\(\int (e x)^{-2-2 p} (c+d x)^3 (a+b x^2)^p \, dx\) [318]

Optimal result

Integrand size = 26, antiderivative size = 233 \[ \int (e x)^{-2-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {3 c d^2 (e x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{b e}+\frac {d^3 (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{2 b e^2}-\frac {c \left (3 a d^2+\frac {b c^2}{1+2 p}\right ) (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{b e}-\frac {d \left (3 b c^2+a d^2 p\right ) (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 b e^2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.92 \[ \int (e x)^{-2-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=-\frac {x (e x)^{-2 (1+p)} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 c^3 p \left (1-3 p+2 p^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+d (1+2 p) x \left (6 c d (-1+p) p x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )+(-1+2 p) \left (d^2 p x^2 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+3 c^2 (-1+p) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )\right )}{2 (-1+p) p (-1+2 p) (1+2 p)} \] Input:

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

Output:

-1/2*(x*(a + b*x^2)^p*(2*c^3*p*(1 - 3*p + 2*p^2)*Hypergeometric2F1[-1/2 - 
p, -p, 1/2 - p, -((b*x^2)/a)] + d*(1 + 2*p)*x*(6*c*d*(-1 + p)*p*x*Hypergeo 
metric2F1[1/2 - p, -p, 3/2 - p, -((b*x^2)/a)] + (-1 + 2*p)*(d^2*p*x^2*Hype 
rgeometric2F1[1 - p, -p, 2 - p, -((b*x^2)/a)] + 3*c^2*(-1 + p)*Hypergeomet 
ric2F1[-p, -p, 1 - p, -((b*x^2)/a)]))))/((-1 + p)*p*(-1 + 2*p)*(1 + 2*p)*( 
e*x)^(2*(1 + p))*(1 + (b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {559, 27, 2340, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

rule 2340

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 95.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00 \[ \int (e x)^{-2-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{3} e^{- 2 p - 2} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {3 a^{p} c^{2} d e^{- 2 p - 2} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} + \frac {3 a^{p} c d^{2} e^{- 2 p - 2} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right )} + \frac {a^{p} d^{3} e^{- 2 p - 2} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} \] Input:

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

Output:

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

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-2-2 p} (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {-12 \left (b \,x^{2}+a \right )^{p} a c \,d^{2} p^{2}-2 \left (b \,x^{2}+a \right )^{p} b \,c^{3} p -3 \left (b \,x^{2}+a \right )^{p} b \,c^{2} d x +6 \left (b \,x^{2}+a \right )^{p} b c \,d^{2} p \,x^{2}+\left (b \,x^{2}+a \right )^{p} b \,d^{3} p \,x^{3}-24 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c \,d^{2} p^{3} x -12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c \,d^{2} p^{2} x -4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a b \,c^{3} p^{2} x +2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b \,d^{3} p^{2} x +6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) b^{2} c^{2} d p x}{2 x^{2 p} e^{2 p} b \,e^{2} p x} \] Input:

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

Output:

( - 12*(a + b*x**2)**p*a*c*d**2*p**2 - 2*(a + b*x**2)**p*b*c**3*p - 3*(a + 
 b*x**2)**p*b*c**2*d*x + 6*(a + b*x**2)**p*b*c*d**2*p*x**2 + (a + b*x**2)* 
*p*b*d**3*p*x**3 - 24*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**3*x - 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*d**2*p**2*x - 4*x**(2*p)*int((a + 
 b*x**2)**p/(x**(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a*b*c**3*p**2*x + 2*x** 
(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*b*d**3*p 
**2*x + 6*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2), 
x)*b**2*c**2*d*p*x)/(2*x**(2*p)*e**(2*p)*b*e**2*p*x)