3.3084 \(\int \frac{(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx\)

Optimal. Leaf size=345 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (d e-c f)^3}-\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)+b (2 d e-c f (m+5)))}{(m+2) (m+3) (b c-a d)^2 (d e-c f)^2} \]

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)*(d*e - c*f)*(3 + m)) + (d*
(a*d*f*(3 + m) + b*(2*d*e - c*f*(5 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/
((b*c - a*d)^2*(d*e - c*f)^2*(2 + m)*(3 + m)) + (d*(a^2*d^2*f^2*(6 + 5*m + m^2)
+ a*b*d*f*(3 + m)*(d*e - c*f*(5 + 2*m)) + b^2*(2*d^2*e^2 - c*d*e*f*(7 + m) + c^2
*f^2*(11 + 6*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^3*(d*
e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - (f^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*
Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x
))])/((b*e - a*f)*(d*e - c*f)^3*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 1.44726, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3 (d e-c f)^3}-\frac{f^3 (a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f) (d e-c f)^3}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}+\frac{d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (m+3) (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)*(d*e - c*f)*(3 + m)) + (d*
(2*b*d*e + a*d*f*(3 + m) - b*c*f*(5 + m))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/
((b*c - a*d)^2*(d*e - c*f)^2*(2 + m)*(3 + m)) + (d*(a^2*d^2*f^2*(6 + 5*m + m^2)
+ a*b*d*f*(3 + m)*(d*e - c*f*(5 + 2*m)) + b^2*(2*d^2*e^2 - c*d*e*f*(7 + m) + c^2
*f^2*(11 + 6*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m))/((b*c - a*d)^3*(d*
e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - (f^3*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*
Hypergeometric2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x
))])/((b*e - a*f)*(d*e - c*f)^3*(1 + m))

_______________________________________________________________________________________

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)**(-4-m)/(f*x+e),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 35.8479, size = 26263, normalized size = 76.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-4-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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