Optimal. Leaf size=84 \[ -\frac {d \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {514, 446, 80, 68} \[ -\frac {d \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 80
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x^2 (c+d x)} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right ) x^3} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x (a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (1+n)}+\frac {d \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (1+n)}-\frac {d \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 (a c-b d) (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 77, normalized size = 0.92 \[ \frac {(a x+b) \left (a+\frac {b}{x}\right )^n \left (b d \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+a c-b d\right )}{b c (n+1) x (b d-a c)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{n}}{d x^{3} + c x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right ) x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{2} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________