Optimal. Leaf size=52 \[ -\frac {(a c-b c x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {a-b x}{2 a}\right )}{4 a^2 b c (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {68} \[ -\frac {(a c-b c x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {a-b x}{2 a}\right )}{4 a^2 b c (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rubi steps
\begin {align*} \int \frac {(a c-b c x)^n}{(a+b x)^2} \, dx &=-\frac {(a c-b c x)^{1+n} \, _2F_1\left (2,1+n;2+n;\frac {a-b x}{2 a}\right )}{4 a^2 b c (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 52, normalized size = 1.00 \[ -\frac {(a-b x) (c (a-b x))^n \, _2F_1\left (2,n+1;n+2;\frac {a-b x}{2 a}\right )}{4 a^2 b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b c x + a c\right )}^{n}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b c x + a c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b c x +a c \right )^{n}}{\left (b x +a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b c x + a c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a\,c-b\,c\,x\right )}^n}{{\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (- a + b x\right )\right )^{n}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________