Optimal. Leaf size=107 \[ \frac {\left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{2 e (1-n)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2118, 2116, 12, 14} \[ \frac {\left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{2 e (1-n)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2116
Rule 2118
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n \, dx &=\int \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {x^{-2+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )}{4 e^2} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )\\ &=\frac {\operatorname {Subst}\left (\int x^{-2+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right ) \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{2 e^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-e \left (d^2-a f^2\right ) x^{-2+n}+e x^n\right ) \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{2 e^2}\\ &=\frac {\left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-1+n}}{2 e (1-n)}+\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 0.83 \[ \frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^{n-1} \left (\frac {a f^2-d^2}{n-1}+\frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^2}{n+1}\right )}{2 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 80, normalized size = 0.75 \[ \frac {{\left (f n \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} - e x - d\right )} {\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e n^{2} - e} \]
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 \[ \int {\left (e x + f \sqrt {\frac {a f^{2} + {\left (e x + 2 \, d\right )} e x}{f^{2}}} + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (e x +d +\sqrt {\frac {a \,f^{2}+\left (e x +2 d \right ) e x}{f^{2}}}\, f \right )^{n}\, dx \]
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 \[ \int {\left (e x + d + \sqrt {a f^{2} + {\left (e x + 2 \, d\right )} e x}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d+e\,x+f\,\sqrt {\frac {a\,f^2+e\,x\,\left (2\,d+e\,x\right )}{f^2}}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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