Integrand size = 23, antiderivative size = 199 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=-\frac {(b e g (2+p)-c (e f+d g) (3+2 p)-2 c e g (1+p) x) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}+\frac {2^{-2 (1+p)} \left (c (2 c d f-b e f-b d g) (3+2 p)-e g \left (2 a c-b^2 (2+p)\right )\right ) (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^3 (3+2 p)} \] Output:
-1/2*(b*e*g*(2+p)-c*(d*g+e*f)*(3+2*p)-2*c*e*g*(p+1)*x)*(c*x^2+b*x+a)^(p+1) /c^2/(p+1)/(3+2*p)+(c*(-b*d*g-b*e*f+2*c*d*f)*(3+2*p)-e*g*(2*a*c-b^2*(2+p)) )*(2*c*x+b)*(c*x^2+b*x+a)^p*hypergeom([1/2, -p],[3/2],(2*c*x+b)^2/(-4*a*c+ b^2))/(2^(2*p+2))/c^3/(3+2*p)/((-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^p)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 1.50 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.10 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\frac {1}{6} (a+x (b+c x))^p \left (3 (e f+d g) x^2 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (2,-p,-p,3,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+2 e g x^3 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\frac {3\ 2^p d f \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c (1+p)}\right ) \] Input:
Integrate[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^p,x]
Output:
((a + x*(b + c*x))^p*((3*(e*f + d*g)*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/( b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2* c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (2*e*g*x^3*AppellF1[3, -p, -p, 4, (-2*c *x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqr t[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^p) + (3*2^p*d*f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c *x)/(2*Sqrt[b^2 - 4*a*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sq rt[b^2 - 4*a*c])^p)))/6
Time = 0.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1225, 1096}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle -\frac {\left (2 a c e g+b^2 (-e) g (p+2)-c (2 p+3) (2 c d f-b (d g+e f))\right ) \int \left (c x^2+b x+a\right )^pdx}{2 c^2 (2 p+3)}-\frac {\left (a+b x+c x^2\right )^{p+1} (b e g (p+2)-c (2 p+3) (d g+e f)-2 c e g (p+1) x)}{2 c^2 (p+1) (2 p+3)}\) |
| \(\Big \downarrow \) 1096 |
| \(\displaystyle \frac {2^p \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right ) \left (2 a c e g+b^2 (-e) g (p+2)-c (2 p+3) (2 c d f-b (d g+e f))\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {\left (a+b x+c x^2\right )^{p+1} (b e g (p+2)-c (2 p+3) (d g+e f)-2 c e g (p+1) x)}{2 c^2 (p+1) (2 p+3)}\) |
Input:
Int[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^p,x]
Output:
-1/2*((b*e*g*(2 + p) - c*(e*f + d*g)*(3 + 2*p) - 2*c*e*g*(1 + p)*x)*(a + b *x + c*x^2)^(1 + p))/(c^2*(1 + p)*(3 + 2*p)) + (2^p*(2*a*c*e*g - b^2*e*g*( 2 + p) - c*(2*c*d*f - b*(e*f + d*g))*(3 + 2*p))*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hypergeome tric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4* a*c])])/(c^2*Sqrt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) /(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) ], x]] /; FreeQ[{a, b, c, p}, x] && !IntegerQ[4*p] && !IntegerQ[3*p]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
\[\int \left (e x +d \right ) \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{p}d x\]
Input:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x)
Output:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (g x + f\right )} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x, algorithm="fricas")
Output:
integral((e*g*x^2 + d*f + (e*f + d*g)*x)*(c*x^2 + b*x + a)^p, x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\int \left (d + e x\right ) \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{p}\, dx \] Input:
integrate((e*x+d)*(g*x+f)*(c*x**2+b*x+a)**p,x)
Output:
Integral((d + e*x)*(f + g*x)*(a + b*x + c*x**2)**p, x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (g x + f\right )} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x, algorithm="maxima")
Output:
integrate((e*x + d)*(g*x + f)*(c*x^2 + b*x + a)^p, x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (g x + f\right )} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x, algorithm="giac")
Output:
integrate((e*x + d)*(g*x + f)*(c*x^2 + b*x + a)^p, x)
Timed out. \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\int \left (f+g\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \] Input:
int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^p,x)
Output:
int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^p, x)
\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^p \, dx=\text {too large to display} \] Input:
int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^p,x)
Output:
( - 4*(a + b*x + c*x**2)**p*a**2*c*e*g*p - 4*(a + b*x + c*x**2)**p*a**2*c* e*g + (a + b*x + c*x**2)**p*a*b**2*e*g*p + 2*(a + b*x + c*x**2)**p*a*b**2* e*g - 2*(a + b*x + c*x**2)**p*a*b*c*d*g*p - 3*(a + b*x + c*x**2)**p*a*b*c* d*g - 2*(a + b*x + c*x**2)**p*a*b*c*e*f*p - 3*(a + b*x + c*x**2)**p*a*b*c* e*f + 4*(a + b*x + c*x**2)**p*a*b*c*e*g*p**2*x + 4*(a + b*x + c*x**2)**p*a *b*c*e*g*p*x + 8*(a + b*x + c*x**2)**p*a*c**2*d*f*p**2 + 20*(a + b*x + c*x **2)**p*a*c**2*d*f*p + 12*(a + b*x + c*x**2)**p*a*c**2*d*f - (a + b*x + c* x**2)**p*b**3*e*g*p**2*x - 2*(a + b*x + c*x**2)**p*b**3*e*g*p*x + 2*(a + b *x + c*x**2)**p*b**2*c*d*g*p**2*x + 3*(a + b*x + c*x**2)**p*b**2*c*d*g*p*x + 2*(a + b*x + c*x**2)**p*b**2*c*e*f*p**2*x + 3*(a + b*x + c*x**2)**p*b** 2*c*e*f*p*x + 2*(a + b*x + c*x**2)**p*b**2*c*e*g*p**2*x**2 + (a + b*x + c* x**2)**p*b**2*c*e*g*p*x**2 + 4*(a + b*x + c*x**2)**p*b*c**2*d*f*p**2*x + 1 0*(a + b*x + c*x**2)**p*b*c**2*d*f*p*x + 6*(a + b*x + c*x**2)**p*b*c**2*d* f*x + 4*(a + b*x + c*x**2)**p*b*c**2*d*g*p**2*x**2 + 8*(a + b*x + c*x**2)* *p*b*c**2*d*g*p*x**2 + 3*(a + b*x + c*x**2)**p*b*c**2*d*g*x**2 + 4*(a + b* x + c*x**2)**p*b*c**2*e*f*p**2*x**2 + 8*(a + b*x + c*x**2)**p*b*c**2*e*f*p *x**2 + 3*(a + b*x + c*x**2)**p*b*c**2*e*f*x**2 + 4*(a + b*x + c*x**2)**p* b*c**2*e*g*p**2*x**3 + 6*(a + b*x + c*x**2)**p*b*c**2*e*g*p*x**3 + 2*(a + b*x + c*x**2)**p*b*c**2*e*g*x**3 + 32*int(((a + b*x + c*x**2)**p*x)/(4*a*p **2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 8*c*...