3.932 \(\int \frac{x (a+b x)^n}{(c+d x)^2} \, dx\)

Optimal. Leaf size=99 \[ -\frac{(a+b x)^{n+1} (a d-b c (n+1)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac{c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.104318, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(a+b x)^{n+1} (a d-b c (n+1)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac{c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(x*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8464, size = 73, normalized size = 0.74 \[ \frac{c \left (a + b x\right )^{n + 1}}{d \left (c + d x\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a d - b c \left (n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{d \left (n + 1\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**n/(d*x+c)**2,x)

[Out]

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

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Mathematica [C]  time = 0.328823, size = 126, normalized size = 1.27 \[ \frac{3 a c x^2 (a+b x)^n F_1\left (2;-n,2;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{2 (c+d x)^2 \left (3 a c F_1\left (2;-n,2;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c n x F_1\left (3;1-n,2;4;-\frac{b x}{a},-\frac{d x}{c}\right )-2 a d x F_1\left (3;-n,3;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

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

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{x \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n/(d*x+c)^2,x)

[Out]

int(x*(b*x+a)^n/(d*x+c)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x/(d*x + c)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x/(d*x + c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x/(d*x + c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**n/(d*x+c)**2,x)

[Out]

Integral(x*(a + b*x)**n/(c + d*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x/(d*x + c)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x/(d*x + c)^2, x)