Optimal. Leaf size=44 \[ -\frac{(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0154593, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 68} \[ -\frac{(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^m}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^{-2+m}}{(a-b x)^2} \, dx\\ &=-\frac{(a+b x)^{-1+m} \, _2F_1\left (2,-1+m;m;\frac{a+b x}{2 a}\right )}{4 a^2 b (1-m)}\\ \end{align*}
Mathematica [B] time = 0.150645, size = 102, normalized size = 2.32 \[ \frac{(a+b x)^m \left (\frac{2 (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{(a+b x) \, _2F_1\left (2,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+4 a \left (\frac{a}{(m-1) (a+b x)}+\frac{1}{m}\right )\right )}{16 a^4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.505, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{b^{4} x^{4} - 2 \, a^{2} b^{2} x^{2} + a^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{m}}{\left (- a + b x\right )^{2} \left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]