3.3114 \(\int \frac{(a+b x)^m (c+d x)^{2-m}}{(e+f x)^5} \, dx\)

Optimal. Leaf size=176 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} (b (4 d e-c f (3-m))-a d f (m+1)) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{4 (m+1) (b e-a f)^5 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 (e+f x)^4 (b e-a f) (d e-c f)} \]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(4*(b*e - a*f)*(d*e - c*f)*(e + f*x)^4)
+ ((b*c - a*d)^3*(b*(4*d*e - c*f*(3 - m)) - a*d*f*(1 + m))*(a + b*x)^(1 + m)*(c
+ d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*
e - a*f)*(c + d*x))])/(4*(b*e - a*f)^5*(d*e - c*f)*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.250943, antiderivative size = 175, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+1)-b c f (3-m)+4 b d e) \, _2F_1\left (4,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{4 (m+1) (b e-a f)^5 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{3-m}}{4 (e+f x)^4 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^5,x]

[Out]

-(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m))/(4*(b*e - a*f)*(d*e - c*f)*(e + f*x)^4)
+ ((b*c - a*d)^3*(4*b*d*e - b*c*f*(3 - m) - a*d*f*(1 + m))*(a + b*x)^(1 + m)*(c
+ d*x)^(-1 - m)*Hypergeometric2F1[4, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*
e - a*f)*(c + d*x))])/(4*(b*e - a*f)^5*(d*e - c*f)*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.5742, size = 143, normalized size = 0.81 \[ - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 3}}{4 \left (e + f x\right )^{4} \left (a f - b e\right ) \left (c f - d e\right )} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (a d - b c\right )^{3} \left (- a d f \left (m + 1\right ) - b c f \left (- m + 3\right ) + 4 b d e\right ){{}_{2}F_{1}\left (\begin{matrix} m + 1, 4 \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{4 \left (m + 1\right ) \left (a f - b e\right )^{5} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**(2-m)/(f*x+e)**5,x)

[Out]

-f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 3)/(4*(e + f*x)**4*(a*f - b*e)*(c*f - d*e
)) - (a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(a*d - b*c)**3*(-a*d*f*(m + 1) - b*c
*f*(-m + 3) + 4*b*d*e)*hyper((m + 1, 4), (m + 2,), (-a - b*x)*(-c*f + d*e)/((c +
d*x)*(a*f - b*e)))/(4*(m + 1)*(a*f - b*e)**5*(c*f - d*e))

_______________________________________________________________________________________

Mathematica [C]  time = 17.2614, size = 3314, normalized size = 18.83 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^m*(c + d*x)^(2 - m))/(e + f*x)^5,x]

[Out]

-((a + b*x)^(1 + m)*(c + d*x)^(3 - m)*((-4*b*e + a*f*(1 + m) + b*f*(-3 + m)*x)*H
urwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, -2 + m] + 4*(
3*b*e - a*f*(1 + m) - b*f*(-2 + m)*x)*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((
b*e - a*f)*(c + d*x)), 1, -1 + m] - 12*b*e*HurwitzLerchPhi[((d*e - c*f)*(a + b*x
))/((b*e - a*f)*(c + d*x)), 1, m] + 6*a*f*HurwitzLerchPhi[((d*e - c*f)*(a + b*x)
)/((b*e - a*f)*(c + d*x)), 1, m] + 6*a*f*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x
))/((b*e - a*f)*(c + d*x)), 1, m] - 6*b*f*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*
x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*b*f*m*x*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 4*b*e*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - 4*a*f*HurwitzLerchPhi[((d*e - c*f)*(a
+ b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - 4*a*f*m*HurwitzLerchPhi[((d*e - c*
f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - 4*b*f*m*x*HurwitzLerchPhi[((d
*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + a*f*HurwitzLerchPhi[((
d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + a*f*m*HurwitzLerchPhi
[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + b*f*x*HurwitzLerch
Phi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + b*f*m*x*Hurwitz
LerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m]))/(4*(e + f*
x)^4*((b*e - a*f)*(c + d*x)*(3*b*e - a*f*(1 + m) - b*f*(-2 + m)*x)*HurwitzLerchP
hi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, -2 + m] + (a^2*f*(1 + m)*
(-4*c*f + d*(e - 3*f*x)) - b^2*(d*e*x*(9*e + f*(5 - 4*m)*x) + c*(6*e^2 - 3*e*f*m
*x + f^2*(-2 + m)*x^2)) + a*b*(c*f*(3*e*(4 + m) + f*(4 - 5*m)*x) + d*(-3*e^2 + e
*f*(8 + 5*m)*x - 3*f^2*(-1 + m)*x^2)))*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/(
(b*e - a*f)*(c + d*x)), 1, -1 + m] + 3*b^2*c*e^2*HurwitzLerchPhi[((d*e - c*f)*(a
+ b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*a*b*d*e^2*HurwitzLerchPhi[((d*e - c*
f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 12*a*b*c*e*f*HurwitzLerchPhi[((d*
e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 3*a^2*d*e*f*HurwitzLerchPhi
[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*a^2*c*f^2*HurwitzLer
chPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 3*a*b*c*e*f*m*Hur
witzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 3*a^2*d*e*
f*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 6*a
^2*c*f^2*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m
] + 9*b^2*d*e^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)
), 1, m] - 6*b^2*c*e*f*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c
+ d*x)), 1, m] - 6*a*b*d*e*f*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e -
a*f)*(c + d*x)), 1, m] + 3*a^2*d*f^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/(
(b*e - a*f)*(c + d*x)), 1, m] - 3*b^2*c*e*f*m*x*HurwitzLerchPhi[((d*e - c*f)*(a
+ b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 9*a*b*d*e*f*m*x*HurwitzLerchPhi[((d*e -
c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 9*a*b*c*f^2*m*x*HurwitzLerchPh
i[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 3*a^2*d*f^2*m*x*Hurwi
tzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] + 3*b^2*d*e*f*
x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 3*b
^2*c*f^2*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1,
m] - 6*b^2*d*e*f*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c
+ d*x)), 1, m] + 3*b^2*c*f^2*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e
- a*f)*(c + d*x)), 1, m] + 3*a*b*d*f^2*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, m] - 3*a*b*d*e^2*HurwitzLerchPhi[((d*e - c*f)*
(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*a*b*c*e*f*HurwitzLerchPhi[((d*
e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 3*a^2*d*e*f*HurwitzLerc
hPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - 4*a^2*c*f^2*Hu
rwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + a*b*c
*e*f*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m
] + 3*a^2*d*e*f*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)
), 1, 1 + m] - 4*a^2*c*f^2*m*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f
)*(c + d*x)), 1, 1 + m] - 3*b^2*d*e^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/
((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*b^2*c*e*f*x*HurwitzLerchPhi[((d*e - c*f)*
(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*a*b*d*e*f*x*HurwitzLerchPhi[((
d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - 4*a*b*c*f^2*x*Hurwitz
LerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - a^2*d*f^2*
x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + b
^2*c*e*f*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1,
1 + m] + 7*a*b*d*e*f*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(
c + d*x)), 1, 1 + m] - 7*a*b*c*f^2*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/(
(b*e - a*f)*(c + d*x)), 1, 1 + m] - a^2*d*f^2*m*x*HurwitzLerchPhi[((d*e - c*f)*(
a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + b^2*d*e*f*x^2*HurwitzLerchPhi[((d
*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] - a*b*d*f^2*x^2*HurwitzL
erchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m] + 4*b^2*d*e*f
*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 1 + m
] - 3*b^2*c*f^2*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x)), 1, 1 + m] - a*b*d*f^2*m*x^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e
- a*f)*(c + d*x)), 1, 1 + m] - a^2*d*e*f*HurwitzLerchPhi[((d*e - c*f)*(a + b*x)
)/((b*e - a*f)*(c + d*x)), 1, 2 + m] + a^2*c*f^2*HurwitzLerchPhi[((d*e - c*f)*(a
+ b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] - a^2*d*e*f*m*HurwitzLerchPhi[((d*e
- c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + a^2*c*f^2*m*HurwitzLerchP
hi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] - 2*a*b*d*e*f*x*Hu
rwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + 2*a*b
*c*f^2*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 +
m] - 2*a*b*d*e*f*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c +
d*x)), 1, 2 + m] + 2*a*b*c*f^2*m*x*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e
- a*f)*(c + d*x)), 1, 2 + m] - b^2*d*e*f*x^2*HurwitzLerchPhi[((d*e - c*f)*(a +
b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + b^2*c*f^2*x^2*HurwitzLerchPhi[((d*e -
c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] - b^2*d*e*f*m*x^2*HurwitzLer
chPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m] + b^2*c*f^2*m*x
^2*HurwitzLerchPhi[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)), 1, 2 + m]))

_______________________________________________________________________________________

Maple [F]  time = 0.273, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{2-m}}{ \left ( fx+e \right ) ^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^5,x)

[Out]

int((b*x+a)^m*(d*x+c)^(2-m)/(f*x+e)^5,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^5,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^5, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{f^{5} x^{5} + 5 \, e f^{4} x^{4} + 10 \, e^{2} f^{3} x^{3} + 10 \, e^{3} f^{2} x^{2} + 5 \, e^{4} f x + e^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^5,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 2)/(f^5*x^5 + 5*e*f^4*x^4 + 10*e^2*f^3*x^3
+ 10*e^3*f^2*x^2 + 5*e^4*f*x + e^5), x)

_______________________________________________________________________________________

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)**(2-m)/(f*x+e)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 2}}{{\left (f x + e\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^5,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 2)/(f*x + e)^5, x)