Optimal. Leaf size=124 \[ \frac {a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)}-\frac {c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {156, 68} \[ \frac {a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)}-\frac {c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 156
Rubi steps
\begin {align*} \int \frac {x (e+f x)^n}{(a+b x) (c+d x)} \, dx &=-\frac {a \int \frac {(e+f x)^n}{a+b x} \, dx}{b c-a d}+\frac {c \int \frac {(e+f x)^n}{c+d x} \, dx}{b c-a d}\\ &=\frac {a (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f) (1+n)}-\frac {c (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (d e-c f) (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 116, normalized size = 0.94 \[ \frac {(e+f x)^{n+1} \left (a (c f-d e) \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )+c (b e-a f) \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )\right )}{(n+1) (b c-a d) (b e-a f) (c f-d e)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n} x}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n} x}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {x \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{n} x}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________