\(\int (e x)^{-3-2 p} (c+d x)^3 (a+b x^2)^p \, dx\) [326]

Optimal result

Integrand size = 26, antiderivative size = 230 \[ \int (e x)^{-3-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {d^3 (e x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{b e^2}-\frac {c^3 (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{2 a e (1+p)}-\frac {d \left (3 b c^2+a d^2 (1+2 p)\right ) (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{b e^2 (1+2 p)}-\frac {3 c d^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 e^3 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.97 \[ \int (e x)^{-3-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (6 a c^2 d p \left (-1+p+2 p^2\right ) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+(1+2 p) \left (2 a d^3 p (1+p) x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+c (-1+2 p) \left (c^2 p \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+3 a d^2 (1+p) x^2 \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )\right )}{2 a e^3 p (1+p) (-1+2 p) (1+2 p) x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {559, 2339, 27, 279, 278, 545, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 2339

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 545

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

(d^3*(e*x)^(-1 - 2*p)*(a + b*x^2)^(1 + p))/(b*e^2) + (-1/2*(b*c^3*(a + b*x 
^2)^(1 + p))/(a*e*(1 + p)*(e*x)^(2*(1 + p))) - (d*(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, (1 
- 2*p)/2, -((b*x^2)/a)])/(e^2*(1 + 2*p)*(1 + (b*x^2)/a)^p) - (3*b*c*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))/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 545

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]
 

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]
 

rule 2339

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With 
[{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/c^q   Int[(c*x)^(m + q)*(a + b*x^ 
2)^p, x], x] + Simp[1/c^q   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq - 
Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0] 
] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] &&  !(IGtQ[m, 0] && ILtQ[p + 
 1/2, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-3-2*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 - 3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 74.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00 \[ \int (e x)^{-3-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{3} e^{- 2 p - 3} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{2 \Gamma \left (- p\right )} + \frac {3 a^{p} c^{2} d e^{- 2 p - 3} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {3 a^{p} c d^{2} e^{- 2 p - 3} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} + \frac {a^{p} d^{3} e^{- 2 p - 3} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*(b*x^2 + a)*c^3*e^(-2*p - 3)*e^(p*log(b*x^2 + a) - 2*p*log(x))/(a*(p 
+ 1)*x^2) + e^(-2*p - 3)*integrate((d^3*x^2 + 3*c*d^2*x + 3*c^2*d)*e^(p*lo 
g(b*x^2 + a) - 2*p*log(x))/x^2, x)
 

Giac [F]

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

integrate((e*x)^(-3-2*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 - 3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-3-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {-\left (b \,x^{2}+a \right )^{p} a \,c^{3}-6 \left (b \,x^{2}+a \right )^{p} a \,c^{2} d p x -6 \left (b \,x^{2}+a \right )^{p} a \,c^{2} d x +2 \left (b \,x^{2}+a \right )^{p} a \,d^{3} p \,x^{3}+2 \left (b \,x^{2}+a \right )^{p} a \,d^{3} x^{3}-\left (b \,x^{2}+a \right )^{p} b \,c^{3} x^{2}-12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c^{2} d \,p^{2} x^{2}-12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c^{2} d p \,x^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{3} p^{2} x^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{3} p \,x^{2}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a c \,d^{2} p \,x^{2}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a c \,d^{2} x^{2}}{2 x^{2 p} e^{2 p} a \,e^{3} x^{2} \left (p +1\right )} \] Input:

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

Output:

( - (a + b*x**2)**p*a*c**3 - 6*(a + b*x**2)**p*a*c**2*d*p*x - 6*(a + b*x** 
2)**p*a*c**2*d*x + 2*(a + b*x**2)**p*a*d**3*p*x**3 + 2*(a + b*x**2)**p*a*d 
**3*x**3 - (a + b*x**2)**p*b*c**3*x**2 - 12*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**2*d*p**2*x**2 - 12*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**2*d*p*x 
**2 + 4*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a** 
2*d**3*p**2*x**2 + 4*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b 
*x**2),x)*a**2*d**3*p*x**2 + 6*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*x),x 
)*a*c*d**2*p*x**2 + 6*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*x),x)*a*c*d** 
2*x**2)/(2*x**(2*p)*e**(2*p)*a*e**3*x**2*(p + 1))