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

Optimal. Leaf size=78 \[ \frac{(a+b x)^{n+1}}{b d (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.06958, antiderivative size = 78, 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 \[ \frac{(a+b x)^{n+1}}{b d (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4009, size = 54, normalized size = 0.69 \[ \frac{c \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{d \left (n + 1\right ) \left (a d - b c\right )} + \frac{\left (a + b x\right )^{n + 1}}{b d \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.16385, size = 86, normalized size = 1.1 \[ \frac{(a+b x)^n \left (\frac{a d+b d x}{b n+b}-\frac{c \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b c-a d}{b c+b d x}\right )}{n}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{x \left ( bx+a \right ) ^{n}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(a + b*x)**n/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)