\(\int (e x)^{-1-2 p} (c+d x)^2 (a+b x^2)^p \, dx\) [311]

Optimal result

Integrand size = 26, antiderivative size = 185 \[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{2 b e}+\frac {2 c d (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{e^2 (1-2 p)}-\frac {\left (b c^2+a d^2 p\right ) (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 b e p} \] Output:

1/2*d^2*(b*x^2+a)^(p+1)/b/e/((e*x)^(2*p))+2*c*d*(e*x)^(1-2*p)*(b*x^2+a)^p* 
hypergeom([-p, 1/2-p],[3/2-p],-b*x^2/a)/e^2/(1-2*p)/((1+b*x^2/a)^p)-1/2*(a 
*d^2*p+b*c^2)*(b*x^2+a)^p*hypergeom([-p, -p],[1-p],-b*x^2/a)/b/e/p/((e*x)^ 
(2*p))/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.84 \[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (4 c d (-1+p) p x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+(-1+2 p) \left (d^2 p x^2 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+c^2 (-1+p) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )}{2 e (-1+p) p (-1+2 p)} \] Input:

Integrate[(e*x)^(-1 - 2*p)*(c + d*x)^2*(a + b*x^2)^p,x]
 

Output:

-1/2*((a + b*x^2)^p*(4*c*d*(-1 + p)*p*x*Hypergeometric2F1[1/2 - p, -p, 3/2 
 - p, -((b*x^2)/a)] + (-1 + 2*p)*(d^2*p*x^2*Hypergeometric2F1[1 - p, -p, 2 
 - p, -((b*x^2)/a)] + c^2*(-1 + p)*Hypergeometric2F1[-p, -p, 1 - p, -((b*x 
^2)/a)])))/(e*(-1 + p)*p*(-1 + 2*p)*(e*x)^(2*p)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {559, 27, 557, 279, 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.

\(\displaystyle \int (c+d x)^2 (e x)^{-2 p-1} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int 2 (e x)^{-2 p-1} \left (b c^2+2 b d x c+a d^2 p\right ) \left (b x^2+a\right )^pdx}{2 b}+\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{2 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{-2 p-1} \left (b c^2+2 b d x c+a d^2 p\right ) \left (b x^2+a\right )^pdx}{b}+\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{2 b e}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\left (a d^2 p+b c^2\right ) \int (e x)^{-2 p-1} \left (b x^2+a\right )^pdx+\frac {2 b c d \int (e x)^{-2 p} \left (b x^2+a\right )^pdx}{e}}{b}+\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{2 b e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a d^2 p+b c^2\right ) \int (e x)^{-2 p-1} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {2 b c d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{e}}{b}+\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{2 b e}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {2 b c d (e x)^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{e^2 (1-2 p)}-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a d^2 p+b c^2\right ) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e p}}{b}+\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{2 b e}\)

Input:

Int[(e*x)^(-1 - 2*p)*(c + d*x)^2*(a + b*x^2)^p,x]
 

Output:

(d^2*(a + b*x^2)^(1 + p))/(2*b*e*(e*x)^(2*p)) + ((2*b*c*d*(e*x)^(1 - 2*p)* 
(a + b*x^2)^p*Hypergeometric2F1[(1 - 2*p)/2, -p, (3 - 2*p)/2, -((b*x^2)/a) 
])/(e^2*(1 - 2*p)*(1 + (b*x^2)/a)^p) - ((b*c^2 + a*d^2*p)*(a + b*x^2)^p*Hy 
pergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(2*e*p*(e*x)^(2*p)*(1 + (b*x 
^2)/a)^p))/b
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

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])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], 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]
 

rule 559

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( 
m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1))   Int[(e*x)^m*(a + b* 
x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 
)*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && IGtQ[n, 1] &&  !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

Maple [F]

\[\int \left (e x \right )^{-1-2 p} \left (d x +c \right )^{2} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 57.41 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.86 \[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{2} e^{- 2 p - 1} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} + \frac {a^{p} c d e^{- 2 p - 1} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {3}{2} - p\right )} + \frac {a^{p} d^{2} e^{- 2 p - 1} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} \] Input:

integrate((e*x)**(-1-2*p)*(d*x+c)**2*(b*x**2+a)**p,x)
 

Output:

a**p*c**2*e**(-2*p - 1)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_pol 
ar(I*pi)/a)/(2*x**(2*p)*gamma(1 - p)) + a**p*c*d*e**(-2*p - 1)*x**(1 - 2*p 
)*gamma(1/2 - p)*hyper((-p, 1/2 - p), (3/2 - p,), b*x**2*exp_polar(I*pi)/a 
)/gamma(3/2 - p) + a**p*d**2*e**(-2*p - 1)*x**(2 - 2*p)*gamma(1 - p)*hyper 
((-p, 1 - p), (2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(2 - p))
 

Maxima [F]

\[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^2*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Giac [F]

\[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((d*x + c)^2*(b*x^2 + a)^p*(e*x)^(-2*p - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^2}{{\left (e\,x\right )}^{2\,p+1}} \,d x \] Input:

int(((a + b*x^2)^p*(c + d*x)^2)/(e*x)^(2*p + 1),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x)^2)/(e*x)^(2*p + 1), x)
 

Reduce [F]

\[ \int (e x)^{-1-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {-\left (b \,x^{2}+a \right )^{p} c^{2}+4 \left (b \,x^{2}+a \right )^{p} c d p x +\left (b \,x^{2}+a \right )^{p} d^{2} p \,x^{2}+8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a c d \,p^{2}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a \,d^{2} p^{2}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) b \,c^{2} p}{2 x^{2 p} e^{2 p} e p} \] Input:

int((e*x)^(-1-2*p)*(d*x+c)^2*(b*x^2+a)^p,x)
 

Output:

( - (a + b*x**2)**p*c**2 + 4*(a + b*x**2)**p*c*d*p*x + (a + b*x**2)**p*d** 
2*p*x**2 + 8*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x 
)*a*c*d*p**2 + 2*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b 
*x**2),x)*a*d**2*p**2 + 2*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x 
**(2*p)*b*x**2),x)*b*c**2*p)/(2*x**(2*p)*e**(2*p)*e*p)