3.1009 \(\int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^4} \, dx\)

Optimal. Leaf size=101 \[ -\frac{4 b^3 (2 n+1) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{3 a^2 (1-n)}-\frac{(a+b x)^{n+2} (a-b x)^{1-n}}{3 a^2 x^3} \]

[Out]

-((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(3*a^2*x^3) - (4*b^3*(1 + 2*n)*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hyp
ergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(3*a^2*(1 - n))

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Rubi [A]  time = 0.0378193, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {96, 131} \[ -\frac{4 b^3 (2 n+1) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{3 a^2 (1-n)}-\frac{(a+b x)^{n+2} (a-b x)^{1-n}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1 + n)/(x^4*(a - b*x)^n),x]

[Out]

-((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(3*a^2*x^3) - (4*b^3*(1 + 2*n)*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hyp
ergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)])/(3*a^2*(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} (a+b x)^{1+n}}{x^4} \, dx &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}+\frac{(b (1+2 n)) \int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{3 a}\\ &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}-\frac{4 b^3 (1+2 n) (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{3 a^2 (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.0411565, size = 88, normalized size = 0.87 \[ \frac{(a-b x)^{1-n} (a+b x)^{n-1} \left (4 b^3 (2 n+1) x^3 \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )-(n-1) (a+b x)^3\right )}{3 a^2 (n-1) x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1 + n)/(x^4*(a - b*x)^n),x]

[Out]

((a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*(-((-1 + n)*(a + b*x)^3) + 4*b^3*(1 + 2*n)*x^3*Hypergeometric2F1[3, 1 -
n, 2 - n, (a - b*x)/(a + b*x)]))/(3*a^2*(-1 + n)*x^3)

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{1+n}}{{x}^{4} \left ( -bx+a \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1+n)/x^4/((-b*x+a)^n),x)

[Out]

int((b*x+a)^(1+n)/x^4/((-b*x+a)^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^4/((-b*x+a)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^4/((-b*x+a)^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n + 1)/((-b*x + a)^n*x^4), x)

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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)**(1+n)/x**4/((-b*x+a)**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^4/((-b*x+a)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^4), x)