3.3051 \(\int \frac{(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=309 \[ -\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (3 d e-c f (1-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-3 m+2\right )-6 c d e f (1-m)+6 d^2 e^2\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 (m+1) (b e-a f)^4 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (b (4 d e-c f (2-m))-a d f (m+2))}{6 (e+f x)^2 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{3 (e+f x)^3 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3)
- (f*(b*(4*d*e - c*f*(2 - m)) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(1 -
m))/(6*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(2*a*b*d*f*(3*d*
e - c*f*(1 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(6*d^2*e^2 - 6*c*d*
e*f*(1 - m) + c^2*f^2*(2 - 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hyp
ergeometric2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]
)/(6*(b*e - a*f)^4*(d*e - c*f)^2*(1 + m))

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Rubi [A]  time = 0.892802, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (3 d e-c f (1-m))+b^2 \left (-\left (c^2 f^2 \left (m^2-3 m+2\right )-6 c d e f (1-m)+6 d^2 e^2\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 (m+1) (b e-a f)^4 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (-a d f (m+2)-b c f (2-m)+4 b d e)}{6 (e+f x)^2 (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m}}{3 (e+f x)^3 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/((c + d*x)^m*(e + f*x)^4),x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(1 - m))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3)
- (f*(4*b*d*e - b*c*f*(2 - m) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(1 -
m))/(6*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(2*a*b*d*f*(3*d*
e - c*f*(1 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(6*d^2*e^2 - 6*c*d*
e*f*(1 - m) + c^2*f^2*(2 - 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hyp
ergeometric2F1[2, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]
)/(6*(b*e - a*f)^4*(d*e - c*f)^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)**m/((d*x+c)**m)/(f*x+e)**4,x)

[Out]

Timed out

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Mathematica [C]  time = 7.46575, size = 1697, normalized size = 5.49 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m/((c + d*x)^m*(e + f*x)^4),x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(1 - m)*(6*(b*e - a*f)^2*HurwitzLerchPhi[((d*e - c*
f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*(b*e - a*f)^2*m*HurwitzLerchPhi
[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*f*(b*e - a*f)*(a + b
*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*f
*(-(b*e) + a*f)*m^2*(a + b*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*
f)*(c + d*x)), 1, m] + 2*f^2*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))
/((b*e - a*f)*(c + d*x)), 1, m] - f^2*m*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)
*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 2*f^2*m^2*(a + b*x)^2*HurwitzLerchP
hi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + f^2*m^3*(a + b*x)^2*
HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 6*(b*e
- a*f)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 +
m] - 6*(b*e - a*f)^2*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c
+ d*x)), 1, 1 + m] + 12*f*(b*e - a*f)*m*(a + b*x)*HurwitzLerchPhi[((d*e - c*f)*(
a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 12*f*(b*e - a*f)*m^2*(a + b*x)*Hu
rwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 3*f^2
*m*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)),
1, 1 + m] - 3*f^2*m^3*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e
- a*f)*(c + d*x)), 1, 1 + m] + 6*f*(-(b*e) + a*f)*(a + b*x)*HurwitzLerchPhi[((d*
e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + 12*f*(-(b*e) + a*f)*m*(
a + b*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 +
m] + 6*f*(-(b*e) + a*f)*m^2*(a + b*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/(
(b*e - a*f)*(c + d*x)), 1, 2 + m] + 3*f^2*m*(a + b*x)^2*HurwitzLerchPhi[((d*e -
c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + 6*f^2*m^2*(a + b*x)^2*Hurwi
tzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + 3*f^2*m^
3*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1
, 2 + m] - 2*f^2*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f
)*(c + d*x)), 1, 3 + m] - 5*f^2*m*(a + b*x)^2*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 4*f^2*m^2*(a + b*x)^2*HurwitzLerchPhi
[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - f^2*m^3*(a + b*x)^
2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m]))/(
3*(1 + m)*(e + f*x)^3*((b*e - a*f)*(c + d*x)*(a^2*f^2*(2 + 3*m + m^2) + 2*a*b*f*
(1 + m)*(-2*e + f*m*x) + b^2*(2*e^2 - 4*e*f*m*x + f^2*(-1 + m)*m*x^2))*HurwitzLe
rchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - (a + b*x)*((a^2*
f^2*(2 + 3*m + m^2)*(-3*c*f + d*(e - 2*f*x)) - 2*a*b*f*(1 + m)*(c*f*(-(e*(6 + m)
) + 2*f*m*x) + d*(2*e^2 - 2*e*f*(2 + m)*x + f^2*m*x^2)) + b^2*(c*f*(-2*e^2*(3 +
2*m) + 2*e*f*m*(3 + m)*x - f^2*(-1 + m)*m*x^2) + d*e*(2*e^2 - 4*e*f*(1 + 2*m)*x
+ f^2*m*(1 + 3*m)*x^2)))*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c
+ d*x)), 1, 1 + m] + f*(1 + m)*(a + b*x)*((a*f*(2 + m)*(-2*d*e + 3*c*f + d*f*x)
+ b*c*f*(-(e*(6 + m)) + 2*f*m*x) + b*d*e*(4*e - f*(2 + 3*m)*x))*HurwitzLerchPhi
[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + f*(d*e - c*f)*(2 +
m)*(a + b*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1
, 3 + m]))))

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Maple [F]  time = 0.167, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/((d*x+c)^m)/(f*x+e)^4,x)

[Out]

int((b*x+a)^m/((d*x+c)^m)/(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m}}{{\left (f x + e\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^4*(d*x + c)^m),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m)/(f*x + e)^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^4*(d*x + c)^m),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{4}{\left (d x + c\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m/((f*x + e)^4*(d*x + c)^m),x, algorithm="giac")

[Out]

integrate((b*x + a)^m/((f*x + e)^4*(d*x + c)^m), x)