Optimal. Leaf size=38 \[ \frac {(a+b x)^m \, _2F_1\left (1,m;1+m;\frac {a+b x}{2 a}\right )}{2 a b m} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {641, 70}
\begin {gather*} \frac {(a+b x)^m \, _2F_1\left (1,m;m+1;\frac {a+b x}{2 a}\right )}{2 a b m} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^m}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^{-1+m}}{a-b x} \, dx\\ &=\frac {(a+b x)^m \, _2F_1\left (1,m;1+m;\frac {a+b x}{2 a}\right )}{2 a b m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 59, normalized size = 1.55 \begin {gather*} \frac {(a+b x)^m \left (2 a (1+m)+m (a+b x) \, _2F_1\left (1,1+m;2+m;\frac {a+b x}{2 a}\right )\right )}{4 a^2 b m (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m}}{-b^{2} x^{2}+a^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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 + b x\right )^{m}}{- a^{2} + b^{2} x^{2}}\, 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 = -0.03 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{a^2-b^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________