\(\int (e x)^{-4-2 p} (c+d x)^3 (a+b x^2)^p \, dx\) [334]

Optimal result

Integrand size = 26, antiderivative size = 226 \[ \int (e x)^{-4-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=-\frac {3 c d^2 (e x)^{-3-2 p} \left (a+b x^2\right )^{1+p}}{b e}-\frac {3 c^2 d (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a e^2 (1+p)}+\frac {c \left (\frac {3 a d^2}{b}-\frac {c^2}{3+2 p}\right ) (e x)^{-3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-p,\frac {1}{2} (-1-2 p),-\frac {b x^2}{a}\right )}{e}-\frac {d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e^4 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.97 \[ \int (e x)^{-4-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 a c^3 p \left (1+3 p+2 p^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-p,-\frac {1}{2}-p,-\frac {b x^2}{a}\right )+d (3+2 p) x \left (6 a c d p (1+p) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+(1+2 p) \left (3 c^2 p \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+a d^2 (1+p) x^2 \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )\right )}{2 a e^4 p (1+p) (1+2 p) (3+2 p) x^3} \] Input:

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

Output:

-1/2*((a + b*x^2)^p*(2*a*c^3*p*(1 + 3*p + 2*p^2)*Hypergeometric2F1[-3/2 - 
p, -p, -1/2 - p, -((b*x^2)/a)] + d*(3 + 2*p)*x*(6*a*c*d*p*(1 + p)*x*Hyperg 
eometric2F1[-1/2 - p, -p, 1/2 - p, -((b*x^2)/a)] + (1 + 2*p)*(3*c^2*p*(a + 
 b*x^2)*(1 + (b*x^2)/a)^p + a*d^2*(1 + p)*x^2*Hypergeometric2F1[-p, -p, 1 
- p, -((b*x^2)/a)]))))/(a*e^4*p*(1 + p)*(1 + 2*p)*(3 + 2*p)*x^3*(e*x)^(2*p 
)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {560, 27, 358, 279, 278, 363, 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.

\(\displaystyle \int (c+d x)^3 (e x)^{-2 p-4} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 560

\(\displaystyle \int (e x)^{-2 p-4} \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\frac {\int d (e x)^{-2 p-3} \left (b x^2+a\right )^p \left (3 c^2+d^2 x^2\right )dx}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle \int (e x)^{-2 p-4} \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\frac {d \int (e x)^{-2 p-3} \left (b x^2+a\right )^p \left (3 c^2+d^2 x^2\right )dx}{e}\)

\(\Big \downarrow \) 358

\(\displaystyle \int (e x)^{-2 p-4} \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\frac {d \left (\frac {d^2 \int (e x)^{-2 p-1} \left (b x^2+a\right )^pdx}{e^2}-\frac {3 c^2 (e x)^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a e (p+1)}\right )}{e}\)

\(\Big \downarrow \) 279

\(\displaystyle \int (e x)^{-2 p-4} \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\frac {d \left (\frac {d^2 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-2 p-1} \left (\frac {b x^2}{a}+1\right )^pdx}{e^2}-\frac {3 c^2 (e x)^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a e (p+1)}\right )}{e}\)

\(\Big \downarrow \) 278

\(\displaystyle \int (e x)^{-2 p-4} \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\frac {d \left (-\frac {3 c^2 (e x)^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{2 a e (p+1)}-\frac {d^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e^3 p}\right )}{e}\)

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

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 358

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ 
Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S 
imp[d/e^2   Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, 
 -1]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 560

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

Maple [F]

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

Input:

int((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(b*x^2 + a)^p*(e*x)^(-2 
*p - 4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 67.57 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.05 \[ \int (e x)^{-4-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{3} e^{- 2 p - 4} x^{- 2 p - 3} \Gamma \left (- p - \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {3}{2} \\ - p - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (- p - \frac {1}{2}\right )} + \frac {3 a^{p} c^{2} d e^{- 2 p - 4} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} + \frac {3 a^{p} c d^{2} e^{- 2 p - 4} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {a^{p} d^{3} e^{- 2 p - 4} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} \] Input:

integrate((e*x)**(-4-2*p)*(d*x+c)**3*(b*x**2+a)**p,x)
 

Output:

a**p*c**3*e**(-2*p - 4)*x**(-2*p - 3)*gamma(-p - 3/2)*hyper((-p, -p - 3/2) 
, (-p - 1/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(-p - 1/2)) + 3*a**p*c**2 
*d*e**(-2*p - 4)*x**(-2*p - 2)*(1 + b*x**2/a)**(p + 1)*gamma(-p - 1)/(2*ga 
mma(-p)) + 3*a**p*c*d**2*e**(-2*p - 4)*x**(-2*p - 1)*gamma(-p - 1/2)*hyper 
((-p, -p - 1/2), (1/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(1/2 - p)) 
+ a**p*d**3*e**(-2*p - 4)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_p 
olar(I*pi)/a)/(2*x**(2*p)*gamma(1 - p))
 

Maxima [F]

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

integrate((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x + c)^3*(b*x^2 + a)^p*(e*x)^(-2*p - 4), x)
 

Giac [F]

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

integrate((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*(b*x^2 + a)^p*(e*x)^(-2*p - 4), x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x)^3)/(e*x)^(2*p + 4),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x)^3)/(e*x)^(2*p + 4), x)
 

Reduce [F]

\[ \int (e x)^{-4-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {-2 \left (b \,x^{2}+a \right )^{p} a \,c^{3} p -2 \left (b \,x^{2}+a \right )^{p} a \,c^{3}-9 \left (b \,x^{2}+a \right )^{p} a \,c^{2} d x -18 \left (b \,x^{2}+a \right )^{p} a c \,d^{2} p \,x^{2}-18 \left (b \,x^{2}+a \right )^{p} a c \,d^{2} x^{2}-9 \left (b \,x^{2}+a \right )^{p} b \,c^{2} d \,x^{3}-4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} c^{3} p^{2} x^{3}-4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} c^{3} p \,x^{3}-36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c \,d^{2} p^{2} x^{3}-36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c \,d^{2} p \,x^{3}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a \,d^{3} p \,x^{3}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a \,d^{3} x^{3}}{6 x^{2 p} e^{2 p} a \,e^{4} x^{3} \left (p +1\right )} \] Input:

int((e*x)^(-4-2*p)*(d*x+c)^3*(b*x^2+a)^p,x)
 

Output:

( - 2*(a + b*x**2)**p*a*c**3*p - 2*(a + b*x**2)**p*a*c**3 - 9*(a + b*x**2) 
**p*a*c**2*d*x - 18*(a + b*x**2)**p*a*c*d**2*p*x**2 - 18*(a + b*x**2)**p*a 
*c*d**2*x**2 - 9*(a + b*x**2)**p*b*c**2*d*x**3 - 4*x**(2*p)*int((a + b*x** 
2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**2*c**3*p**2*x**3 - 4*x**(2 
*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**2*c**3*p 
*x**3 - 36*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**2 + x**(2*p)*b*x**4 
),x)*a**2*c*d**2*p**2*x**3 - 36*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x 
**2 + x**(2*p)*b*x**4),x)*a**2*c*d**2*p*x**3 + 6*x**(2*p)*int((a + b*x**2) 
**p/(x**(2*p)*x),x)*a*d**3*p*x**3 + 6*x**(2*p)*int((a + b*x**2)**p/(x**(2* 
p)*x),x)*a*d**3*x**3)/(6*x**(2*p)*e**(2*p)*a*e**4*x**3*(p + 1))