3.1856 \(\int \frac{(c+d x)^n}{a+b x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0352512, 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 \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^n/(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.43047, size = 37, normalized size = 0.73 \[ \frac{\left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{\left (n + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0334955, size = 51, normalized size = 1. \[ -\frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) ^{n}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^n/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)^n/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**n/(b*x+a),x)

[Out]

Integral((c + d*x)**n/(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^n/(b*x + a), x)