3.291 \(\int \frac{(a+\frac{b}{x})^n}{x^5 (c+d x)} \, dx\)

Optimal. Leaf size=207 \[ \frac{(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{n+1}}{b^4 c^4 (n+1)}-\frac{\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{n+2}}{b^4 c^3 (n+2)}+\frac{(3 a c+b d) \left (a+\frac{b}{x}\right )^{n+3}}{b^4 c^2 (n+3)}-\frac{\left (a+\frac{b}{x}\right )^{n+4}}{b^4 c (n+4)}+\frac{d^4 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^4 (n+1) (a c-b d)} \]

[Out]

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

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Rubi [A]  time = 0.157829, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {514, 446, 88, 68} \[ \frac{(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{n+1}}{b^4 c^4 (n+1)}-\frac{\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{n+2}}{b^4 c^3 (n+2)}+\frac{(3 a c+b d) \left (a+\frac{b}{x}\right )^{n+3}}{b^4 c^2 (n+3)}-\frac{\left (a+\frac{b}{x}\right )^{n+4}}{b^4 c (n+4)}+\frac{d^4 \left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^4 (n+1) (a c-b d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^n/(x^5*(c + d*x)),x]

[Out]

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

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n}{x^5 (c+d x)} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n}{\left (d+\frac{c}{x}\right ) x^6} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x^4 (a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{(a c+b d) \left (-a^2 c^2-b^2 d^2\right ) (a+b x)^n}{b^3 c^4}+\frac{\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) (a+b x)^{1+n}}{b^3 c^3}+\frac{(-3 a c-b d) (a+b x)^{2+n}}{b^3 c^2}+\frac{(a+b x)^{3+n}}{b^3 c}+\frac{d^4 (a+b x)^n}{c^4 (d+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{1+n}}{b^4 c^4 (1+n)}-\frac{\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{2+n}}{b^4 c^3 (2+n)}+\frac{(3 a c+b d) \left (a+\frac{b}{x}\right )^{3+n}}{b^4 c^2 (3+n)}-\frac{\left (a+\frac{b}{x}\right )^{4+n}}{b^4 c (4+n)}-\frac{d^4 \operatorname{Subst}\left (\int \frac{(a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=\frac{(a c+b d) \left (a^2 c^2+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{1+n}}{b^4 c^4 (1+n)}-\frac{\left (3 a^2 c^2+2 a b c d+b^2 d^2\right ) \left (a+\frac{b}{x}\right )^{2+n}}{b^4 c^3 (2+n)}+\frac{(3 a c+b d) \left (a+\frac{b}{x}\right )^{3+n}}{b^4 c^2 (3+n)}-\frac{\left (a+\frac{b}{x}\right )^{4+n}}{b^4 c (4+n)}+\frac{d^4 \left (a+\frac{b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^4 (a c-b d) (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.162881, size = 184, normalized size = 0.89 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \left (-\frac{c \left (a+\frac{b}{x}\right ) \left (3 a^2 c^2+2 a b c d+b^2 d^2\right )}{b^4 (n+2)}+\frac{(a c+b d) \left (a^2 c^2+b^2 d^2\right )}{b^4 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^2 (3 a c+b d)}{b^4 (n+3)}-\frac{c^3 \left (a+\frac{b}{x}\right )^3}{b^4 (n+4)}+\frac{d^4 \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)}\right )}{c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^n/(x^5*(c + d*x)),x]

[Out]

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

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Maple [F]  time = 0.511, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( dx+c \right ) } \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^n/x^5/(d*x+c),x)

[Out]

int((a+b/x)^n/x^5/(d*x+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x)^n/((d*x + c)*x^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (\frac{a x + b}{x}\right )^{n}}{d x^{6} + c x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((a*x + b)/x)^n/(d*x^6 + c*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((a+b/x)**n/x**5/(d*x+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x)^n/((d*x + c)*x^5), x)