Optimal. Leaf size=150 \[ -\frac {c \left (a+\frac {b}{x}\right )^{1+n}}{d (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {c (a c-b d (1-n)) \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (a c-b d)^2 (1+n)}+\frac {\left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a d^2 (1+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {528, 457, 105,
162, 67, 70} \begin {gather*} -\frac {c \left (a+\frac {b}{x}\right )^{n+1} (a c-b d (1-n)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)^2}-\frac {c \left (a+\frac {b}{x}\right )^{n+1}}{d \left (\frac {c}{x}+d\right ) (a c-b d)}+\frac {\left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{a d^2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 70
Rule 105
Rule 162
Rule 457
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n x}{(c+d x)^2} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right )^2 x} \, dx\\ &=-\text {Subst}\left (\int \frac {(a+b x)^n}{x (d+c x)^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c \left (a+\frac {b}{x}\right )^{1+n}}{d (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {\text {Subst}\left (\int \frac {(a+b x)^n (a c-b d-b c n x)}{x (d+c x)} \, dx,x,\frac {1}{x}\right )}{d (a c-b d)}\\ &=-\frac {c \left (a+\frac {b}{x}\right )^{1+n}}{d (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {\text {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(c (a c-b d (1-n))) \text {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{d^2 (a c-b d)}\\ &=-\frac {c \left (a+\frac {b}{x}\right )^{1+n}}{d (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {c (a c-b d (1-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 )}{d^2 (a c-b d)^2 (1+n)}+\frac {\left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a d^2 (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 120, normalized size = 0.80 \begin {gather*} \frac {\left (a+\frac {b}{x}\right )^{1+n} \left (-\frac {c d x}{(a c-b d) (c+d x)}-\frac {c (a c+b d (-1+n)) \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(a c-b d)^2 (1+n)}+\frac {\, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a (1+n)}\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n} x}{\left (d x +c \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + \frac {b}{x}\right )^{n}}{\left (c + d x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (a+\frac {b}{x}\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________