Optimal. Leaf size=217 \[ -\frac {\left (a+\frac {b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac {c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac {c}{x}+d\right ) (a c-b d)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac {c}{x}+d\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {514, 446, 100, 146, 68} \[ -\frac {\left (a+\frac {b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac {c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac {c}{x}+d\right ) (a c-b d)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac {c}{x}+d\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 100
Rule 146
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)^2} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right )^2 x^5} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x^3 (a+b x)^n}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac {c}{x}\right ) x^2}-\frac {\operatorname {Subst}\left (\int \frac {x (a+b x)^n (-2 a d+(-a c-b d (3+n)) x)}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )}{b c (2+n)}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (d (b d (2+n) (a c+b d (3+n))-a c (a c+b d (5+3 n)))-\frac {c (a c-b d) (a c+b d (3+n))}{x}\right )}{b^2 c^3 (a c-b d) (1+n) (2+n) \left (d+\frac {c}{x}\right )}-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac {c}{x}\right ) x^2}-\frac {\left (d^2 (3 a c-b d (3+n))\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c^3 (a c-b d)}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n} \left (d (b d (2+n) (a c+b d (3+n))-a c (a c+b d (5+3 n)))-\frac {c (a c-b d) (a c+b d (3+n))}{x}\right )}{b^2 c^3 (a c-b d) (1+n) (2+n) \left (d+\frac {c}{x}\right )}-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac {c}{x}\right ) x^2}+\frac {d^2 (3 a c-b d (3+n)) \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^3 (a c-b d)^2 (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.35, size = 182, normalized size = 0.84 \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} \left (\frac {-a^2 c^2 (c+d x)-a b c d (c (n+2)+d (2 n+3) x)+b^2 d^2 (n+3) (c+d (n+2) x)}{b c^2 (n+1) (c+d x) (b d-a c)}-\frac {b d^2 (n+2) (b d (n+3)-3 a c) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac {1}{x (c+d x)}\right )}{b c (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{n}}{d^{2} x^{5} + 2 \, c d x^{4} + c^{2} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right )^{2} x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^3\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{3} \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________