3.2543 \(\int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=424 \[ \frac{c (d+e x)^{m+1} \left (-2 c e \left (d m \sqrt{b^2-4 a c}-2 a e (1-m)+2 b d\right )+b e^2 m \left (\sqrt{b^2-4 a c}+b\right )+4 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (b^2-4 a c\right )^{3/2} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c (d+e x)^{m+1} \left (\frac{-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{(d+e x)^{m+1} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(((d + e*x)^(1 + m)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (c*(4*c^2*d^2 + b*(b + Sqrt[b^2
- 4*a*c])*e^2*m - 2*c*e*(2*b*d - 2*a*e*(1 - m) + Sqrt[b^2 - 4*a*c]*d*m))*(d + e
*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e)])/((b^2 - 4*a*c)^(3/2)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c
*d^2 - b*d*e + a*e^2)*(1 + m)) - (c*(e*(2*c*d - b*e)*m + (4*c^2*d^2 - 4*c*e*(b*d
- a*e*(1 - m)) + b^2*e^2*m)/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + m)*Hypergeometric
2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((b^2
- 4*a*c)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + m))

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Rubi [A]  time = 2.76382, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{c (d+e x)^{m+1} \left (-2 c e \left (d m \sqrt{b^2-4 a c}-2 a e (1-m)+2 b d\right )+b e^2 m \left (\sqrt{b^2-4 a c}+b\right )+4 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (b^2-4 a c\right )^{3/2} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c (d+e x)^{m+1} \left (\frac{-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{(d+e x)^{m+1} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(((d + e*x)^(1 + m)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (c*(4*c^2*d^2 + b*(b + Sqrt[b^2
- 4*a*c])*e^2*m - 2*c*e*(2*b*d - 2*a*e*(1 - m) + Sqrt[b^2 - 4*a*c]*d*m))*(d + e
*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e)])/((b^2 - 4*a*c)^(3/2)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c
*d^2 - b*d*e + a*e^2)*(1 + m)) - (c*(e*(2*c*d - b*e)*m + (4*c^2*d^2 - 4*c*e*(b*d
- a*e*(1 - m)) + b^2*e^2*m)/Sqrt[b^2 - 4*a*c])*(d + e*x)^(1 + m)*Hypergeometric
2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/((b^2
- 4*a*c)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)*(1 + m))

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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/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.114901, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.179, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^m/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2),
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/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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