3.1010 \(\int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx\)

Optimal. Leaf size=139 \[ -\frac{4 b^4 \left (n^2+n+1\right ) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{3 a^3 (1-n)}-\frac{b (2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{12 a^3 x^3}-\frac{(a+b x)^{n+2} (a-b x)^{1-n}}{4 a^2 x^4} \]

[Out]

-((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(4*a^2*x^4) - (b*(1 + 2*n)*(a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(12*a^3
*x^3) - (4*b^4*(1 + n + n^2)*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)
/(a + b*x)])/(3*a^3*(1 - n))

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Rubi [A]  time = 0.0696251, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {129, 151, 12, 131} \[ -\frac{4 b^4 \left (n^2+n+1\right ) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )}{3 a^3 (1-n)}-\frac{b (2 n+1) (a+b x)^{n+2} (a-b x)^{1-n}}{12 a^3 x^3}-\frac{(a+b x)^{n+2} (a-b x)^{1-n}}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1 + n)/(x^5*(a - b*x)^n),x]

[Out]

-((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(4*a^2*x^4) - (b*(1 + 2*n)*(a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(12*a^3
*x^3) - (4*b^4*(1 + n + n^2)*(a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)
/(a + b*x)])/(3*a^3*(1 - n))

Rule 129

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^5} \, dx &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac{\int \frac{(a-b x)^{-n} (a+b x)^{1+n} \left (-a b (1+2 n)-b^2 x\right )}{x^4} \, dx}{4 a^2}\\ &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac{b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}+\frac{\int \frac{4 a^2 b^2 \left (1+n+n^2\right ) (a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{12 a^4}\\ &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac{b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}+\frac{\left (b^2 \left (1+n+n^2\right )\right ) \int \frac{(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{3 a^2}\\ &=-\frac{(a-b x)^{1-n} (a+b x)^{2+n}}{4 a^2 x^4}-\frac{b (1+2 n) (a-b x)^{1-n} (a+b x)^{2+n}}{12 a^3 x^3}-\frac{4 b^4 \left (1+n+n^2\right ) (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^3 (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.0512396, size = 101, normalized size = 0.73 \[ \frac{(a-b x)^{1-n} (a+b x)^{n-1} \left (16 b^4 \left (n^2+n+1\right ) x^4 \, _2F_1\left (3,1-n;2-n;\frac{a-b x}{a+b x}\right )-(n-1) (a+b x)^3 (3 a+b (2 n+1) x)\right )}{12 a^3 (n-1) x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1 + n)/(x^5*(a - b*x)^n),x]

[Out]

((a - b*x)^(1 - n)*(a + b*x)^(-1 + n)*(-((-1 + n)*(a + b*x)^3*(3*a + b*(1 + 2*n)*x)) + 16*b^4*(1 + n + n^2)*x^
4*Hypergeometric2F1[3, 1 - n, 2 - n, (a - b*x)/(a + b*x)]))/(12*a^3*(-1 + n)*x^4)

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{1+n}}{{x}^{5} \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^5/((-b*x+a)^n),x)

[Out]

int((b*x+a)^(1+n)/x^5/((-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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), 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^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), 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**5/((-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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1+n)/x^5/((-b*x+a)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n + 1)/((-b*x + a)^n*x^5), x)