3.123 \(\int \frac{x^m (e+f x)^n}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=140 \[ \frac{b x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{b x}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{d x}{c}\right )}{c (m+1) (b c-a d)} \]

[Out]

(b*x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((b*x)/a)])/
(a*(b*c - a*d)*(1 + m)*(1 + (f*x)/e)^n) - (d*x^(1 + m)*(e + f*x)^n*AppellF1[1 +
m, -n, 1, 2 + m, -((f*x)/e), -((d*x)/c)])/(c*(b*c - a*d)*(1 + m)*(1 + (f*x)/e)^n
)

_______________________________________________________________________________________

Rubi [A]  time = 0.383137, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{b x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{b x}{a}\right )}{a (m+1) (b c-a d)}-\frac{d x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{d x}{c}\right )}{c (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((b*x)/a)])/
(a*(b*c - a*d)*(1 + m)*(1 + (f*x)/e)^n) - (d*x^(1 + m)*(e + f*x)^n*AppellF1[1 +
m, -n, 1, 2 + m, -((f*x)/e), -((d*x)/c)])/(c*(b*c - a*d)*(1 + m)*(1 + (f*x)/e)^n
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.0696, size = 104, normalized size = 0.74 \[ \frac{d x^{m + 1} \left (1 + \frac{f x}{e}\right )^{- n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{d x}{c},- \frac{f x}{e} \right )}}{c \left (m + 1\right ) \left (a d - b c\right )} - \frac{b x^{m + 1} \left (1 + \frac{f x}{e}\right )^{- n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{b x}{a},- \frac{f x}{e} \right )}}{a \left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x**(m + 1)*(1 + f*x/e)**(-n)*(e + f*x)**n*appellf1(m + 1, 1, -n, m + 2, -d*x/c
, -f*x/e)/(c*(m + 1)*(a*d - b*c)) - b*x**(m + 1)*(1 + f*x/e)**(-n)*(e + f*x)**n*
appellf1(m + 1, 1, -n, m + 2, -b*x/a, -f*x/e)/(a*(m + 1)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [B]  time = 1.21839, size = 309, normalized size = 2.21 \[ \frac{e (m+2) x^{m+1} (e+f x)^n \left (-\frac{a b F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{b x}{a}\right )}{(a+b x) (a d-b c) \left (a e (m+2) F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{b x}{a}\right )+x \left (a f n F_1\left (m+2;1-n,1;m+3;-\frac{f x}{e},-\frac{b x}{a}\right )-b e F_1\left (m+2;-n,2;m+3;-\frac{f x}{e},-\frac{b x}{a}\right )\right )\right )}-\frac{c d F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{d x}{c}\right )}{(c+d x) (b c-a d) \left (c e (m+2) F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{d x}{c}\right )+x \left (c f n F_1\left (m+2;1-n,1;m+3;-\frac{f x}{e},-\frac{d x}{c}\right )-d e F_1\left (m+2;-n,2;m+3;-\frac{f x}{e},-\frac{d x}{c}\right )\right )\right )}\right )}{m+1} \]

Warning: Unable to verify antiderivative.

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

[Out]

(e*(2 + m)*x^(1 + m)*(e + f*x)^n*(-((a*b*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e
), -((b*x)/a)])/((-(b*c) + a*d)*(a + b*x)*(a*e*(2 + m)*AppellF1[1 + m, -n, 1, 2
+ m, -((f*x)/e), -((b*x)/a)] + x*(a*f*n*AppellF1[2 + m, 1 - n, 1, 3 + m, -((f*x)
/e), -((b*x)/a)] - b*e*AppellF1[2 + m, -n, 2, 3 + m, -((f*x)/e), -((b*x)/a)]))))
- (c*d*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((d*x)/c)])/((b*c - a*d)*(c +
d*x)*(c*e*(2 + m)*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((d*x)/c)] + x*(c*
f*n*AppellF1[2 + m, 1 - n, 1, 3 + m, -((f*x)/e), -((d*x)/c)] - d*e*AppellF1[2 +
m, -n, 2, 3 + m, -((f*x)/e), -((d*x)/c)])))))/(1 + m)

_______________________________________________________________________________________

Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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