3.3060 \(\int \frac{(a+b x)^m (c+d x)^{-1-m}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=300 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f m+2 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+4 c d e f m+2 d^2 e^2\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 m (m+1) (b e-a f)^3 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (m+2)-b (c f m+2 d e))}{2 m (e+f x)^2 (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x)^2 (b c-a d) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.868818, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (c f m+2 d e)+b^2 \left (-\left (-c^2 f^2 (1-m) m+4 c d e f m+2 d^2 e^2\right )\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{2 m (m+1) (b e-a f)^3 (d e-c f)^2}-\frac{f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (m+2)-b (c f m+2 d e))}{2 m (e+f x)^2 (b c-a d) (b e-a f) (d e-c f)^2}+\frac{d (a+b x)^{m+1} (c+d x)^{-m}}{m (e+f x)^2 (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x)^3,x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 5.73855, size = 2361, normalized size = 7.87 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x)^3,x]

[Out]

-((b*e - a*f)^3*(a + b*x)^(1 + m)*(2*(b*e - a*f)^2*HurwitzLerchPhi[((d*e - c*f)*
(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 2*(b*e - a*f)^2*m*HurwitzLerchPh
i[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*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] + 4*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, 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, 1 + m] + f^2*m^3*(a + b*x)^2*Hurw
itzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*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] + 8*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] + 4*f*(b*e - a*f)*m^2*(a + b*x)*HurwitzLe
rchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] - 2*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] - 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, 2 + m] - 2*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*Hu
rwitzLerchPhi[((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]))/(2*(-(b*e) + a*f)^3*(1 + m)*(c + d*x)^m*(e + f*x)
^2*(b^3*c*e^3 - a*b^2*c*e^2*f + b^3*d*e^3*x + 2*b^3*c*e^2*f*x - a*b^2*d*e^2*f*x
- 2*a*b^2*c*e*f^2*x + 2*b^3*d*e^2*f*x^2 + b^3*c*e*f^2*x^2 - 2*a*b^2*d*e*f^2*x^2
- a*b^2*c*f^3*x^2 + b^3*d*e*f^2*x^3 - a*b^2*d*f^3*x^3 - f*(-(b*e) + a*f)*(1 + m)
*(a + b*x)*(c + d*x)*(a*f*(2 + m) + b*(-2*e + f*m*x))*HurwitzLerchPhi[((d*e - c*
f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + f*(1 + m)*(a + b*x)^2*(a*f*(2
+ m)*(-(d*e) + 2*c*f + d*f*x) + b*c*f*(-(e*(4 + m)) + f*m*x) + 2*b*d*e*(e - f*(
1 + m)*x))*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2
+ m] + 2*a^3*d*e*f^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x)), 1, 3 + m] - 2*a^3*c*f^3*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a
*f)*(c + d*x)), 1, 3 + m] + 3*a^3*d*e*f^2*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*
x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 3*a^3*c*f^3*m*HurwitzLerchPhi[((d*e - c
*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + a^3*d*e*f^2*m^2*HurwitzLerch
Phi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - a^3*c*f^3*m^2*H
urwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + 6*a^
2*b*d*e*f^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1
, 3 + m] - 6*a^2*b*c*f^3*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*
(c + d*x)), 1, 3 + m] + 9*a^2*b*d*e*f^2*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*
x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 9*a^2*b*c*f^3*m*x*HurwitzLerchPhi[((d*e
- c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + 3*a^2*b*d*e*f^2*m^2*x*Hu
rwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 3*a^2
*b*c*f^3*m^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)),
1, 3 + m] + 6*a*b^2*d*e*f^2*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e -
a*f)*(c + d*x)), 1, 3 + m] - 6*a*b^2*c*f^3*x^2*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + 9*a*b^2*d*e*f^2*m*x^2*HurwitzLerchPh
i[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 9*a*b^2*c*f^3*m*x
^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] +
3*a*b^2*d*e*f^2*m^2*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c
+ d*x)), 1, 3 + m] - 3*a*b^2*c*f^3*m^2*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x
))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + 2*b^3*d*e*f^2*x^3*HurwitzLerchPhi[((d*e
- c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] - 2*b^3*c*f^3*x^3*HurwitzLe
rchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 + m] + 3*b^3*d*e*f^
2*m*x^3*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 3 +
m] - 3*b^3*c*f^3*m*x^3*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x)), 1, 3 + m] + b^3*d*e*f^2*m^2*x^3*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/
((b*e - a*f)*(c + d*x)), 1, 3 + m] - b^3*c*f^3*m^2*x^3*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 = 0.125, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m}}{ \left ( fx+e \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 1)/(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3
), 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)**(-1-m)/(f*x+e)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{{\left (f x + e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e)^3, x)