Optimal. Leaf size=115 \[ \frac {(a c+b d) \left (a+\frac {b}{x}\right )^{1+n}}{b^2 c^2 (1+n)}-\frac {\left (a+\frac {b}{x}\right )^{2+n}}{b^2 c (2+n)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (a c-b d) (1+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 115, 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 = {528, 457, 90,
70} \begin {gather*} \frac {(a c+b d) \left (a+\frac {b}{x}\right )^{n+1}}{b^2 c^2 (n+1)}-\frac {\left (a+\frac {b}{x}\right )^{n+2}}{b^2 c (n+2)}+\frac {d^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 )}{c^2 (n+1) (a c-b d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 90
Rule 457
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x^3 (c+d x)} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right ) x^4} \, dx\\ &=-\text {Subst}\left (\int \frac {x^2 (a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (\frac {(-a c-b d) (a+b x)^n}{b c^2}+\frac {(a+b x)^{1+n}}{b c}+\frac {d^2 (a+b x)^n}{c^2 (d+c x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {(a c+b d) \left (a+\frac {b}{x}\right )^{1+n}}{b^2 c^2 (1+n)}-\frac {\left (a+\frac {b}{x}\right )^{2+n}}{b^2 c (2+n)}-\frac {d^2 \text {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {(a c+b d) \left (a+\frac {b}{x}\right )^{1+n}}{b^2 c^2 (1+n)}-\frac {\left (a+\frac {b}{x}\right )^{2+n}}{b^2 c (2+n)}+\frac {d^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 )}{c^2 (a c-b d) (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 112, normalized size = 0.97 \begin {gather*} -\frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left ((a c-b d) (-b c (1+n)+a c x+b d (2+n) x)+b^2 d^2 (2+n) x \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )\right )}{b^2 c^2 (-a c+b d) (1+n) (2+n) x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x^{3} \left (d x +c \right )}\, dx\]
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 {\left (a + \frac {b}{x}\right )^{n}}{x^{3} \left (c + d x\right )}\, 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 {{\left (a+\frac {b}{x}\right )}^n}{x^3\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________