Integrand size = 18, antiderivative size = 92 \[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {d x^2 \left (a+b x^3\right )^{1+p}}{b (5+3 p)}+\frac {1}{2} \left (c-\frac {2 a d}{5 b+3 b p}\right ) x^2 \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b x^3}{a}\right ) \] Output:
d*x^2*(b*x^3+a)^(p+1)/b/(5+3*p)+1/2*(c-2*a*d/(3*b*p+5*b))*x^2*(b*x^3+a)^p* hypergeom([2/3, -p],[5/3],-b*x^3/a)/((1+b*x^3/a)^p)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {1}{10} x^2 \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (5 c \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b x^3}{a}\right )+2 d x^3 \operatorname {Hypergeometric2F1}\left (\frac {5}{3},-p,\frac {8}{3},-\frac {b x^3}{a}\right )\right ) \] Input:
Integrate[x*(a + b*x^3)^p*(c + d*x^3),x]
Output:
(x^2*(a + b*x^3)^p*(5*c*Hypergeometric2F1[2/3, -p, 5/3, -((b*x^3)/a)] + 2* d*x^3*Hypergeometric2F1[5/3, -p, 8/3, -((b*x^3)/a)]))/(10*(1 + (b*x^3)/a)^ p)
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {959, 889, 888}
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.
| \(\displaystyle \int x \left (c+d x^3\right ) \left (a+b x^3\right )^p \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \left (c-\frac {2 a d}{3 b p+5 b}\right ) \int x \left (b x^3+a\right )^pdx+\frac {d x^2 \left (a+b x^3\right )^{p+1}}{b (3 p+5)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c-\frac {2 a d}{3 b p+5 b}\right ) \int x \left (\frac {b x^3}{a}+1\right )^pdx+\frac {d x^2 \left (a+b x^3\right )^{p+1}}{b (3 p+5)}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c-\frac {2 a d}{3 b p+5 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},-\frac {b x^3}{a}\right )+\frac {d x^2 \left (a+b x^3\right )^{p+1}}{b (3 p+5)}\) |
Input:
Int[x*(a + b*x^3)^p*(c + d*x^3),x]
Output:
(d*x^2*(a + b*x^3)^(1 + p))/(b*(5 + 3*p)) + ((c - (2*a*d)/(5*b + 3*b*p))*x ^2*(a + b*x^3)^p*Hypergeometric2F1[2/3, -p, 5/3, -((b*x^3)/a)])/(2*(1 + (b *x^3)/a)^p)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
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]
\[\int x \left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )d x\]
Input:
int(x*(b*x^3+a)^p*(d*x^3+c),x)
Output:
int(x*(b*x^3+a)^p*(d*x^3+c),x)
\[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\int { {\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p} x \,d x } \] Input:
integrate(x*(b*x^3+a)^p*(d*x^3+c),x, algorithm="fricas")
Output:
integral((d*x^4 + c*x)*(b*x^3 + a)^p, x)
Result contains complex when optimal does not.
Time = 64.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {a^{p} c x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, - p \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {a^{p} d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{3}, - p \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate(x*(b*x**3+a)**p*(d*x**3+c),x)
Output:
a**p*c*x**2*gamma(2/3)*hyper((2/3, -p), (5/3,), b*x**3*exp_polar(I*pi)/a)/ (3*gamma(5/3)) + a**p*d*x**5*gamma(5/3)*hyper((5/3, -p), (8/3,), b*x**3*ex p_polar(I*pi)/a)/(3*gamma(8/3))
\[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\int { {\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p} x \,d x } \] Input:
integrate(x*(b*x^3+a)^p*(d*x^3+c),x, algorithm="maxima")
Output:
integrate((d*x^3 + c)*(b*x^3 + a)^p*x, x)
\[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\int { {\left (d x^{3} + c\right )} {\left (b x^{3} + a\right )}^{p} x \,d x } \] Input:
integrate(x*(b*x^3+a)^p*(d*x^3+c),x, algorithm="giac")
Output:
integrate((d*x^3 + c)*(b*x^3 + a)^p*x, x)
Timed out. \[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\int x\,{\left (b\,x^3+a\right )}^p\,\left (d\,x^3+c\right ) \,d x \] Input:
int(x*(a + b*x^3)^p*(c + d*x^3),x)
Output:
int(x*(a + b*x^3)^p*(c + d*x^3), x)
\[ \int x \left (a+b x^3\right )^p \left (c+d x^3\right ) \, dx=\frac {3 \left (b \,x^{3}+a \right )^{p} a d p \,x^{2}+3 \left (b \,x^{3}+a \right )^{p} b c p \,x^{2}+5 \left (b \,x^{3}+a \right )^{p} b c \,x^{2}+3 \left (b \,x^{3}+a \right )^{p} b d p \,x^{5}+2 \left (b \,x^{3}+a \right )^{p} b d \,x^{5}-54 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a^{2} d \,p^{3}-126 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a^{2} d \,p^{2}-60 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a^{2} d p +81 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a b c \,p^{4}+324 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a b c \,p^{3}+405 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a b c \,p^{2}+150 \left (\int \frac {\left (b \,x^{3}+a \right )^{p} x}{9 b \,p^{2} x^{3}+21 b p \,x^{3}+10 b \,x^{3}+9 a \,p^{2}+21 a p +10 a}d x \right ) a b c p}{b \left (9 p^{2}+21 p +10\right )} \] Input:
int(x*(b*x^3+a)^p*(d*x^3+c),x)
Output:
(3*(a + b*x**3)**p*a*d*p*x**2 + 3*(a + b*x**3)**p*b*c*p*x**2 + 5*(a + b*x* *3)**p*b*c*x**2 + 3*(a + b*x**3)**p*b*d*p*x**5 + 2*(a + b*x**3)**p*b*d*x** 5 - 54*int(((a + b*x**3)**p*x)/(9*a*p**2 + 21*a*p + 10*a + 9*b*p**2*x**3 + 21*b*p*x**3 + 10*b*x**3),x)*a**2*d*p**3 - 126*int(((a + b*x**3)**p*x)/(9* a*p**2 + 21*a*p + 10*a + 9*b*p**2*x**3 + 21*b*p*x**3 + 10*b*x**3),x)*a**2* d*p**2 - 60*int(((a + b*x**3)**p*x)/(9*a*p**2 + 21*a*p + 10*a + 9*b*p**2*x **3 + 21*b*p*x**3 + 10*b*x**3),x)*a**2*d*p + 81*int(((a + b*x**3)**p*x)/(9 *a*p**2 + 21*a*p + 10*a + 9*b*p**2*x**3 + 21*b*p*x**3 + 10*b*x**3),x)*a*b* c*p**4 + 324*int(((a + b*x**3)**p*x)/(9*a*p**2 + 21*a*p + 10*a + 9*b*p**2* x**3 + 21*b*p*x**3 + 10*b*x**3),x)*a*b*c*p**3 + 405*int(((a + b*x**3)**p*x )/(9*a*p**2 + 21*a*p + 10*a + 9*b*p**2*x**3 + 21*b*p*x**3 + 10*b*x**3),x)* a*b*c*p**2 + 150*int(((a + b*x**3)**p*x)/(9*a*p**2 + 21*a*p + 10*a + 9*b*p **2*x**3 + 21*b*p*x**3 + 10*b*x**3),x)*a*b*c*p)/(b*(9*p**2 + 21*p + 10))