3.2643 \(\int \frac{(A+B x) (d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=537 \[ \frac{c (d+e x)^{m+1} \left (e m (-2 a B e+A b e-2 A c d+b B d)-\frac{2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )+b^2 (-e) (A e m+B d (2-m))}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}+\frac{c (d+e x)^{m+1} \left (\frac{2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (A \left (a e^2 (1-m)+c d^2\right )+a B d e m\right )+b^2 (-e) (A e m+B d (2-m))}{\sqrt{b^2-4 a c}}+e m (-2 a B e+A b e-2 A c d+b B d)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}+\frac{(d+e x)^{m+1} \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

((d + e*x)^(1 + m)*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d -
2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x
+ c*x^2)) + (c*(e*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*m - (2*b*(B*c*d^2 + 2*A*c*
d*e + a*B*e^2) - b^2*e*(B*d*(2 - m) + A*e*m) - 4*c*(A*(c*d^2 + a*e^2*(1 - m)) +
a*B*d*e*m))/Sqrt[b^2 - 4*a*c])*(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)])/((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)) + (c*(e*(b*B*d - 2
*A*c*d + A*b*e - 2*a*B*e)*m + (2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2) - b^2*e*(B*d*
(2 - m) + A*e*m) - 4*c*(A*(c*d^2 + a*e^2*(1 - m)) + a*B*d*e*m))/Sqrt[b^2 - 4*a*c
])*(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)])/((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 = 7.73398, antiderivative size = 535, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{c (d+e x)^{m+1} \left (e m (-2 a B e+A b e-2 A c d+b B d)-\frac{2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (a A e^2 (1-m)+a B d e m+A c d^2\right )+b^2 (-e) (A e m+B d (2-m))}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}+\frac{c (d+e x)^{m+1} \left (\frac{2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (a A e^2 (1-m)+a B d e m+A c d^2\right )+b^2 (-e) (A e m+B d (2-m))}{\sqrt{b^2-4 a c}}+e m (-2 a B e+A b e-2 A c d+b B d)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}+\frac{(d+e x)^{m+1} \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.206651, size = 0, normalized size = 0. \[ \int \frac{(A+B x) (d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((B*x+A)*(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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)*(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. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*(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((B*x+A)*(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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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