Optimal. Leaf size=231 \[ \frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} (g (m+n+2) (a e g (m+n+3)-c d (d g (n+1)+e f (m+2)))+c (m+2) (e f-d g) (d g (n+1)+e f (m+1))) \, _2F_1\left (m+1,-n;m+2;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g^2 (m+1) (m+n+2) (m+n+3)}-\frac {c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 227, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {951, 80, 70, 69} \[ \frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (a e g (m+n+3)+\frac {c (m+2) (e f-d g) (d g (n+1)+e f (m+1))}{g (m+n+2)}-c d (d g (n+1)+e f (m+2))\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g (m+1) (m+n+3)}-\frac {c (m+2) (e f-d g) (d+e x)^{m+1} (f+g x)^{n+1}}{e g^2 (m+n+2) (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e g (m+n+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 80
Rule 951
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\int (d+e x)^m (f+g x)^n \left (e (a e g (3+m+n)-c d (e f (2+m)+d g (1+n)))-c e^2 (e f-d g) (2+m) x\right ) \, dx}{e^2 g (3+m+n)}\\ &=-\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) \int (d+e x)^m (f+g x)^n \, dx}{e g (3+m+n)}\\ &=-\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n}\right ) \int (d+e x)^m \left (\frac {e f}{e f-d g}+\frac {e g x}{e f-d g}\right )^n \, dx}{e g (3+m+n)}\\ &=-\frac {c (e f-d g) (2+m) (d+e x)^{1+m} (f+g x)^{1+n}}{e g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e g (3+m+n)}+\frac {\left (a e g (3+m+n)+\frac {c (e f-d g) (2+m) (e f (1+m)+d g (1+n))}{g (2+m+n)}-c d (e f (2+m)+d g (1+n))\right ) (d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g (1+m) (3+m+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 179, normalized size = 0.77 \[ \frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (e \left (a g^2+c f (e f-2 d g)\right ) \, _2F_1\left (m+1,-n;m+2;\frac {g (d+e x)}{d g-e f}\right )+c (e f-d g)^2 \, _2F_1\left (m+1,-n-2;m+2;\frac {g (d+e x)}{d g-e f}\right )-2 c (e f-d g)^2 \, _2F_1\left (m+1,-n-1;m+2;\frac {g (d+e x)}{d g-e f}\right )\right )}{e^2 g^2 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (c e \,x^{2}+2 c d x +a \right ) \left (e x +d \right )^{m} \left (g x +f \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c e x^{2} + 2 \, c d x + a\right )} {\left (e x + d\right )}^{m} {\left (g x + f\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^m\,\left (c\,e\,x^2+2\,c\,d\,x+a\right ) \,d x \]
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]
________________________________________________________________________________________