\(\int x^{2-3 p} (a+b x^3)^p (c+d x^3) \, dx\) [885]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int x^{2-3 p} \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {d x^{3-3 p} \left (a+b x^3\right )^{1+p}}{6 b}-\frac {1}{6} \left (\frac {a d}{b}-\frac {2 c}{1-p}\right ) x^{3-3 p} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^3}{a}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int x^{2-3 p} \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=-\frac {x^{3-3 p} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (c (-2+p) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^3}{a}\right )+d (-1+p) x^3 \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^3}{a}\right )\right )}{3 (-2+p) (-1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {959, 882, 74}

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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 882

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

\(\Big \downarrow \) 74

\(\displaystyle \frac {a x^{3-3 p} \left (a+b x^3\right )^{p-1} (2 b c-a d (1-p)) \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {b x^3}{b x^3+a}\right )}{6 b (1-p)}+\frac {d x^{3-3 p} \left (a+b x^3\right )^{p+1}}{6 b}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 882

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ 
(m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p 
]))   Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), 
x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli 
fy[(m + 1)/n + p]]
 

rule 959

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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