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

Optimal. Leaf size=525 \[ \frac{(d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (e g^2 (m+1) (b d-a e)+c \left (2 d^2 g^2 (p+1)-2 d e f g (m+2 p+3)+e^2 f^2 (m+2 p+3)\right )\right )}{c e^3 (m+1) (m+2 p+3)}-\frac{g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} (b e g (m+p+2)+2 c (d g (p+1)-e f (m+2 p+3))) F_1\left (m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^3 (m+2) (m+2 p+3)}+\frac{g^2 (d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)} \]

[Out]

(g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(1 + p))/(c*e*(3 + m + 2*p)) + ((e*(b*d
- a*e)*g^2*(1 + m) + c*(2*d^2*g^2*(1 + p) + e^2*f^2*(3 + m + 2*p) - 2*d*e*f*g*(
3 + m + 2*p)))*(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)])/(c*e^3*(1 + m)*(3 + m + 2*p)*(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) - (g*(b*e*g*(2 + m + p) + 2*c*(d*g*(1 + p) - e*f*(3 + m + 2
*p)))*(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 + Sq
rt[b^2 - 4*a*c])*e)])/(c*e^3*(2 + m)*(3 + m + 2*p)*(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 = 1.71325, antiderivative size = 523, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{(d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (g^2 (b d-a e)+\frac{c \left (2 d^2 g^2 (p+1)-2 d e f g (m+2 p+3)+e^2 f^2 (m+2 p+3)\right )}{e (m+1)}\right )}{c e^2 (m+2 p+3)}-\frac{g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )^{-p} (b e g (m+p+2)+2 c d g (p+1)-2 c e f (m+2 p+3)) F_1\left (m+2;-p,-p;m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^3 (m+2) (m+2 p+3)}+\frac{g^2 (d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(1 + p))/(c*e*(3 + m + 2*p)) + (((b*d -
a*e)*g^2 + (c*(2*d^2*g^2*(1 + p) + e^2*f^2*(3 + m + 2*p) - 2*d*e*f*g*(3 + m + 2
*p)))/(e*(1 + m)))*(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)])/(c*e^2*(3 + m + 2*p)*(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) - (g*(2*c*d*g*(1 + p) + b*e*g*(2 + m + p) - 2*c*e*f*(3 + m + 2*
p))*(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)])/(c*e^3*(2 + m)*(3 + m + 2*p)*(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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 2.70802, size = 0, normalized size = 0. \[ \int (d+e x)^m (f+g x)^2 \left (a+b x+c x^2\right )^p \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((g*x + f)^2*(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. \[{\rm integral}\left ({\left (g^{2} x^{2} + 2 \, f g x + f^{2}\right )}{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((g^2*x^2 + 2*f*g*x + f^2)*(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)**2*(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. \[ \int{\left (g x + f\right )}^{2}{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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