] time = 0.73, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2401, 2389, 2300, 2180, 2204, 2390, 2310} \[ \frac {\sqrt {3 \pi } g^2 e^{-\frac {3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {3 \sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^4 \sqrt {n}}+\frac {\sqrt {\pi } g^3 e^{-\frac {4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {Erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
Rubi steps
\begin {align*} \int \frac {(f+g x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \left (\frac {(e f-d g)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g (e f-d g)^2 (d+e x)}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {3 g^2 (e f-d g) (d+e x)^2}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^3 (d+e x)^3}{e^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx\\ &=\frac {g^3 \int \frac {(d+e x)^3}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {\left (3 g^2 (e f-d g)\right ) \int \frac {(d+e x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {\left (3 g (e f-d g)^2\right ) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e^3}\\ &=\frac {g^3 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g^2 (e f-d g)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {\left (3 g (e f-d g)^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}+\frac {(e f-d g)^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^4}\\ &=\frac {\left (g^3 (d+e x)^4 \left (c (d+e 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 (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e 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 (d+e x)^n\right )\right )}{e^4 n}+\frac {\left (3 g (e f-d g)^2 (d+e x)^2 \left (c (d+e 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 (d+e x)^n\right )\right )}{e^4 n}+\frac {\left ((e f-d g)^3 (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 )}{e^4 n}\\ &=\frac {\left (2 g^3 (d+e x)^4 \left (c (d+e 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 (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (6 g^2 (e f-d g) (d+e x)^3 \left (c (d+e 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 (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (6 g (e f-d g)^2 (d+e x)^2 \left (c (d+e 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 (d+e x)^n\right )}\right )}{b e^4 n}+\frac {\left (2 (e f-d g)^3 (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 )}{b e^4 n}\\ &=\frac {e^{-\frac {a}{b n}} (e f-d g)^3 \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 )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {4 a}{b n}} g^3 \sqrt {\pi } (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 \sqrt {b} e^4 \sqrt {n}}+\frac {3 e^{-\frac {2 a}{b n}} g (e f-d g)^2 \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 )}{\sqrt {b} e^4 \sqrt {n}}+\frac {e^{-\frac {3 a}{b n}} g^2 (e f-d g) \sqrt {3 \pi } (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 )}{\sqrt {b} e^4 \sqrt {n}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 331, normalized size = 0.86 \[ \frac {\sqrt {\pi } e^{-\frac {4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (2 \sqrt {3} g^2 e^{\frac {a}{b n}} (d+e x)^2 (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+3 \sqrt {2} g e^{\frac {2 a}{b n}} (d+e x) (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 e^{\frac {3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+g^3 (d+e x)^3 \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{2 \sqrt {b} e^4 \sqrt {n}} \]
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 \[ \int \frac {{\left (g x + f\right )}^{3}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{3}}{\sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}\, dx \]
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 \[ \int \frac {{\left (g x + f\right )}^{3}}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^3}{\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{3}}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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