Integrand size = 27, antiderivative size = 87 \[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=-\frac {(e x)^{-2 (1+p)} \left (1-b^2 x^2\right )^{1+p}}{2 e (1+p)}-\frac {b (e x)^{-1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),b^2 x^2\right )}{e^2 (1+2 p)} \] Output:
-1/2*(-b^2*x^2+1)^(p+1)/e/(p+1)/((e*x)^(2*p+2))-b*(e*x)^(-1-2*p)*hypergeom ([-p, -1/2-p],[1/2-p],b^2*x^2)/e^2/(1+2*p)
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\frac {(e x)^{-2 p} \left (-\left ((1+2 p) \left (1-b^2 x^2\right )^{1+p}\right )-2 b (1+p) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,b^2 x^2\right )\right )}{2 e^3 (1+p) (1+2 p) x^2} \] Input:
Integrate[(e*x)^(-3 - 2*p)*(1 + b*x)*(1 - b^2*x^2)^p,x]
Output:
(-((1 + 2*p)*(1 - b^2*x^2)^(1 + p)) - 2*b*(1 + p)*x*Hypergeometric2F1[-1/2 - p, -p, 1/2 - p, b^2*x^2])/(2*e^3*(1 + p)*(1 + 2*p)*x^2*(e*x)^(2*p))
Time = 0.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {545, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (b x+1) \left (1-b^2 x^2\right )^p (e x)^{-2 p-3} \, dx\) |
| \(\Big \downarrow \) 545 |
| \(\displaystyle \frac {b \int (e x)^{-2 (p+1)} \left (1-b^2 x^2\right )^pdx}{e}-\frac {\left (1-b^2 x^2\right )^{p+1} (e x)^{-2 (p+1)}}{2 e (p+1)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle -\frac {b (e x)^{-2 p-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 p-1),-p,\frac {1}{2} (1-2 p),b^2 x^2\right )}{e^2 (2 p+1)}-\frac {\left (1-b^2 x^2\right )^{p+1} (e x)^{-2 (p+1)}}{2 e (p+1)}\) |
Input:
Int[(e*x)^(-3 - 2*p)*(1 + b*x)*(1 - b^2*x^2)^p,x]
Output:
-1/2*(1 - b^2*x^2)^(1 + p)/(e*(1 + p)*(e*x)^(2*(1 + p))) - (b*(e*x)^(-1 - 2*p)*Hypergeometric2F1[(-1 - 2*p)/2, -p, (1 - 2*p)/2, b^2*x^2])/(e^2*(1 + 2*p))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(-c)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[d/e Int[(e*x)^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 1.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90
| method | result | size |
| meijerg | \(-\frac {b \left (e x \right )^{-3-2 p} x^{2} \operatorname {hypergeom}\left (\left [-p , -\frac {1}{2}-p \right ], \left [\frac {1}{2}-p \right ], b^{2} x^{2}\right )}{2 p +1}-\frac {\left (e x \right )^{-3-2 p} x \left (-b^{2} x^{2}+1\right )^{p +1}}{2 \left (p +1\right )}\) | \(78\) |
Input:
int((e*x)^(-3-2*p)*(b*x+1)*(-b^2*x^2+1)^p,x,method=_RETURNVERBOSE)
Output:
-b*(e*x)^(-3-2*p)*x^2/(2*p+1)*hypergeom([-p,-1/2-p],[1/2-p],b^2*x^2)-1/2*( e*x)^(-3-2*p)*x*(-b^2*x^2+1)^(p+1)/(p+1)
\[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\int { {\left (b x + 1\right )} {\left (-b^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{-2 \, p - 3} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x+1)*(-b^2*x^2+1)^p,x, algorithm="fricas")
Output:
integral((b*x + 1)*(-b^2*x^2 + 1)^p*(e*x)^(-2*p - 3), x)
Result contains complex when optimal does not.
Time = 3.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.84 \[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\frac {b e^{- 2 p - 3} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {b^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \begin {cases} - \frac {e^{- 2 p - 3} x^{- 2 p - 2} \left (b^{2} x^{2} - 1\right )^{p + 1} e^{i \pi p} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} & \text {for}\: \left |{b^{2} x^{2}}\right | > 1 \\\frac {e^{- 2 p - 3} x^{- 2 p - 2} \left (- b^{2} x^{2} + 1\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} & \text {otherwise} \end {cases} \] Input:
integrate((e*x)**(-3-2*p)*(b*x+1)*(-b**2*x**2+1)**p,x)
Output:
b*e**(-2*p - 3)*x**(-2*p - 1)*gamma(-p - 1/2)*hyper((-p, -p - 1/2), (1/2 - p,), b**2*x**2*exp_polar(2*I*pi))/(2*gamma(1/2 - p)) + Piecewise((-e**(-2 *p - 3)*x**(-2*p - 2)*(b**2*x**2 - 1)**(p + 1)*exp(I*pi*p)*gamma(-p - 1)/( 2*gamma(-p)), Abs(b**2*x**2) > 1), (e**(-2*p - 3)*x**(-2*p - 2)*(-b**2*x** 2 + 1)**(p + 1)*gamma(-p - 1)/(2*gamma(-p)), True))
\[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\int { {\left (b x + 1\right )} {\left (-b^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{-2 \, p - 3} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x+1)*(-b^2*x^2+1)^p,x, algorithm="maxima")
Output:
b*e^(-2*p - 3)*integrate(e^(p*log(b*x + 1) + p*log(-b*x + 1) - 2*p*log(x)) /x^2, x) + 1/2*(b^2*x^2 - 1)*e^(-2*p - 3)*e^(p*log(b*x + 1) + p*log(-b*x + 1) - 2*p*log(x))/((p + 1)*x^2)
\[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\int { {\left (b x + 1\right )} {\left (-b^{2} x^{2} + 1\right )}^{p} \left (e x\right )^{-2 \, p - 3} \,d x } \] Input:
integrate((e*x)^(-3-2*p)*(b*x+1)*(-b^2*x^2+1)^p,x, algorithm="giac")
Output:
integrate((b*x + 1)*(-b^2*x^2 + 1)^p*(e*x)^(-2*p - 3), x)
Timed out. \[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\int \frac {{\left (1-b^2\,x^2\right )}^p\,\left (b\,x+1\right )}{{\left (e\,x\right )}^{2\,p+3}} \,d x \] Input:
int(((1 - b^2*x^2)^p*(b*x + 1))/(e*x)^(2*p + 3),x)
Output:
int(((1 - b^2*x^2)^p*(b*x + 1))/(e*x)^(2*p + 3), x)
\[ \int (e x)^{-3-2 p} (1+b x) \left (1-b^2 x^2\right )^p \, dx=\frac {\left (-b^{2} x^{2}+1\right )^{p} b^{2} x^{2}-2 \left (-b^{2} x^{2}+1\right )^{p} b p x -2 \left (-b^{2} x^{2}+1\right )^{p} b x -\left (-b^{2} x^{2}+1\right )^{p}+4 x^{2 p} \left (\int \frac {\left (-b^{2} x^{2}+1\right )^{p}}{x^{2 p} b^{2} x^{4}-x^{2 p} x^{2}}d x \right ) b \,p^{2} x^{2}+4 x^{2 p} \left (\int \frac {\left (-b^{2} x^{2}+1\right )^{p}}{x^{2 p} b^{2} x^{4}-x^{2 p} x^{2}}d x \right ) b p \,x^{2}}{2 x^{2 p} e^{2 p} e^{3} x^{2} \left (p +1\right )} \] Input:
int((e*x)^(-3-2*p)*(b*x+1)*(-b^2*x^2+1)^p,x)
Output:
(( - b**2*x**2 + 1)**p*b**2*x**2 - 2*( - b**2*x**2 + 1)**p*b*p*x - 2*( - b **2*x**2 + 1)**p*b*x - ( - b**2*x**2 + 1)**p + 4*x**(2*p)*int(( - b**2*x** 2 + 1)**p/(x**(2*p)*b**2*x**4 - x**(2*p)*x**2),x)*b*p**2*x**2 + 4*x**(2*p) *int(( - b**2*x**2 + 1)**p/(x**(2*p)*b**2*x**4 - x**(2*p)*x**2),x)*b*p*x** 2)/(2*x**(2*p)*e**(2*p)*e**3*x**2*(p + 1))