Optimal. Leaf size=239 \[ \frac {\left (d^2-a f^2\right )^3 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac {3 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac {3 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]
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Rubi [A] time = 0.25, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2121, 12, 270} \[ \frac {\left (d^2-a f^2\right )^3 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac {3 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac {3 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2121
Rubi steps
\begin {align*} \int \left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{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 )^n \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^{-4+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3}{16 e^4} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{f^2}\\ &=\frac {\operatorname {Subst}\left (\int x^{-4+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3 \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{8 e^4 f^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-e^3 \left (d^2-a f^2\right )^3 x^{-4+n}+3 e^3 \left (d^2-a f^2\right )^2 x^{-2+n}-3 e^3 \left (d^2-a f^2\right ) x^n+e^3 x^{2+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 )}{8 e^4 f^2}\\ &=\frac {\left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-3+n}}{8 e f^2 (3-n)}-\frac {3 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-1+n}}{8 e f^2 (1-n)}-\frac {3 \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}}{8 e f^2 (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 )^{3+n}}{8 e f^2 (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 186, normalized size = 0.78 \[ \frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^{n-3} \left (-\frac {3 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^4}{n+1}+\frac {3 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^2}{n-1}-\frac {\left (d^2-a f^2\right )^3}{n-3}+\frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^6}{n+3}\right )}{8 e f^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 239, normalized size = 1.00 \[ -\frac {{\left (3 \, a d f^{2} n^{2} - 9 \, a d f^{2} + 3 \, {\left (e^{3} n^{2} - e^{3}\right )} x^{3} + 6 \, d^{3} + 9 \, {\left (d e^{2} n^{2} - d e^{2}\right )} x^{2} - 3 \, {\left (3 \, a e f^{2} - {\left (a e f^{2} + 2 \, d^{2} e\right )} n^{2}\right )} x - {\left (a f^{3} n^{3} + {\left (e^{2} f n^{3} - e^{2} f n\right )} x^{2} - {\left (7 \, a f^{3} - 6 \, d^{2} f\right )} n + 2 \, {\left (d e f n^{3} - d e f n\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )} {\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{2} n^{4} - 10 \, e f^{2} n^{2} + 9 \, e f^{2}} \]
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 \[ \int {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
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 \[ \int \left (\frac {e^{2} x^{2}}{f^{2}}+a +\frac {2 d e x}{f^{2}}\right ) \left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a +\frac {2 d e x}{f^{2}}}\, f \right )^{n}\, dx \]
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 \[ \int {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n\,\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right ) \,d x \]
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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