3.2544 \(\int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=988 \[ -\frac{c \left (24 c^4 d^4-4 c^3 e \left (12 b d+3 \sqrt{b^2-4 a c} m d-2 a e \left (-m^2-2 m+6\right )\right ) d^2-b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4 (1-m) m+2 b c e^3 m \left (-d (m+2) b^2+\left (3 a e (2-m)-\sqrt{b^2-4 a c} d (m+2)\right ) b+a \sqrt{b^2-4 a c} e (5-2 m)\right )+2 c^2 e^2 \left (b^2 \left (m^2+2 m+12\right ) d^2+b \left (9 \sqrt{b^2-4 a c} d m-4 a e \left (-m^2-2 m+6\right )\right ) d-2 a e \left (\sqrt{b^2-4 a c} d (5-2 m) m-2 a e \left (m^2-4 m+3\right )\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^{5/2} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^2 (m+1)}+\frac{c \left (24 c^4 d^4-4 c^3 e \left (12 b d-3 \sqrt{b^2-4 a c} m d-2 a e \left (-m^2-2 m+6\right )\right ) d^2-b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4 (1-m) m-2 b c e^3 m \left (d (m+2) b^2-\left (3 a e (2-m)+\sqrt{b^2-4 a c} d (m+2)\right ) b+a \sqrt{b^2-4 a c} e (5-2 m)\right )+2 c^2 e^2 \left (b^2 \left (m^2+2 m+12\right ) d^2-b \left (9 \sqrt{b^2-4 a c} d m+4 a e \left (-m^2-2 m+6\right )\right ) d+2 a e \left (\sqrt{b^2-4 a c} d (5-2 m) m+2 a e \left (m^2-4 m+3\right )\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^{5/2} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^2 (m+1)}-\frac{\left (a c e (2-m) (2 c d-b e)^2-c \left (6 c^2 d^2-2 c e (3 b d-a e (5-2 m))-b^2 e^2 (1-m)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 d^2-b^2 e^2 (1-m)+c e (2 a e (3-m)-b d (m+4))\right )\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}-\frac{\left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2} \]

[Out]

-((d + e*x)^(1 + m)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - ((d + e*x)^(1 + m)*(a*c*e*(2*
c*d - b*e)^2*(2 - m) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2 - b^2*e^2*(1 - m) +
c*e*(2*a*e*(3 - m) - b*d*(4 + m))) - c*(2*c*d - b*e)*(6*c^2*d^2 - 2*c*e*(3*b*d -
a*e*(5 - 2*m)) - b^2*e^2*(1 - m))*x))/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2
)^2*(a + b*x + c*x^2)) - (c*(24*c^4*d^4 - b^3*(b + Sqrt[b^2 - 4*a*c])*e^4*(1 - m
)*m - 4*c^3*d^2*e*(12*b*d + 3*Sqrt[b^2 - 4*a*c]*d*m - 2*a*e*(6 - 2*m - m^2)) + 2
*b*c*e^3*m*(a*Sqrt[b^2 - 4*a*c]*e*(5 - 2*m) - b^2*d*(2 + m) + b*(3*a*e*(2 - m) -
Sqrt[b^2 - 4*a*c]*d*(2 + m))) + 2*c^2*e^2*(b^2*d^2*(12 + 2*m + m^2) + b*d*(9*Sq
rt[b^2 - 4*a*c]*d*m - 4*a*e*(6 - 2*m - m^2)) - 2*a*e*(Sqrt[b^2 - 4*a*c]*d*(5 - 2
*m)*m - 2*a*e*(3 - 4*m + m^2))))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2
+ m, (2*c*(d + e*x))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)])/(2*(b^2 - 4*a*c)^(5/
2)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^2*(1 + m)) + (c*(
24*c^4*d^4 - b^3*(b - Sqrt[b^2 - 4*a*c])*e^4*(1 - m)*m - 4*c^3*d^2*e*(12*b*d - 3
*Sqrt[b^2 - 4*a*c]*d*m - 2*a*e*(6 - 2*m - m^2)) - 2*b*c*e^3*m*(a*Sqrt[b^2 - 4*a*
c]*e*(5 - 2*m) + b^2*d*(2 + m) - b*(3*a*e*(2 - m) + Sqrt[b^2 - 4*a*c]*d*(2 + m))
) + 2*c^2*e^2*(b^2*d^2*(12 + 2*m + m^2) - b*d*(9*Sqrt[b^2 - 4*a*c]*d*m + 4*a*e*(
6 - 2*m - m^2)) + 2*a*e*(Sqrt[b^2 - 4*a*c]*d*(5 - 2*m)*m + 2*a*e*(3 - 4*m + m^2)
)))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(2*(b^2 - 4*a*c)^(5/2)*(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^2*(1 + m))

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Rubi [A]  time = 24.0073, antiderivative size = 986, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{c \left (24 c^4 d^4-4 c^3 e \left (12 b d+3 \sqrt{b^2-4 a c} m d-2 a e \left (-m^2-2 m+6\right )\right ) d^2-b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4 (1-m) m+2 b c e^3 m \left (-d (m+2) b^2+3 a e (2-m) b-\sqrt{b^2-4 a c} d (m+2) b+a \sqrt{b^2-4 a c} e (5-2 m)\right )+2 c^2 e^2 \left (b^2 \left (m^2+2 m+12\right ) d^2+b \left (9 \sqrt{b^2-4 a c} d m-4 a e \left (-m^2-2 m+6\right )\right ) d-2 a e \left (\sqrt{b^2-4 a c} d (5-2 m) m-2 a e \left (m^2-4 m+3\right )\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^{5/2} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^2 (m+1)}+\frac{c \left (24 c^4 d^4-4 c^3 e \left (12 b d-3 \sqrt{b^2-4 a c} m d-2 a e \left (-m^2-2 m+6\right )\right ) d^2-b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4 (1-m) m-2 b c e^3 m \left (d (m+2) b^2-3 a e (2-m) b-\sqrt{b^2-4 a c} d (m+2) b+a \sqrt{b^2-4 a c} e (5-2 m)\right )+2 c^2 e^2 \left (b^2 \left (m^2+2 m+12\right ) d^2-b \left (9 \sqrt{b^2-4 a c} d m+4 a e \left (-m^2-2 m+6\right )\right ) d+2 a e \left (\sqrt{b^2-4 a c} d (5-2 m) m+2 a e \left (m^2-4 m+3\right )\right )\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^{5/2} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^2 (m+1)}-\frac{\left (a c e (2-m) (2 c d-b e)^2-c \left (6 c^2 d^2-2 c e (3 b d-a e (5-2 m))-b^2 e^2 (1-m)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 d^2-b^2 e^2 (1-m)+c e (2 a e (3-m)-b d (m+4))\right )\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}-\frac{\left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) (d+e x)^{m+1}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/(a + b*x + c*x^2)^3,x]

[Out]

-((d + e*x)^(1 + m)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - ((d + e*x)^(1 + m)*(a*c*e*(2*
c*d - b*e)^2*(2 - m) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2 - b^2*e^2*(1 - m) +
c*e*(2*a*e*(3 - m) - b*d*(4 + m))) - c*(2*c*d - b*e)*(6*c^2*d^2 - 2*c*e*(3*b*d -
a*e*(5 - 2*m)) - b^2*e^2*(1 - m))*x))/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2
)^2*(a + b*x + c*x^2)) - (c*(24*c^4*d^4 - b^3*(b + Sqrt[b^2 - 4*a*c])*e^4*(1 - m
)*m + 2*b*c*e^3*m*(a*Sqrt[b^2 - 4*a*c]*e*(5 - 2*m) + 3*a*b*e*(2 - m) - b^2*d*(2
+ m) - b*Sqrt[b^2 - 4*a*c]*d*(2 + m)) - 4*c^3*d^2*e*(12*b*d + 3*Sqrt[b^2 - 4*a*c
]*d*m - 2*a*e*(6 - 2*m - m^2)) + 2*c^2*e^2*(b^2*d^2*(12 + 2*m + m^2) + b*d*(9*Sq
rt[b^2 - 4*a*c]*d*m - 4*a*e*(6 - 2*m - m^2)) - 2*a*e*(Sqrt[b^2 - 4*a*c]*d*(5 - 2
*m)*m - 2*a*e*(3 - 4*m + m^2))))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2
+ m, (2*c*(d + e*x))/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)])/(2*(b^2 - 4*a*c)^(5/
2)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^2*(1 + m)) + (c*(
24*c^4*d^4 - b^3*(b - Sqrt[b^2 - 4*a*c])*e^4*(1 - m)*m - 2*b*c*e^3*m*(a*Sqrt[b^2
- 4*a*c]*e*(5 - 2*m) - 3*a*b*e*(2 - m) + b^2*d*(2 + m) - b*Sqrt[b^2 - 4*a*c]*d*
(2 + m)) - 4*c^3*d^2*e*(12*b*d - 3*Sqrt[b^2 - 4*a*c]*d*m - 2*a*e*(6 - 2*m - m^2)
) + 2*c^2*e^2*(b^2*d^2*(12 + 2*m + m^2) - b*d*(9*Sqrt[b^2 - 4*a*c]*d*m + 4*a*e*(
6 - 2*m - m^2)) + 2*a*e*(Sqrt[b^2 - 4*a*c]*d*(5 - 2*m)*m + 2*a*e*(3 - 4*m + m^2)
)))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*c*(d + e*x))/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(2*(b^2 - 4*a*c)^(5/2)*(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.351329, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m/(a + b*x + c*x^2)^3,x]

[Out]

Integrate[(d + e*x)^m/(a + b*x + c*x^2)^3, x]

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Maple [F]  time = 0.266, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+b*x+a)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x
+ (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

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