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

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))

________________________________________________________________________________________

Rubi [A]  time = 0.0462151, 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, Rules used = {96, 131} \[ -\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} (a d (n+1)+b (c-c n)) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{2 a c x^2} \]

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))

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x)^n (c+d x)^{-n}}{x^3} \, dx &=-\frac{(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac{(a d (1+n)+b (c-c n)) \int \frac{(a+b x)^n (c+d x)^{-n}}{x^2} \, dx}{2 a c}\\ &=-\frac{(a+b x)^{1+n} (c+d x)^{1-n}}{2 a c x^2}-\frac{(b c-a d) (a d (1+n)+b (c-c n)) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac{c (a+b x)}{a (c+d x)}\right )}{2 a^3 c (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0500335, size = 99, normalized size = 0.85 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-1} \left (\frac{(b c-a d) (b c (n-1)-a d (n+1)) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{n+1}-\frac{a^2 (c+d x)^2}{x^2}\right )}{2 a^3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-1 - n)*(-((a^2*(c + d*x)^2)/x^2) + ((b*c - a*d)*(b*c*(-1 + n) - a*d*(1 + n))*Hy
pergeometric2F1[2, 1 + n, 2 + n, (c*(a + b*x))/(a*(c + d*x))])/(1 + n)))/(2*a^3*c)

________________________________________________________________________________________

Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3} \left ( dx+c \right ) ^{n}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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