3.945 \(\int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=509 \[ \frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (e g^2 (m+1) (b d-a e)+c \left (3 d^2 g^2-2 d e f g (m+4)+e^2 f^2 (m+4)\right )\right )}{c e^3 (m+1) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} (b e g (2 m+5)+2 c (3 d g-2 e f (m+4))) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c e^3 (m+2) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (d+e x)^{m+1}}{c e (m+4)} \]

[Out]

(g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(3/2))/(c*e*(4 + m)) + ((e*(b*d - a*e)*
g^2*(1 + m) + c*(3*d^2*g^2 + e^2*f^2*(4 + m) - 2*d*e*f*g*(4 + m)))*(d + e*x)^(1
+ m)*Sqrt[a + b*x + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 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)*(4 + m)*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2
- 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]) - (
g*(b*e*g*(5 + 2*m) + 2*c*(3*d*g - 2*e*f*(4 + m)))*(d + e*x)^(2 + m)*Sqrt[a + b*x
+ c*x^2]*AppellF1[2 + m, -1/2, -1/2, 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)])/(2*c*e^3
*(2 + m)*(4 + m)*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*S
qrt[1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])

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Rubi [A]  time = 2.36314, antiderivative size = 506, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\sqrt{a+b x+c x^2} (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) \left (g^2 (b d-a e)+\frac{c \left (3 d^2 g^2-2 d e f g (m+4)+e^2 f^2 (m+4)\right )}{e (m+1)}\right )}{c e^2 (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} (b e g (2 m+5)+6 c d g-4 c e f (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c e^3 (m+2) (m+4) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (d+e x)^{m+1}}{c e (m+4)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(3/2))/(c*e*(4 + m)) + (((b*d - a*e)*g^
2 + (c*(3*d^2*g^2 + e^2*f^2*(4 + m) - 2*d*e*f*g*(4 + m)))/(e*(1 + m)))*(d + e*x)
^(1 + m)*Sqrt[a + b*x + c*x^2]*AppellF1[1 + m, -1/2, -1/2, 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*(4 + m)*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*
a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]) - (g*(6
*c*d*g - 4*c*e*f*(4 + m) + b*e*g*(5 + 2*m))*(d + e*x)^(2 + m)*Sqrt[a + b*x + c*x
^2]*AppellF1[2 + m, -1/2, -1/2, 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)])/(2*c*e^3*(2 +
m)*(4 + m)*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1
- (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])

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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)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.12638, size = 0, normalized size = 0. \[ \int (d+e x)^m (f+g x)^2 \sqrt{a+b x+c x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{2}\sqrt{c{x}^{2}+bx+a}\, dx \]

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)^(1/2),x)

[Out]

int((e*x+d)^m*(g*x+f)^2*(c*x^2+b*x+a)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((g^2*x^2 + 2*f*g*x + f^2)*sqrt(c*x^2 + b*x + a)*(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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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