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

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

[Out]

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

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Rubi [A]  time = 0.0305174, 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 \[ \frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/(c + d*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.40442, size = 37, normalized size = 0.73 \[ - \frac{\left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (n + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0241373, size = 66, normalized size = 1.29 \[ \frac{(a+b x)^n \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b c-a d}{b c+b d x}\right )}{d n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/(d*x+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/(d*x+c),x)

[Out]

Integral((a + b*x)**n/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/(d*x + c), x)