Optimal. Leaf size=131 \[ \frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {c^2 \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (a c-b d) (1+n)}-\frac {(a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a^2 d^2 (1+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 131, 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, 382, 105,
162, 67, 70} \begin {gather*} -\frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac {c^2 \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 )}{d^2 (n+1) (a c-b d)}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 70
Rule 105
Rule 162
Rule 382
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{d+\frac {c}{x}} \, dx\\ &=-\text {Subst}\left (\int \frac {(a+b x)^n}{x^2 (d+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {\text {Subst}\left (\int \frac {(a+b x)^n (a c-b d n-b c n x)}{x (d+c x)} \, dx,x,\frac {1}{x}\right )}{a d}\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}-\frac {c^2 \text {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(a c-b d n) \text {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\frac {1}{x}\right )}{a d^2}\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {c^2 \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) (1+n)}-\frac {(a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a^2 d^2 (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 119, normalized size = 0.91 \begin {gather*} \frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left (a^2 c^2 \, _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) \left (a d (1+n) x+(-a c+b d n) \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )\right )\right )}{a^2 d^2 (a c-b d) (1+n) x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n} x}{d x +c}\, dx\]
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}}{c + d x}\, 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}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________