Optimal. Leaf size=122 \[ -\frac {8 f^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{3+n} \, _2F_1\left (3,\frac {3+n}{2};\frac {5+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2146, 12, 371}
\begin {gather*} -\frac {8 f^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+3} \, _2F_1\left (3,\frac {n+3}{2};\frac {n+5}{2};\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+3) \left (d^2-a f^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 371
Rule 2146
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}{\left (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}\right )^2} \, dx &=\left (2 f^4\right ) \text {Subst}\left (\int \frac {4 e^2 x^{2+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 )\\ &=\left (8 e^2 f^4\right ) \text {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 )^3} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )\\ &=-\frac {8 f^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{3+n} \, _2F_1\left (3,\frac {3+n}{2};\frac {5+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.43, size = 112, normalized size = 0.92 \begin {gather*} -\frac {8 f^4 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{3+n} \, _2F_1\left (3,\frac {3+n}{2};\frac {5+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right )^3 (3+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )^{n}}{\left (a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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}{{\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________