Optimal. Leaf size=157 \[ \frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {g (d+e x)}{e f-d g}\right ) (c f (2 d g-e f (m+2))-g (a e g m+b (d g-e f (m+1))))}{g^2 (m+1) (e f-d g)^2}+\frac {(d+e x)^{m+1} \left (a+\frac {f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac {c (d+e x)^{m+1}}{e g^2 (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {949, 80, 68} \[ -\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {g (d+e x)}{e f-d g}\right ) (g (a e g m+b d g-b e f (m+1))-c f (2 d g-e f (m+2)))}{g^2 (m+1) (e f-d g)^2}+\frac {(d+e x)^{m+1} \left (a+\frac {f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac {c (d+e x)^{m+1}}{e g^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 80
Rule 949
Rubi steps
\begin {align*} \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx &=\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}+\frac {\int \frac {(d+e x)^m \left (\frac {c d f g-a e g^2 m-c e f^2 (1+m)-b g (d g-e f (1+m))}{g^2}-c \left (d-\frac {e f}{g}\right ) x\right )}{f+g x} \, dx}{e f-d g}\\ &=\frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) \int \frac {(d+e x)^m}{f+g x} \, dx}{g^2 (e f-d g)}\\ &=\frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g)^2 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 134, normalized size = 0.85 \[ \frac {(d+e x)^{m+1} \left (e^2 \left (g (a g-b f)+c f^2\right ) \, _2F_1\left (2,m+1;m+2;\frac {g (d+e x)}{d g-e f}\right )-e (2 c f-b g) (e f-d g) \, _2F_1\left (1,m+1;m+2;\frac {g (d+e x)}{d g-e f}\right )+c (e f-d g)^2\right )}{e g^2 (m+1) (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{2}+b x +a \right ) \left (e x +d \right )^{m}}{\left (g x +f \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{{\left (f+g\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________