3.1857 \(\int \frac{(c+d x)^n}{(a+b x)^2} \, dx\)

Optimal. Leaf size=51 \[ \frac{d (c+d x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0383093, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{d (c+d x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^n/(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.33273, size = 41, normalized size = 0.8 \[ \frac{d \left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{\left (n + 1\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0458257, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^n}{(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(c + d*x)^n/(a + b*x)^2, x]

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) ^{n}}{ \left ( bx+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^n/(b*x+a)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n/(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n/(b*x + a)^2, x)