\(\int x^5 (c+d x)^3 (a+b x^2)^p \, dx\) [19]

Optimal result

Integrand size = 20, antiderivative size = 247 \[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {a^2 c \left (b c^2-3 a d^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^4 (1+p)}+\frac {d^3 x^7 \left (a+b x^2\right )^{1+p}}{b (9+2 p)}-\frac {a c \left (2 b c^2-9 a d^2\right ) \left (a+b x^2\right )^{2+p}}{2 b^4 (2+p)}+\frac {c \left (b c^2-9 a d^2\right ) \left (a+b x^2\right )^{3+p}}{2 b^4 (3+p)}+\frac {3 c d^2 \left (a+b x^2\right )^{4+p}}{2 b^4 (4+p)}-\frac {d \left (7 a d^2-3 b c^2 (9+2 p)\right ) x^7 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},-\frac {b x^2}{a}\right )}{7 b (9+2 p)} \] Output:

1/2*a^2*c*(-3*a*d^2+b*c^2)*(b*x^2+a)^(p+1)/b^4/(p+1)+d^3*x^7*(b*x^2+a)^(p+ 
1)/b/(9+2*p)-1/2*a*c*(-9*a*d^2+2*b*c^2)*(b*x^2+a)^(2+p)/b^4/(2+p)+1/2*c*(- 
9*a*d^2+b*c^2)*(b*x^2+a)^(3+p)/b^4/(3+p)+3/2*c*d^2*(b*x^2+a)^(4+p)/b^4/(4+ 
p)-1/7*d*(7*a*d^2-3*b*c^2*(9+2*p))*x^7*(b*x^2+a)^p*hypergeom([7/2, -p],[9/ 
2],-b*x^2/a)/b/(9+2*p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.01 \[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\frac {1}{126} \left (a+b x^2\right )^p \left (\frac {63 c^3 \left (a+b x^2\right ) \left (2 a^2-2 a b (1+p) x^2+b^2 \left (2+3 p+p^2\right ) x^4\right )}{b^3 (1+p) (2+p) (3+p)}+\frac {189 c d^2 \left (a+b x^2\right ) \left (-6 a^3+6 a^2 b (1+p) x^2-3 a b^2 \left (2+3 p+p^2\right ) x^4+b^3 \left (6+11 p+6 p^2+p^3\right ) x^6\right )}{b^4 (1+p) (2+p) (3+p) (4+p)}+54 c^2 d x^7 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},-\frac {b x^2}{a}\right )+14 d^3 x^9 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-p,\frac {11}{2},-\frac {b x^2}{a}\right )\right ) \] Input:

Integrate[x^5*(c + d*x)^3*(a + b*x^2)^p,x]
 

Output:

((a + b*x^2)^p*((63*c^3*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 
3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p)*(3 + p)) + (189*c*d^2*(a + b*x^2)*(- 
6*a^3 + 6*a^2*b*(1 + p)*x^2 - 3*a*b^2*(2 + 3*p + p^2)*x^4 + b^3*(6 + 11*p 
+ 6*p^2 + p^3)*x^6))/(b^4*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (54*c^2*d*x^7 
*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (14*d^ 
3*x^9*Hypergeometric2F1[9/2, -p, 11/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/ 
126
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {543, 354, 27, 86, 363, 279, 278, 2009}

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.

\(\displaystyle \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 543

\(\displaystyle \int x^5 \left (b x^2+a\right )^p \left (c^3+3 d^2 x^2 c\right )dx+\int x^6 \left (b x^2+a\right )^p \left (x^2 d^3+3 c^2 d\right )dx\)

\(\Big \downarrow \) 354

\(\displaystyle \int x^6 \left (b x^2+a\right )^p \left (x^2 d^3+3 c^2 d\right )dx+\frac {1}{2} \int c x^4 \left (b x^2+a\right )^p \left (c^2+3 d^2 x^2\right )dx^2\)

\(\Big \downarrow \) 27

\(\displaystyle \int x^6 \left (b x^2+a\right )^p \left (x^2 d^3+3 c^2 d\right )dx+\frac {1}{2} c \int x^4 \left (b x^2+a\right )^p \left (c^2+3 d^2 x^2\right )dx^2\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{2} c \int \left (-\frac {a^2 \left (3 a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b^3}+\frac {a \left (9 a d^2-2 b c^2\right ) \left (b x^2+a\right )^{p+1}}{b^3}+\frac {\left (b c^2-9 a d^2\right ) \left (b x^2+a\right )^{p+2}}{b^3}+\frac {3 d^2 \left (b x^2+a\right )^{p+3}}{b^3}\right )dx^2+\int x^6 \left (b x^2+a\right )^p \left (x^2 d^3+3 c^2 d\right )dx\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {1}{2} c \int \left (-\frac {a^2 \left (3 a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b^3}+\frac {a \left (9 a d^2-2 b c^2\right ) \left (b x^2+a\right )^{p+1}}{b^3}+\frac {\left (b c^2-9 a d^2\right ) \left (b x^2+a\right )^{p+2}}{b^3}+\frac {3 d^2 \left (b x^2+a\right )^{p+3}}{b^3}\right )dx^2+d \left (3 c^2-\frac {7 a d^2}{2 b p+9 b}\right ) \int x^6 \left (b x^2+a\right )^pdx+\frac {d^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {1}{2} c \int \left (-\frac {a^2 \left (3 a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b^3}+\frac {a \left (9 a d^2-2 b c^2\right ) \left (b x^2+a\right )^{p+1}}{b^3}+\frac {\left (b c^2-9 a d^2\right ) \left (b x^2+a\right )^{p+2}}{b^3}+\frac {3 d^2 \left (b x^2+a\right )^{p+3}}{b^3}\right )dx^2+d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 c^2-\frac {7 a d^2}{2 b p+9 b}\right ) \int x^6 \left (\frac {b x^2}{a}+1\right )^pdx+\frac {d^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {1}{2} c \int \left (-\frac {a^2 \left (3 a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b^3}+\frac {a \left (9 a d^2-2 b c^2\right ) \left (b x^2+a\right )^{p+1}}{b^3}+\frac {\left (b c^2-9 a d^2\right ) \left (b x^2+a\right )^{p+2}}{b^3}+\frac {3 d^2 \left (b x^2+a\right )^{p+3}}{b^3}\right )dx^2+\frac {1}{7} d x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 c^2-\frac {7 a d^2}{2 b p+9 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},-\frac {b x^2}{a}\right )+\frac {d^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} c \left (\frac {a^2 \left (b c^2-3 a d^2\right ) \left (a+b x^2\right )^{p+1}}{b^4 (p+1)}-\frac {a \left (2 b c^2-9 a d^2\right ) \left (a+b x^2\right )^{p+2}}{b^4 (p+2)}+\frac {\left (b c^2-9 a d^2\right ) \left (a+b x^2\right )^{p+3}}{b^4 (p+3)}+\frac {3 d^2 \left (a+b x^2\right )^{p+4}}{b^4 (p+4)}\right )+\frac {1}{7} d x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 c^2-\frac {7 a d^2}{2 b p+9 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},-\frac {b x^2}{a}\right )+\frac {d^3 x^7 \left (a+b x^2\right )^{p+1}}{b (2 p+9)}\)

Input:

Int[x^5*(c + d*x)^3*(a + b*x^2)^p,x]
 

Output:

(d^3*x^7*(a + b*x^2)^(1 + p))/(b*(9 + 2*p)) + (c*((a^2*(b*c^2 - 3*a*d^2)*( 
a + b*x^2)^(1 + p))/(b^4*(1 + p)) - (a*(2*b*c^2 - 9*a*d^2)*(a + b*x^2)^(2 
+ p))/(b^4*(2 + p)) + ((b*c^2 - 9*a*d^2)*(a + b*x^2)^(3 + p))/(b^4*(3 + p) 
) + (3*d^2*(a + b*x^2)^(4 + p))/(b^4*(4 + p))))/2 + (d*(3*c^2 - (7*a*d^2)/ 
(9*b + 2*b*p))*x^7*(a + b*x^2)^p*Hypergeometric2F1[7/2, -p, 9/2, -((b*x^2) 
/a)])/(7*(1 + (b*x^2)/a)^p)
 

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

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 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

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 543

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int x^{5} \left (d x +c \right )^{3} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int(x^5*(d*x+c)^3*(b*x^2+a)^p,x)
 

Output:

int(x^5*(d*x+c)^3*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((d^3*x^8 + 3*c*d^2*x^7 + 3*c^2*d*x^6 + c^3*x^5)*(b*x^2 + a)^p, x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (218) = 436\).

Time = 31.35 (sec) , antiderivative size = 2919, normalized size of antiderivative = 11.82 \[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\text {Too large to display} \] Input:

integrate(x**5*(d*x+c)**3*(b*x**2+a)**p,x)
 

Output:

3*a**p*c**2*d*x**7*hyper((7/2, -p), (9/2,), b*x**2*exp_polar(I*pi)/a)/7 + 
a**p*d**3*x**9*hyper((9/2, -p), (11/2,), b*x**2*exp_polar(I*pi)/a)/9 + c** 
3*Piecewise((a**p*x**6/6, Eq(b, 0)), (2*a**2*log(x - sqrt(-a/b))/(4*a**2*b 
**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*a**2*log(x + sqrt(-a/b))/(4*a**2*b* 
*3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 3*a**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 
4*b**5*x**4) + 4*a*b*x**2*log(x - sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 
 + 4*b**5*x**4) + 4*a*b*x**2*log(x + sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x 
**2 + 4*b**5*x**4) + 4*a*b*x**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4 
) + 2*b**2*x**4*log(x - sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5* 
x**4) + 2*b**2*x**4*log(x + sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b 
**5*x**4), Eq(p, -3)), (-2*a**2*log(x - sqrt(-a/b))/(2*a*b**3 + 2*b**4*x** 
2) - 2*a**2*log(x + sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) - 2*a**2/(2*a*b** 
3 + 2*b**4*x**2) - 2*a*b*x**2*log(x - sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) 
 - 2*a*b*x**2*log(x + sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) + b**2*x**4/(2* 
a*b**3 + 2*b**4*x**2), Eq(p, -2)), (a**2*log(x - sqrt(-a/b))/(2*b**3) + a* 
*2*log(x + sqrt(-a/b))/(2*b**3) - a*x**2/(2*b**2) + x**4/(4*b), Eq(p, -1)) 
, (2*a**3*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b** 
3) - 2*a**2*b*p*x**2*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3 
*p + 12*b**3) + a*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p* 
*2 + 22*b**3*p + 12*b**3) + a*b**2*p*x**4*(a + b*x**2)**p/(2*b**3*p**3 ...
 

Maxima [F]

\[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

1/2*((p^2 + 3*p + 2)*b^3*x^6 + (p^2 + p)*a*b^2*x^4 - 2*a^2*b*p*x^2 + 2*a^3 
)*(b*x^2 + a)^p*c^3/((p^3 + 6*p^2 + 11*p + 6)*b^3) + integrate((d^3*x^8 + 
3*c*d^2*x^7 + 3*c^2*d*x^6)*(b*x^2 + a)^p, x)
 

Giac [F]

\[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} x^{5} \,d x } \] Input:

integrate(x^5*(d*x+c)^3*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*(b*x^2 + a)^p*x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\int x^5\,{\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int(x^5*(a + b*x^2)^p*(c + d*x)^3,x)
 

Output:

int(x^5*(a + b*x^2)^p*(c + d*x)^3, x)
 

Reduce [F]

\[ \int x^5 (c+d x)^3 \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:

int(x^5*(d*x+c)^3*(b*x^2+a)^p,x)
 

Output:

( - 576*(a + b*x**2)**p*a**4*c*d**2*p**5 - 7200*(a + b*x**2)**p*a**4*c*d** 
2*p**4 - 33120*(a + b*x**2)**p*a**4*c*d**2*p**3 - 68400*(a + b*x**2)**p*a* 
*4*c*d**2*p**2 - 60804*(a + b*x**2)**p*a**4*c*d**2*p - 17010*(a + b*x**2)* 
*p*a**4*c*d**2 - 420*(a + b*x**2)**p*a**4*d**3*p**5*x - 4200*(a + b*x**2)* 
*p*a**4*d**3*p**4*x - 14700*(a + b*x**2)**p*a**4*d**3*p**3*x - 21000*(a + 
b*x**2)**p*a**4*d**3*p**2*x - 10080*(a + b*x**2)**p*a**4*d**3*p*x + 64*(a 
+ b*x**2)**p*a**3*b*c**3*p**6 + 1056*(a + b*x**2)**p*a**3*b*c**3*p**5 + 68 
80*(a + b*x**2)**p*a**3*b*c**3*p**4 + 22320*(a + b*x**2)**p*a**3*b*c**3*p* 
*3 + 37156*(a + b*x**2)**p*a**3*b*c**3*p**2 + 28914*(a + b*x**2)**p*a**3*b 
*c**3*p + 7560*(a + b*x**2)**p*a**3*b*c**3 + 360*(a + b*x**2)**p*a**3*b*c* 
*2*d*p**6*x + 5220*(a + b*x**2)**p*a**3*b*c**2*d*p**5*x + 28800*(a + b*x** 
2)**p*a**3*b*c**2*d*p**4*x + 74700*(a + b*x**2)**p*a**3*b*c**2*d*p**3*x + 
89640*(a + b*x**2)**p*a**3*b*c**2*d*p**2*x + 38880*(a + b*x**2)**p*a**3*b* 
c**2*d*p*x + 576*(a + b*x**2)**p*a**3*b*c*d**2*p**6*x**2 + 7200*(a + b*x** 
2)**p*a**3*b*c*d**2*p**5*x**2 + 33120*(a + b*x**2)**p*a**3*b*c*d**2*p**4*x 
**2 + 68400*(a + b*x**2)**p*a**3*b*c*d**2*p**3*x**2 + 60804*(a + b*x**2)** 
p*a**3*b*c*d**2*p**2*x**2 + 17010*(a + b*x**2)**p*a**3*b*c*d**2*p*x**2 + 2 
80*(a + b*x**2)**p*a**3*b*d**3*p**6*x**3 + 2940*(a + b*x**2)**p*a**3*b*d** 
3*p**5*x**3 + 11200*(a + b*x**2)**p*a**3*b*d**3*p**4*x**3 + 18900*(a + b*x 
**2)**p*a**3*b*d**3*p**3*x**3 + 13720*(a + b*x**2)**p*a**3*b*d**3*p**2*...