Optimal. Leaf size=297 \[ -\frac {\left (d^2-a f^2\right )^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-4}}{16 e f^3 (4-n)}+\frac {\left (d^2-a f^2\right )^3 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-2}}{4 e f^3 (2-n)}+\frac {3 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{8 e f^3 n}-\frac {\left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+2}}{4 e f^3 (n+2)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+4}}{16 e f^3 (n+4)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2121, 12, 270} \[ -\frac {\left (d^2-a f^2\right )^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-4}}{16 e f^3 (4-n)}+\frac {\left (d^2-a f^2\right )^3 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-2}}{4 e f^3 (2-n)}+\frac {3 \left (d^2-a f^2\right )^2 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^n}{8 e f^3 n}-\frac {\left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+2}}{4 e f^3 (n+2)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+4}}{16 e f^3 (n+4)} \]
Antiderivative was successfully verified.
[In]
[Out]
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 )^{3/2} \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^{-5+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^4}{32 e^5} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{f^3}\\ &=\frac {\operatorname {Subst}\left (\int x^{-5+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^4 \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{16 e^5 f^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (e^4 \left (d^2-a f^2\right )^4 x^{-5+n}-4 e^4 \left (d^2-a f^2\right )^3 x^{-3+n}+6 e^4 \left (d^2-a f^2\right )^2 x^{-1+n}-4 e^4 \left (d^2-a f^2\right ) x^{1+n}+e^4 x^{3+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 )}{16 e^5 f^3}\\ &=-\frac {\left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-4+n}}{16 e f^3 (4-n)}+\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 )^{-2+n}}{4 e f^3 (2-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 )^n}{8 e f^3 n}-\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 )^{2+n}}{4 e f^3 (2+n)}+\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{4+n}}{16 e f^3 (4+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.20, size = 228, normalized size = 0.77 \[ \frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^n \left (\frac {\left (d^2-a f^2\right )^4}{(n-4) \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^4}-\frac {4 \left (d^2-a f^2\right )^3}{(n-2) \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^2}-\frac {4 \left (d^2-a f^2\right ) \left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^2}{n+2}+\frac {6 \left (d^2-a f^2\right )^2}{n}+\frac {\left (f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}+d+e x\right )^4}{n+4}\right )}{16 e f^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 377, normalized size = 1.27 \[ \frac {{\left (a^{2} f^{4} n^{4} + 24 \, a^{2} f^{4} - 48 \, a d^{2} f^{2} + {\left (e^{4} n^{4} - 4 \, e^{4} n^{2}\right )} x^{4} + 24 \, d^{4} + 4 \, {\left (d e^{3} n^{4} - 4 \, d e^{3} n^{2}\right )} x^{3} - 4 \, {\left (4 \, a^{2} f^{4} - 3 \, a d^{2} f^{2}\right )} n^{2} + 2 \, {\left ({\left (a e^{2} f^{2} + 2 \, d^{2} e^{2}\right )} n^{4} - 2 \, {\left (5 \, a e^{2} f^{2} + d^{2} e^{2}\right )} n^{2}\right )} x^{2} + 4 \, {\left (a d e f^{2} n^{4} - 2 \, {\left (5 \, a d e f^{2} - 3 \, d^{3} e\right )} n^{2}\right )} x - 4 \, {\left (a d f^{3} n^{3} + {\left (e^{3} f n^{3} - 4 \, e^{3} f n\right )} x^{3} + 3 \, {\left (d e^{2} f n^{3} - 4 \, d e^{2} f n\right )} x^{2} - 2 \, {\left (5 \, a d f^{3} - 3 \, d^{3} f\right )} n + {\left ({\left (a e f^{3} + 2 \, d^{2} e f\right )} n^{3} - 2 \, {\left (5 \, a e f^{3} + d^{2} e f\right )} n\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )} {\left (e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{3} n^{5} - 20 \, e f^{3} n^{3} + 64 \, e f^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )}^{\frac {3}{2}} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (\frac {e^{2} x^{2}}{f^{2}}+a +\frac {2 d e x}{f^{2}}\right )^{\frac {3}{2}} \left (e x +d +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a +\frac {2 d e x}{f^{2}}}\, f \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}\right )}^{\frac {3}{2}} {\left (e x + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a + \frac {2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n\,{\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________