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

Optimal. Leaf size=117 \[ -\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} (a d (n+1)+b (c-c n)) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.14183, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} (a d (n+1)+b (c-c n)) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.811, size = 92, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1}}{2 a c x^{2}} + \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 1} \left (a d - b c\right ) \left (a d \left (n + 1\right ) + b c \left (- n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n + 1, 2 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + b x\right )}{a \left (c + d x\right )}} \right )}}{2 a^{3} c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.363787, size = 146, normalized size = 1.25 \[ -\frac{3 b d (a+b x)^n (c+d x)^{-n} F_1\left (2;-n,n;3;-\frac{a}{b x},-\frac{c}{d x}\right )}{6 b d x^2 F_1\left (2;-n,n;3;-\frac{a}{b x},-\frac{c}{d x}\right )+2 a d n x F_1\left (3;1-n,n;4;-\frac{a}{b x},-\frac{c}{d x}\right )-2 b c n x F_1\left (3;-n,n+1;4;-\frac{a}{b x},-\frac{c}{d x}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*b*d*(a + b*x)^n*AppellF1[2, -n, n, 3, -(a/(b*x)), -(c/(d*x))])/((c + d*x)^n*
(6*b*d*x^2*AppellF1[2, -n, n, 3, -(a/(b*x)), -(c/(d*x))] + 2*a*d*n*x*AppellF1[3,
1 - n, n, 4, -(a/(b*x)), -(c/(d*x))] - 2*b*c*n*x*AppellF1[3, -n, 1 + n, 4, -(a/
(b*x)), -(c/(d*x))]))

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Maple [F]  time = 0.094, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3} \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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