3.953 \(\int (d+e x)^m (f+g x)^n (a+b x+c x^2) \, dx\)

Optimal. Leaf size=265 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )-(d g (n+1)+e f (m+1)) (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}+\frac{(d+e x)^{m+1} (f+g x)^{n+1} (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]

[Out]

((b*e*g*(3 + m + n) - c*(e*f*(2 + m) + d*g*(4 + m + 2*n)))*(d + e*x)^(1 + m)*(f + g*x)^(1 + n))/(e^2*g^2*(2 +
m + n)*(3 + m + n)) + (c*(d + e*x)^(2 + m)*(f + g*x)^(1 + n))/(e^2*g*(3 + m + n)) + ((g*(2 + m + n)*(a*e^2*g*(
3 + m + n) - c*d*(e*f*(2 + m) + d*g*(1 + n))) - (e*f*(1 + m) + d*g*(1 + n))*(b*e*g*(3 + m + n) - c*(e*f*(2 + m
) + d*g*(4 + m + 2*n))))*(d + e*x)^(1 + m)*(f + g*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((g*(d + e*x))/(e*
f - d*g))])/(e^3*g^2*(1 + m)*(2 + m + n)*(3 + m + n)*((e*(f + g*x))/(e*f - d*g))^n)

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Rubi [A]  time = 0.350857, antiderivative size = 263, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {951, 80, 70, 69} \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )+(d g (n+1)+e f (m+1)) (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}-\frac{(d+e x)^{m+1} (f+g x)^{n+1} (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2),x]

[Out]

-(((c*e*f*(2 + m) - b*e*g*(3 + m + n) + c*d*g*(4 + m + 2*n))*(d + e*x)^(1 + m)*(f + g*x)^(1 + n))/(e^2*g^2*(2
+ m + n)*(3 + m + n))) + (c*(d + e*x)^(2 + m)*(f + g*x)^(1 + n))/(e^2*g*(3 + m + n)) + (((e*f*(1 + m) + d*g*(1
+ n))*(c*e*f*(2 + m) - b*e*g*(3 + m + n) + c*d*g*(4 + m + 2*n)) + g*(2 + m + n)*(a*e^2*g*(3 + m + n) - c*d*(e
*f*(2 + m) + d*g*(1 + n))))*(d + e*x)^(1 + m)*(f + g*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((g*(d + e*x))/
(e*f - d*g))])/(e^3*g^2*(1 + m)*(2 + m + n)*(3 + m + n)*((e*(f + g*x))/(e*f - d*g))^n)

Rule 951

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x
] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int (d+e x)^m (f+g x)^n \left (a+b x+c x^2\right ) \, dx &=\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\int (d+e x)^m (f+g x)^n \left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))-e (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) x\right ) \, dx}{e^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+n)}\right ) \int (d+e x)^m (f+g x)^n \, dx}{e^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+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^2 g (3+m+n)}\\ &=-\frac{(c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac{c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac{\left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))+\frac{(e f (1+m)+d g (1+n)) (c e f (2+m)-b e g (3+m+n)+c d g (4+m+2 n))}{g (2+m+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^3 g (1+m) (3+m+n)}\\ \end{align*}

Mathematica [A]  time = 0.208153, size = 187, normalized size = 0.71 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \left (e \left (e \left (g (a g-b f)+c f^2\right ) \, _2F_1\left (m+1,-n;m+2;\frac{g (d+e x)}{d g-e f}\right )-(2 c f-b g) (e f-d g) \, _2F_1\left (m+1,-n-1;m+2;\frac{g (d+e x)}{d g-e f}\right )\right )+c (e f-d g)^2 \, _2F_1\left (m+1,-n-2;m+2;\frac{g (d+e x)}{d g-e f}\right )\right )}{e^3 g^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2),x]

[Out]

((d + e*x)^(1 + m)*(f + g*x)^n*(c*(e*f - d*g)^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (g*(d + e*x))/(-(e*f)
+ d*g)] + e*(-((2*c*f - b*g)*(e*f - d*g)*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)]
) + e*(c*f^2 + g*(-(b*f) + a*g))*Hypergeometric2F1[1 + m, -n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)])))/(e^3*g^2
*(1 + m)*((e*(f + g*x))/(e*f - d*g))^n)

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Maple [F]  time = 0.623, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{n} \left ( c{x}^{2}+bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(g*x+f)^n*(c*x^2+b*x+a),x)

[Out]

int((e*x+d)^m*(g*x+f)^n*(c*x^2+b*x+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}{\left (g x + f\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)*(e*x + d)^m*(g*x + f)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}{\left (g x + f\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)*(e*x + d)^m*(g*x + f)^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(g*x+f)**n*(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}{\left (g x + f\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^n*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)*(e*x + d)^m*(g*x + f)^n, x)