Optimal. Leaf size=121 \[ \frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2117, 947, 64} \[ \frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 64
Rule 947
Rule 2117
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^n \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^n \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^n+\frac {a f^2 x^n}{(d-x)^2}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac {\left (a f^2\right ) \operatorname {Subst}\left (\int \frac {x^n}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{2 e (1+n)}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^{1+n} \, _2F_1\left (2,1+n;2+n;\frac {d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}}{d}\right )}{2 d^2 e (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 86, normalized size = 0.71 \[ \frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+1} \left (a f^2 \, _2F_1\left (2,n+1;n+2;\frac {d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+a}}{d}\right )+d^2\right )}{2 d^2 e (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}} + d\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, f \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________