3.955 \(\int (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=384 \[ \frac{(e f-d g) (d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1)}+\frac{g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2)} \]

[Out]

((e*f - d*g)*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m
, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)])/(e^2*(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqr
t[b^2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^
p) + (g*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*AppellF1[2 + m, -p, -p, 3 + m, (2*
c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)])/(e^2*(2 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2
- 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

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Rubi [A]  time = 0.883747, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(e f-d g) (d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1)}+\frac{g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((e*f - d*g)*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1[1 + m, -p, -p, 2 + m
, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)])/(e^2*(1 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqr
t[b^2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^
p) + (g*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*AppellF1[2 + m, -p, -p, 3 + m, (2*
c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x))/(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)])/(e^2*(2 + m)*(1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2
- 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^p)

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Rubi in Sympy [A]  time = 99.0257, size = 347, normalized size = 0.9 \[ \frac{g \left (d + e x\right )^{m + 2} \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{- p} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 2,- p,- p,m + 3,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e^{2} \left (m + 2\right )} - \frac{\left (d + e x\right )^{m + 1} \left (d g - e f\right ) \left (\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1\right )^{- p} \left (\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(g*x+f)*(c*x**2+b*x+a)**p,x)

[Out]

g*(d + e*x)**(m + 2)*(c*(-2*d - 2*e*x)/(2*c*d - e*(b + sqrt(-4*a*c + b**2))) + 1
)**(-p)*(c*(2*d + 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)) + 1)**(-p)*(a + b
*x + c*x**2)**p*appellf1(m + 2, -p, -p, m + 3, c*(-2*d - 2*e*x)/(b*e - 2*c*d - e
*sqrt(-4*a*c + b**2)), c*(2*d + 2*e*x)/(2*c*d - e*(b + sqrt(-4*a*c + b**2))))/(e
**2*(m + 2)) - (d + e*x)**(m + 1)*(d*g - e*f)*(c*(-2*d - 2*e*x)/(2*c*d - e*(b +
sqrt(-4*a*c + b**2))) + 1)**(-p)*(c*(2*d + 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c +
b**2)) + 1)**(-p)*(a + b*x + c*x**2)**p*appellf1(m + 1, -p, -p, m + 2, c*(-2*d
- 2*e*x)/(b*e - 2*c*d - e*sqrt(-4*a*c + b**2)), c*(2*d + 2*e*x)/(2*c*d - e*(b +
sqrt(-4*a*c + b**2))))/(e**2*(m + 1))

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Mathematica [A]  time = 1.99338, size = 0, normalized size = 0. \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^p \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(g*x+f)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (g x + f\right )}{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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