3.960 \(\int x (a+b x)^n (c+d x)^{-n} \, dx\)

Optimal. Leaf size=124 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac{(a+b x)^{n+1} (c+d x)^{-n} (a d (1-n)+b c (n+1)) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 d (n+1)} \]

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(2*b*d) - ((a*d*(1 - n) + b*c*(1 + n))*(a
+ b*x)^(1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 + n,
-((d*(a + b*x))/(b*c - a*d))])/(2*b^2*d*(1 + n)*(c + d*x)^n)

_______________________________________________________________________________________

Rubi [A]  time = 0.157965, antiderivative size = 120, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 b d}-\frac{(a+b x)^{n+1} (c+d x)^{-n} \left (\frac{a-a n}{n+1}+\frac{b c}{d}\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(2*b*d) - (((b*c)/d + (a - a*n)/(1 + n))*(
a + b*x)^(1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 + n
, -((d*(a + b*x))/(b*c - a*d))])/(2*b^2*(c + d*x)^n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.1902, size = 95, normalized size = 0.77 \[ \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1}}{2 b d} - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n} \left (a d \left (- n + 1\right ) + b c \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{2 b^{2} d \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(n + 1)*(c + d*x)**(-n + 1)/(2*b*d) - (b*(-c - d*x)/(a*d - b*c))**n*(
a + b*x)**(n + 1)*(c + d*x)**(-n)*(a*d*(-n + 1) + b*c*(n + 1))*hyper((n, n + 1),
(n + 2,), d*(a + b*x)/(a*d - b*c))/(2*b**2*d*(n + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.117774, size = 130, normalized size = 1.05 \[ \frac{3 a c x^2 (a+b x)^n (c+d x)^{-n} F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 n x \left (b c F_1\left (3;1-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-n,n+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*a*c*x^2*(a + b*x)^n*AppellF1[2, -n, n, 3, -((b*x)/a), -((d*x)/c)])/((c + d*x)
^n*(6*a*c*AppellF1[2, -n, n, 3, -((b*x)/a), -((d*x)/c)] + 2*n*x*(b*c*AppellF1[3,
1 - n, n, 4, -((b*x)/a), -((d*x)/c)] - a*d*AppellF1[3, -n, 1 + n, 4, -((b*x)/a)
, -((d*x)/c)])))

_______________________________________________________________________________________

Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{x \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n/((d*x+c)^n),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*(d*x + c)^(-n)*x, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n*x/(d*x + c)^n, 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*(b*x+a)**n/((d*x+c)**n),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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