Optimal. Leaf size=41 \[ \frac {f \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{e n} \]
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Rubi [A]
time = 0.31, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2152, 2146, 12,
30} \begin {gather*} \frac {f \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2146
Rule 2152
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}} \, dx &=\int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}} \, dx\\ &=(2 f) \text {Subst}\left (\int \frac {x^{-1+n}}{2 e} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )\\ &=\frac {f \text {Subst}\left (\int x^{-1+n} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{e}\\ &=\frac {f \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{e n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.88 \begin {gather*} \frac {f \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^n}{e n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (d +e x +f \sqrt {\frac {a \,f^{2}+e x \left (e x +2 d \right )}{f^{2}}}\right )^{n}}{\sqrt {\frac {a \,f^{2}+e x \left (e x +2 d \right )}{f^{2}}}}\, dx\]
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 [A]
time = 0.35, size = 41, normalized size = 1.00 \begin {gather*} \frac {{\left (x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2} + 2 \, d x e}{f^{2}}} + d\right )}^{n} f e^{\left (-1\right )}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 3.08, size = 39, normalized size = 0.95 \begin {gather*} \frac {f\,{\left (d+e\,x+f\,\sqrt {\frac {a\,f^2+e\,x\,\left (2\,d+e\,x\right )}{f^2}}\right )}^n}{e\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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