Optimal. Leaf size=109 \[ \frac {x^{m+1} (-a-b x+1)^{-n/2} (a+b x+1)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (\frac {b x}{a+1}+1\right )^{-n/2} F_1\left (m+1;\frac {n}{2},-\frac {n}{2};m+2;\frac {b x}{1-a},-\frac {b x}{a+1}\right )}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6163, 135, 133} \[ \frac {x^{m+1} (-a-b x+1)^{-n/2} (a+b x+1)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (\frac {b x}{a+1}+1\right )^{-n/2} F_1\left (m+1;\frac {n}{2},-\frac {n}{2};m+2;\frac {b x}{1-a},-\frac {b x}{a+1}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 135
Rule 6163
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a+b x)} x^m \, dx &=\int x^m (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=\left ((1-a-b x)^{-n/2} \left (1-\frac {b x}{1-a}\right )^{n/2}\right ) \int x^m (1+a+b x)^{n/2} \left (1-\frac {b x}{1-a}\right )^{-n/2} \, dx\\ &=\left ((1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (1+\frac {b x}{1+a}\right )^{-n/2}\right ) \int x^m \left (1-\frac {b x}{1-a}\right )^{-n/2} \left (1+\frac {b x}{1+a}\right )^{n/2} \, dx\\ &=\frac {x^{1+m} (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (1+\frac {b x}{1+a}\right )^{-n/2} F_1\left (1+m;\frac {n}{2},-\frac {n}{2};2+m;\frac {b x}{1-a},-\frac {b x}{1+a}\right )}{1+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.84, size = 0, normalized size = 0.00 \[ \int e^{n \tanh ^{-1}(a+b x)} x^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (b x +a \right )} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} e^{n \operatorname {atanh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________