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

Optimal. Leaf size=388 \[ \frac{\sqrt{a+b x+c x^2} (e f-d g) (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

((e*f - d*g)*(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)])/(e^2*(1 + 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*(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)])/(e^2*(2 + 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 [A]  time = 1.40259, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{a+b x+c x^2} (e f-d g) (d+e x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+1) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{g \sqrt{a+b x+c x^2} (d+e x)^{m+2} F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e^2 (m+2) \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((e*f - d*g)*(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)])/(e^2*(1 + 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*(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)])/(e^2*(2 + 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 [A]  time = 93.1527, size = 364, normalized size = 0.94 \[ \frac{g \left (d + e x\right )^{m + 2} \sqrt{a + b x + c x^{2}} \operatorname{appellf_{1}}{\left (m + 2,- \frac{1}{2},- \frac{1}{2},m + 3,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e^{2} \left (m + 2\right ) \sqrt{\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1} \sqrt{\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1}} - \frac{\left (d + e x\right )^{m + 1} \left (d g - e f\right ) \sqrt{a + b x + c x^{2}} \operatorname{appellf_{1}}{\left (m + 1,- \frac{1}{2},- \frac{1}{2},m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e^{2} \left (m + 1\right ) \sqrt{\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1} \sqrt{\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 1.57829, size = 0, normalized size = 0. \[ \int (d+e x)^m (f+g x) \sqrt{a+b x+c x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) \sqrt{c{x}^{2}+bx+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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