Optimal. Leaf size=93 \[ \frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.53, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2125, 2121, 12, 30} \[ \frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2121
Rule 2125
Rubi steps
\begin {align*} \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 g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}} \, dx &=\frac {\sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \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}{\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\\ &=\frac {\left (2 f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right ) \operatorname {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 )}{\sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\\ &=\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right ) \operatorname {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 \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\\ &=\frac {f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \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 \sqrt {a g+\frac {2 d e g x}{f^2}+\frac {e^2 g x^2}{f^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 76, normalized size = 0.82 \[ \frac {f \sqrt {a+\frac {e x (2 d+e x)}{f^2}} \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^n}{e n \sqrt {g \left (a+\frac {e x (2 d+e x)}{f^2}\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 117, normalized size = 1.26 \[ \frac {{\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f^{3} \sqrt {\frac {e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}} \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}}{e^{3} g n x^{2} + a e f^{2} g n + 2 \, d e^{2} g n x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}}{\sqrt {\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a +\frac {2 d e x}{f^{2}}}\, f \right )^{n}}{\sqrt {\frac {e^{2} g \,x^{2}}{f^{2}}+a g +\frac {2 d e g x}{f^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}}{\sqrt {\frac {e^{2} g x^{2}}{f^{2}} + a g + \frac {2 \, d e g x}{f^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n}{\sqrt {a\,g+\frac {e^2\,g\,x^2}{f^2}+\frac {2\,d\,e\,g\,x}{f^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x + f \sqrt {a + \frac {2 d e x}{f^{2}} + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}}{\sqrt {g \left (a + \frac {2 d e x}{f^{2}} + \frac {e^{2} x^{2}}{f^{2}}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________