\(\int (e x)^{-2 p} (c+d x)^2 (a+b x^2)^p \, dx\) [303]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int (e x)^{-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {d^2 (e x)^{1-2 p} \left (a+b x^2\right )^{1+p}}{3 b e}-\frac {\left (\frac {a d^2}{b}-\frac {3 c^2}{1-2 p}\right ) (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{3 e}+\frac {c d (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{e^2 (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int (e x)^{-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=-\frac {x (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c^2 \left (3-5 p+2 p^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+d (-1+2 p) x \left (c (-3+2 p) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+d (-1+p) x \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,-p,\frac {5}{2}-p,-\frac {b x^2}{a}\right )\right )\right )}{(-1+p) (-3+2 p) (-1+2 p)} \] Input:

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

Output:

-((x*(a + b*x^2)^p*(c^2*(3 - 5*p + 2*p^2)*Hypergeometric2F1[1/2 - p, -p, 3 
/2 - p, -((b*x^2)/a)] + d*(-1 + 2*p)*x*(c*(-3 + 2*p)*Hypergeometric2F1[1 - 
 p, -p, 2 - p, -((b*x^2)/a)] + d*(-1 + p)*x*Hypergeometric2F1[3/2 - p, -p, 
 5/2 - p, -((b*x^2)/a)])))/((-1 + p)*(-3 + 2*p)*(-1 + 2*p)*(e*x)^(2*p)*(1 
+ (b*x^2)/a)^p))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {559, 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.

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

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int (e x)^{-2 p} \left (3 b c^2+6 b d x c-a d^2 (1-2 p)\right ) \left (b x^2+a\right )^pdx}{3 b}+\frac {d^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\left (3 b c^2-a d^2 (1-2 p)\right ) \int (e x)^{-2 p} \left (b x^2+a\right )^pdx+\frac {6 b c d \int (e x)^{1-2 p} \left (b x^2+a\right )^pdx}{e}}{3 b}+\frac {d^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 b c^2-a d^2 (1-2 p)\right ) \int (e x)^{-2 p} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {6 b c d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{1-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{e}}{3 b}+\frac {d^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e}\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 48.85 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int (e x)^{-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c d e^{- 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (2 - p\right )} + \frac {a^{p} d^{2} e^{- 2 p} x^{3 - 2 p} \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right )} + \sqrt {b} b^{p - \frac {1}{2}} c^{2} e^{- 2 p} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )} \] Input:

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

Output:

a**p*c*d*x**(2 - 2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), b*x**2*exp 
_polar(I*pi)/a)/(e**(2*p)*gamma(2 - p)) + a**p*d**2*x**(3 - 2*p)*gamma(3/2 
 - p)*hyper((-p, 3/2 - p), (5/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*e**(2* 
p)*gamma(5/2 - p)) + sqrt(b)*b**(p - 1/2)*c**2*x*hyper((-1/2, -p), (1/2,), 
 a*exp_polar(I*pi)/(b*x**2))/e**(2*p)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-2 p} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {2 \left (b \,x^{2}+a \right )^{p} a \,d^{2} p x +3 \left (b \,x^{2}+a \right )^{p} b \,c^{2} x +3 \left (b \,x^{2}+a \right )^{p} b c d \,x^{2}+\left (b \,x^{2}+a \right )^{p} b \,d^{2} x^{3}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{2} p^{2}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{2} p +6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b \,c^{2} p +6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b c d p}{3 x^{2 p} e^{2 p} b} \] Input:

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

Output:

(2*(a + b*x**2)**p*a*d**2*p*x + 3*(a + b*x**2)**p*b*c**2*x + 3*(a + b*x**2 
)**p*b*c*d*x**2 + (a + b*x**2)**p*b*d**2*x**3 + 4*x**(2*p)*int((a + b*x**2 
)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**2*p**2 - 2*x**(2*p)*int((a 
+ b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**2*p + 6*x**(2*p)*in 
t((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*b*c**2*p + 6*x**(2*p 
)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*b*c*d*p)/(3* 
x**(2*p)*e**(2*p)*b)