3.11.10 \(\int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^5} \, dx\) [1010]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 = {105, 156, 12, 133} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((a - b*x)^(1 - n)*(a + b*x)^(2 + n))/(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 12

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

Rule 105

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, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 133

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)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 156

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] && ILtQ[m, -1]

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*}

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Mathematica [A]
time = 0.08, size = 101, normalized size = 0.73 \begin {gather*} \frac {(a-b x)^{1-n} (a+b x)^{-1+n} \left (-\left ((-1+n) (a+b x)^3 (3 a+b (1+2 n) x)\right )+16 b^4 \left (1+n+n^2\right ) x^4 \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )\right )}{12 a^3 (-1+n) x^4} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1+n} \left (-b x +a \right )^{-n}}{x^{5}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{n+1}}{x^5\,{\left (a-b\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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