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)*hypergeom([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 = 0.16, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.24, size = 184, normalized size = 0.89 \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} \left (-\frac {c \left (a+\frac {b}{x}\right ) \left (3 a^2 c^2+2 a b c d+b^2 d^2\right )}{b^4 (n+2)}+\frac {(a c+b d) \left (a^2 c^2+b^2 d^2\right )}{b^4 (n+1)}-\frac {c^3 \left (a+\frac {b}{x}\right )^3}{b^4 (n+4)}+\frac {c^2 \left (a+\frac {b}{x}\right )^2 (3 a c+b d)}{b^4 (n+3)}+\frac {d^4 \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)}\right )}{c^4} \]

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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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{n}}{d x^{6} + c x^{5}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{5}}\,{d x} \]

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)

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maple [F]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right ) x^{5}}\, dx \]

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^5\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^n/(x^5*(c + d*x)),x)

[Out]

int((a + b/x)^n/(x^5*(c + d*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{5} \left (c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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