Optimal. Leaf size=140 \[ \frac {(a+b x)^{m+1} (d g-c h) \, _2F_1\left (1,m+1;m+2;-\frac {h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (f g-e h)}-\frac {(a+b x)^{m+1} (d e-c f) \, _2F_1\left (1,m+1;m+2;-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (f g-e h)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {156, 68} \[ \frac {(a+b x)^{m+1} (d g-c h) \, _2F_1\left (1,m+1;m+2;-\frac {h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (f g-e h)}-\frac {(a+b x)^{m+1} (d e-c f) \, _2F_1\left (1,m+1;m+2;-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (f g-e h)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)}{(e+f x) (g+h x)} \, dx &=-\frac {(d e-c f) \int \frac {(a+b x)^m}{e+f x} \, dx}{f g-e h}+\frac {(d g-c h) \int \frac {(a+b x)^m}{g+h x} \, dx}{f g-e h}\\ &=-\frac {(d e-c f) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f) (f g-e h) (1+m)}+\frac {(d g-c h) (a+b x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {h (a+b x)}{b g-a h}\right )}{(b g-a h) (f g-e h) (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 115, normalized size = 0.82 \[ \frac {(a+b x)^{m+1} \left (\frac {(d g-c h) \, _2F_1\left (1,m+1;m+2;\frac {h (a+b x)}{a h-b g}\right )}{b g-a h}-\frac {(d e-c f) \, _2F_1\left (1,m+1;m+2;\frac {f (a+b x)}{a f-b e}\right )}{b e-a f}\right )}{(m+1) (f g-e h)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{f h x^{2} + e g + {\left (f g + e h\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (h x + g\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right ) \left (b x +a \right )^{m}}{\left (f x +e \right ) \left (h x +g \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{m}}{{\left (f x + e\right )} {\left (h x + g\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m\,\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (g+h\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{m} \left (c + d x\right )}{\left (e + f x\right ) \left (g + h x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________