3.2571 \(\int (d+e x)^m \left (a+b x+c x^2\right )^{-2-\frac{m}{2}} \, dx\)

Optimal. Leaf size=440 \[ -\frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{m+3} \left (a+b x+c x^2\right )^{-\frac{m}{2}-2} \left (4 c e (a e-b d (m+1))+b^2 e^2 m+4 c^2 d^2 (m+1)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{\frac{m+4}{2}} \, _2F_1\left (m+3,\frac{m+4}{2};m+4;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (m+1) (m+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac{e (d+e x)^{m+1} \left (a+b x+c x^2\right )^{-\frac{m}{2}-1}}{(m+1) \left (a e^2-b d e+c d^2\right )}+\frac{e m (2 c d-b e) (d+e x)^{m+2} \left (a+b x+c x^2\right )^{-\frac{m}{2}-1}}{2 (m+1) (m+2) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.966639, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{m+3} \left (a+b x+c x^2\right )^{-\frac{m}{2}-2} \left (4 c e (a e-b d (m+1))+b^2 e^2 m+4 c^2 d^2 (m+1)\right ) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{\frac{m+4}{2}} \, _2F_1\left (m+3,\frac{m+4}{2};m+4;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{4 (m+1) (m+3) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac{e (d+e x)^{m+1} \left (a+b x+c x^2\right )^{-\frac{m}{2}-1}}{(m+1) \left (a e^2-b d e+c d^2\right )}+\frac{e m (2 c d-b e) (d+e x)^{m+2} \left (a+b x+c x^2\right )^{-\frac{m}{2}-1}}{2 (m+1) (m+2) \left (a e^2-b d e+c d^2\right )^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

[Out]

Timed out

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Mathematica [B]  time = 158.96, size = 2324, normalized size = 5.28 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2^(4 + m)*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x)^(2 + m/2)*(-(-b + Sqrt[b^2 - 4*
a*c])/(2*c) + x)^(2 + m/2)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/c)^(-2 - m/2)*((b +
Sqrt[b^2 - 4*a*c] + 2*c*x)/c)^(-2 - m/2)*(d + e*x)^(1 + m)*((-(b*e) - Sqrt[b^2 -
4*a*c]*e - 2*c*e*x)/(2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e))^(2 + m/2)*((-(b*e) + S
qrt[b^2 - 4*a*c]*e - 2*c*e*x)/(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e))^(2 + m/2)*(a
+ b*x + c*x^2)^(-2 - m/2)*(1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])
*e))^(-1 - m/2)*(1 - (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^(-2 -
m/2)*Gamma[2 + m]*(6*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*(b + Sqrt[b^2 - 4*a*
c] + 2*c*x)*Gamma[2 + m/2]*Gamma[5 + m]*Hypergeometric2F1[1, (4 + m)/2, 4 + m, (
4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt
[b^2 - 4*a*c] + 2*c*x))] + 5*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*m*(b + Sqrt[
b^2 - 4*a*c] + 2*c*x)*Gamma[2 + m/2]*Gamma[5 + m]*Hypergeometric2F1[1, (4 + m)/2
, 4 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)
*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] + (2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)^3*m^2*
(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*Gamma[2 + m/2]*Gamma[5 + m]*Hypergeometric2F1[1,
(4 + m)/2, 4 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 12*c*(2*c*d + (-b + Sqrt[b^2 - 4*
a*c])*e)^2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)*Gamma[2 + m/2]*Gamma[5 + m]
*Hypergeometric2F1[1, (4 + m)/2, 4 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*
d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 4*c*(2*c*d +
(-b + Sqrt[b^2 - 4*a*c])*e)^2*m*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)*Gamma
[2 + m/2]*Gamma[5 + m]*Hypergeometric2F1[1, (4 + m)/2, 4 + m, (4*c*Sqrt[b^2 - 4*
a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2
*c*x))] + 8*c^2*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*
c*x)*(d + e*x)^2*Gamma[2 + m/2]*Gamma[5 + m]*Hypergeometric2F1[1, (4 + m)/2, 4 +
m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b +
Sqrt[b^2 - 4*a*c] + 2*c*x))] + 20*c*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)^2*(d + e*x)*Gamma[3 + m/2]*Gamma[4 + m]*Hypergeometric2F1[2, (6 + m)
/2, 5 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*
e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] + 8*c*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt
[b^2 - 4*a*c])*e)^2*m*(d + e*x)*Gamma[3 + m/2]*Gamma[4 + m]*Hypergeometric2F1[2,
(6 + m)/2, 5 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 64*c^2*Sqrt[b^2 - 4*a*c]*(2*c*d +
(-b + Sqrt[b^2 - 4*a*c])*e)*(d + e*x)^2*Gamma[3 + m/2]*Gamma[4 + m]*Hypergeomet
ric2F1[2, (6 + m)/2, 5 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sq
rt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 16*c^2*Sqrt[b^2 - 4*a*c]
*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*m*(d + e*x)^2*Gamma[3 + m/2]*Gamma[4 + m]*
Hypergeometric2F1[2, (6 + m)/2, 5 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d
+ (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] + 48*c^3*Sqrt[b
^2 - 4*a*c]*(d + e*x)^3*Gamma[3 + m/2]*Gamma[4 + m]*Hypergeometric2F1[2, (6 + m)
/2, 5 + m, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*
e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] + 4*c*Sqrt[b^2 - 4*a*c]*(2*c*d + (-b + Sqrt
[b^2 - 4*a*c])*e)^2*(d + e*x)*Gamma[3 + m/2]*Gamma[4 + m]*HypergeometricPFQ[{2,
2, 3 + m/2}, {1, 5 + m}, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[
b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] - 16*c^2*Sqrt[b^2 - 4*a*c]*(2
*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(d + e*x)^2*Gamma[3 + m/2]*Gamma[4 + m]*Hyper
geometricPFQ[{2, 2, 3 + m/2}, {1, 5 + m}, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*
c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))] + 16*c^3*Sqr
t[b^2 - 4*a*c]*(d + e*x)^3*Gamma[3 + m/2]*Gamma[4 + m]*HypergeometricPFQ[{2, 2,
3 + m/2}, {1, 5 + m}, (4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d + (-b + Sqrt[b^2
- 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))]))/(e*(2*c*d + (-b + Sqrt[b^2 - 4
*a*c])*e)^3*(1 + m)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*Gamma[(4 + m)/2]*Gamma[4 + m
]*Gamma[5 + m])

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Maple [F]  time = 0.228, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{-2-{\frac{m}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{-\frac{1}{2} \, m - 2}{\left (e x + d\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/2*m - 2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{-\frac{1}{2} \, m - 2}{\left (e x + d\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/2*m - 2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(-1/2*m - 2)*(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*(c*x**2+b*x+a)**(-2-1/2*m),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(-1/2*m - 2)*(e*x + d)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError