Optimal. Leaf size=79 \[ \frac {d x^2 \left (a+\frac {b}{x}\right )^{m+1}}{2 a}-\frac {b \left (a+\frac {b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac {b}{a x}+1\right )}{2 a^3 (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {434, 446, 78, 65} \[ \frac {d x^2 \left (a+\frac {b}{x}\right )^{m+1}}{2 a}-\frac {b \left (a+\frac {b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac {b}{a x}+1\right )}{2 a^3 (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 78
Rule 434
Rule 446
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^m (c+d x) \, dx &=\int \left (a+\frac {b}{x}\right )^m \left (d+\frac {c}{x}\right ) x \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(a+b x)^m (d+c x)}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {d \left (a+\frac {b}{x}\right )^{1+m} x^2}{2 a}-\frac {(2 a c+b d (-1+m)) \operatorname {Subst}\left (\int \frac {(a+b x)^m}{x^2} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {d \left (a+\frac {b}{x}\right )^{1+m} x^2}{2 a}-\frac {b (2 a c-b d (1-m)) \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;1+\frac {b}{a x}\right )}{2 a^3 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 73, normalized size = 0.92 \[ \frac {(a x+b) \left (a+\frac {b}{x}\right )^m \left (a^2 d (m+1) x^2+b (-2 a c-b d (m-1)) \, _2F_1\left (2,m+1;m+2;\frac {b}{a x}+1\right )\right )}{2 a^3 (m+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x + c\right )} \left (\frac {a x + b}{x}\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \left (a +\frac {b}{x}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+\frac {b}{x}\right )}^m\,\left (c+d\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 4.10, size = 75, normalized size = 0.95 \[ \frac {b^{m} c x x^{- m} \Gamma \left (1 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 1 - m \\ 2 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac {b^{m} d x^{2} x^{- m} \Gamma \left (2 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 2 - m \\ 3 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________