Integrand size = 20, antiderivative size = 287 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{1+p}}{2 c (2+p)}-\frac {e \left (b e (2 c d-b e) (2+p) (3+p)-2 c (3+2 p) \left (c d^2 (5+2 p)-e (a e+b d (2+p))\right )-2 c e (2 c d-b e) (1+p) (3+p) x\right ) \left (a+b x+c x^2\right )^{1+p}}{4 c^3 (1+p) (2+p) (3+2 p)}+\frac {2^{-3-2 p} (2 c d-b e) \left (b^2 e^2 (3+p)+2 c^2 d^2 (3+2 p)-2 c e (3 a e+b d (3+2 p))\right ) (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^4 (3+2 p)} \] Output:
1/2*e*(e*x+d)^2*(c*x^2+b*x+a)^(p+1)/c/(2+p)-1/4*e*(b*e*(-b*e+2*c*d)*(2+p)* (3+p)-2*c*(3+2*p)*(c*d^2*(5+2*p)-e*(a*e+b*d*(2+p)))-2*c*e*(-b*e+2*c*d)*(p+ 1)*(3+p)*x)*(c*x^2+b*x+a)^(p+1)/c^3/(p+1)/(2+p)/(3+2*p)+2^(-3-2*p)*(-b*e+2 *c*d)*(b^2*e^2*(3+p)+2*c^2*d^2*(3+2*p)-2*c*e*(3*a*e+b*d*(3+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))/c^4/ (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 = 1.56 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.94 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\frac {1}{4} (a+x (b+c x))^p \left (6 d^2 e x^2 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (2,-p,-p,3,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+4 d e^2 x^3 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+e^3 x^4 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (4,-p,-p,5,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\frac {2^{1+p} d^3 \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c (1+p)}\right ) \] Input:
Integrate[(d + e*x)^3*(a + b*x + c*x^2)^p,x]
Output:
((a + x*(b + c*x))^p*((6*d^2*e*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sq rt[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) + (4*d*e^2*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 - 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) + (e^3*x^4*AppellF1[4, -p, -p, 5, (-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^(1 + p)*d^3*(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) /Sqrt[b^2 - 4*a*c])^p)))/4
Time = 0.51 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1166, 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.
| \(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {\int (d+e x) \left (2 c (p+2) d^2-e (2 a e+b d (p+1))+e (2 c d-b e) (p+3) x\right ) \left (c x^2+b x+a\right )^pdx}{2 c (p+2)}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {(p+2) (2 c d-b e) \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \int \left (c x^2+b x+a\right )^pdx}{2 c^2 (2 p+3)}-\frac {e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)}\) |
| \(\Big \downarrow \) 1096 |
| \(\displaystyle \frac {-\frac {2^p (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)}\) |
Input:
Int[(d + e*x)^3*(a + b*x + c*x^2)^p,x]
Output:
(e*(d + e*x)^2*(a + b*x + c*x^2)^(1 + p))/(2*c*(2 + p)) + (-1/2*(e*(b*e*(2 *c*d - b*e)*(2 + p)*(3 + p) - 2*c*(3 + 2*p)*(c*d^2*(5 + 2*p) - e*(a*e + b* d*(2 + p))) - 2*c*e*(2*c*d - b*e)*(1 + p)*(3 + p)*x)*(a + b*x + c*x^2)^(1 + p))/(c^2*(1 + p)*(3 + 2*p)) - (2^p*(2*c*d - b*e)*(2 + p)*(b^2*e^2*(3 + p ) + 2*c^2*d^2*(3 + 2*p) - 2*c*e*(3*a*e + b*d*(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)*H ypergeometric2F1[-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)))/(2*c*(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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
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]
\[\int \left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{p}d x\]
Input:
int((e*x+d)^3*(c*x^2+b*x+a)^p,x)
Output:
int((e*x+d)^3*(c*x^2+b*x+a)^p,x)
\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^p,x, algorithm="fricas")
Output:
integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(c*x^2 + b*x + a)^p, x)
\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{p}\, dx \] Input:
integrate((e*x+d)**3*(c*x**2+b*x+a)**p,x)
Output:
Integral((d + e*x)**3*(a + b*x + c*x**2)**p, x)
\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^p,x, algorithm="maxima")
Output:
integrate((e*x + d)^3*(c*x^2 + b*x + a)^p, x)
\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^p,x, algorithm="giac")
Output:
integrate((e*x + d)^3*(c*x^2 + b*x + a)^p, x)
Timed out. \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^p \,d x \] Input:
int((d + e*x)^3*(a + b*x + c*x^2)^p,x)
Output:
int((d + e*x)^3*(a + b*x + c*x^2)^p, x)
\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx=\text {too large to display} \] Input:
int((e*x+d)^3*(c*x^2+b*x+a)^p,x)
Output:
(4*(a + b*x + c*x**2)**p*a**2*b*c*e**3*p**2 + 20*(a + b*x + c*x**2)**p*a** 2*b*c*e**3*p + 18*(a + b*x + c*x**2)**p*a**2*b*c*e**3 - 24*(a + b*x + c*x* *2)**p*a**2*c**2*d*e**2*p**2 - 72*(a + b*x + c*x**2)**p*a**2*c**2*d*e**2*p - 48*(a + b*x + c*x**2)**p*a**2*c**2*d*e**2 - (a + b*x + c*x**2)**p*a*b** 3*e**3*p**2 - 5*(a + b*x + c*x**2)**p*a*b**3*e**3*p - 6*(a + b*x + c*x**2) **p*a*b**3*e**3 + 6*(a + b*x + c*x**2)**p*a*b**2*c*d*e**2*p**2 + 24*(a + b *x + c*x**2)**p*a*b**2*c*d*e**2*p + 24*(a + b*x + c*x**2)**p*a*b**2*c*d*e* *2 - 4*(a + b*x + c*x**2)**p*a*b**2*c*e**3*p**3*x - 20*(a + b*x + c*x**2)* *p*a*b**2*c*e**3*p**2*x - 18*(a + b*x + c*x**2)**p*a*b**2*c*e**3*p*x - 12* (a + b*x + c*x**2)**p*a*b*c**2*d**2*e*p**2 - 42*(a + b*x + c*x**2)**p*a*b* c**2*d**2*e*p - 36*(a + b*x + c*x**2)**p*a*b*c**2*d**2*e + 24*(a + b*x + c *x**2)**p*a*b*c**2*d*e**2*p**3*x + 72*(a + b*x + c*x**2)**p*a*b*c**2*d*e** 2*p**2*x + 48*(a + b*x + c*x**2)**p*a*b*c**2*d*e**2*p*x + 8*(a + b*x + c*x **2)**p*a*b*c**2*e**3*p**3*x**2 + 16*(a + b*x + c*x**2)**p*a*b*c**2*e**3*p **2*x**2 + 6*(a + b*x + c*x**2)**p*a*b*c**2*e**3*p*x**2 + 16*(a + b*x + c* x**2)**p*a*c**3*d**3*p**3 + 72*(a + b*x + c*x**2)**p*a*c**3*d**3*p**2 + 10 4*(a + b*x + c*x**2)**p*a*c**3*d**3*p + 48*(a + b*x + c*x**2)**p*a*c**3*d* *3 + (a + b*x + c*x**2)**p*b**4*e**3*p**3*x + 5*(a + b*x + c*x**2)**p*b**4 *e**3*p**2*x + 6*(a + b*x + c*x**2)**p*b**4*e**3*p*x - 6*(a + b*x + c*x**2 )**p*b**3*c*d*e**2*p**3*x - 24*(a + b*x + c*x**2)**p*b**3*c*d*e**2*p**2...