3.1858 \(\int \frac{(c+d x)^n}{(a+b x)^3} \, dx\)

Optimal. Leaf size=54 \[ -\frac{d^2 (c+d x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^3} \]

[Out]

-((d^2*(c + d*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, (b*(c + d*x))/(b*c -
a*d)])/((b*c - a*d)^3*(1 + n)))

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Rubi [A]  time = 0.0465895, antiderivative size = 54, 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 \[ -\frac{d^2 (c+d x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^n/(a + b*x)^3,x]

[Out]

-((d^2*(c + d*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, (b*(c + d*x))/(b*c -
a*d)])/((b*c - a*d)^3*(1 + n)))

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Rubi in Sympy [A]  time = 5.93743, size = 42, normalized size = 0.78 \[ \frac{d^{2} \left (c + d x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{\left (n + 1\right ) \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**n/(b*x+a)**3,x)

[Out]

d**2*(c + d*x)**(n + 1)*hyper((3, n + 1), (n + 2,), b*(-c - d*x)/(a*d - b*c))/((
n + 1)*(a*d - b*c)**3)

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Mathematica [A]  time = 0.0559148, size = 0, normalized size = 0. \[ \int \frac{(c+d x)^n}{(a+b x)^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(c + d*x)^n/(a + b*x)^3,x]

[Out]

Integrate[(c + d*x)^n/(a + b*x)^3, x]

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) ^{n}}{ \left ( bx+a \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^n/(b*x+a)^3,x)

[Out]

int((d*x+c)^n/(b*x+a)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n/(b*x + a)^3,x, algorithm="maxima")

[Out]

integrate((d*x + c)^n/(b*x + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{n}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n/(b*x + a)^3,x, algorithm="fricas")

[Out]

integral((d*x + c)^n/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x\right )^{n}}{\left (a + b x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**n/(b*x+a)**3,x)

[Out]

Integral((c + d*x)**n/(a + b*x)**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n/(b*x + a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^n/(b*x + a)^3, x)