3.951 \(\int \frac{(d+e x)^m}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \sqrt{a+b x+c x^2}} \]

[Out]

((d + e*x)^(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)]*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*(1 + m)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.64371, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(d+e x)^{m+1} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e (m+1) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(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)]*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*(1 + m)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 37.7253, size = 172, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{\frac{c \left (- 2 d - 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} + 1} \sqrt{\frac{c \left (2 d + 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} + 1} \operatorname{appellf_{1}}{\left (m + 1,\frac{1}{2},\frac{1}{2},m + 2,\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}},\frac{c \left (2 d + 2 e x\right )}{2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )} \right )}}{e \left (m + 1\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(c*x**2+b*x+a)**(1/2),x)

[Out]

(d + e*x)**(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)*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*(m + 1)*sqrt(a + b*x + c*
x**2))

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Mathematica [A]  time = 0.02702, size = 207, normalized size = 1.1 \[ \frac{(d+e x)^{m+1} \sqrt{\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}} \sqrt{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}} F_1\left (m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e},\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e (m+1) \sqrt{a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[(e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]
*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*
(d + e*x)^(1 + m)*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*(1 + m)*Sqrt[a + x*(b + c*x)])

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Maple [F]  time = 0.176, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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