\(\int (a+b x^3)^{7/3} (c+d x^3)^2 \, dx\) [131]

Optimal result

Integrand size = 21, antiderivative size = 131 \[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\frac {2 d (7 b c-a d) x \left (a+b x^3\right )^{10/3}}{77 b^2}+\frac {d^2 x^4 \left (a+b x^3\right )^{10/3}}{14 b}+\frac {a^2 \left (77 b^2 c^2-2 a d (7 b c-a d)\right ) x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{77 b^2 \sqrt [3]{1+\frac {b x^3}{a}}} \] Output:

2/77*d*(-a*d+7*b*c)*x*(b*x^3+a)^(10/3)/b^2+1/14*d^2*x^4*(b*x^3+a)^(10/3)/b 
+1/77*a^2*(77*b^2*c^2-2*a*d*(-a*d+7*b*c))*x*(b*x^3+a)^(1/3)*hypergeom([-7/ 
3, 1/3],[4/3],-b*x^3/a)/b^2/(1+b*x^3/a)^(1/3)
 

Mathematica [A] (warning: unable to verify)

Time = 12.80 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.35 \[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\frac {a x \sqrt [3]{a+b x^3} \left (20 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \operatorname {Gamma}\left (-\frac {7}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {1}{3},\frac {10}{3},-\frac {b x^3}{a}\right )-3 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \operatorname {Gamma}\left (-\frac {4}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {4}{3},\frac {13}{3},-\frac {b x^3}{a}\right )-9 b x^3 \left (c+d x^3\right )^2 \operatorname {Gamma}\left (-\frac {4}{3}\right ) \, _3F_2\left (-\frac {4}{3},\frac {4}{3},2;1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{280 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Gamma}\left (-\frac {7}{3}\right )} \] Input:

Integrate[(a + b*x^3)^(7/3)*(c + d*x^3)^2,x]
 

Output:

(a*x*(a + b*x^3)^(1/3)*(20*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Gamma[-7/3]* 
Hypergeometric2F1[-7/3, 1/3, 10/3, -((b*x^3)/a)] - 3*b*x^3*(11*c^2 + 16*c* 
d*x^3 + 5*d^2*x^6)*Gamma[-4/3]*Hypergeometric2F1[-4/3, 4/3, 13/3, -((b*x^3 
)/a)] - 9*b*x^3*(c + d*x^3)^2*Gamma[-4/3]*HypergeometricPFQ[{-4/3, 4/3, 2} 
, {1, 13/3}, -((b*x^3)/a)]))/(280*(1 + (b*x^3)/a)^(1/3)*Gamma[-7/3])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {933, 913, 779, 778}

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.

\(\displaystyle \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx\)

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int \left (b x^3+a\right )^{7/3} \left (d (17 b c-4 a d) x^3+c (14 b c-a d)\right )dx}{14 b}+\frac {d x \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}{14 b}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {\frac {2 \left (2 a^2 d^2-14 a b c d+77 b^2 c^2\right ) \int \left (b x^3+a\right )^{7/3}dx}{11 b}+\frac {d x \left (a+b x^3\right )^{10/3} (17 b c-4 a d)}{11 b}}{14 b}+\frac {d x \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}{14 b}\)

\(\Big \downarrow \) 779

\(\displaystyle \frac {\frac {2 a^2 \sqrt [3]{a+b x^3} \left (2 a^2 d^2-14 a b c d+77 b^2 c^2\right ) \int \left (\frac {b x^3}{a}+1\right )^{7/3}dx}{11 b \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{10/3} (17 b c-4 a d)}{11 b}}{14 b}+\frac {d x \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}{14 b}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {\frac {2 a^2 x \sqrt [3]{a+b x^3} \left (2 a^2 d^2-14 a b c d+77 b^2 c^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{11 b \sqrt [3]{\frac {b x^3}{a}+1}}+\frac {d x \left (a+b x^3\right )^{10/3} (17 b c-4 a d)}{11 b}}{14 b}+\frac {d x \left (a+b x^3\right )^{10/3} \left (c+d x^3\right )}{14 b}\)

Input:

Int[(a + b*x^3)^(7/3)*(c + d*x^3)^2,x]
 

Output:

(d*x*(a + b*x^3)^(10/3)*(c + d*x^3))/(14*b) + ((d*(17*b*c - 4*a*d)*x*(a + 
b*x^3)^(10/3))/(11*b) + (2*a^2*(77*b^2*c^2 - 14*a*b*c*d + 2*a^2*d^2)*x*(a 
+ b*x^3)^(1/3)*Hypergeometric2F1[-7/3, 1/3, 4/3, -((b*x^3)/a)])/(11*b*(1 + 
 (b*x^3)/a)^(1/3)))/(14*b)
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 913

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

rule 933

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), 
x] + Simp[1/(b*(n*(p + q) + 1))   Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim 
p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q 
- 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d 
, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[ 
a, b, c, d, n, p, q, x]
 

Maple [F]

\[\int \left (b \,x^{3}+a \right )^{\frac {7}{3}} \left (d \,x^{3}+c \right )^{2}d x\]

Input:

int((b*x^3+a)^(7/3)*(d*x^3+c)^2,x)
 

Output:

int((b*x^3+a)^(7/3)*(d*x^3+c)^2,x)
 

Fricas [F]

\[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \] Input:

integrate((b*x^3+a)^(7/3)*(d*x^3+c)^2,x, algorithm="fricas")
 

Output:

integral((b^2*d^2*x^12 + 2*(b^2*c*d + a*b*d^2)*x^9 + (b^2*c^2 + 4*a*b*c*d 
+ a^2*d^2)*x^6 + a^2*c^2 + 2*(a*b*c^2 + a^2*c*d)*x^3)*(b*x^3 + a)^(1/3), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.01 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.19 \[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((b*x**3+a)**(7/3)*(d*x**3+c)**2,x)
 

Output:

a**(7/3)*c**2*x*gamma(1/3)*hyper((-1/3, 1/3), (4/3,), b*x**3*exp_polar(I*p 
i)/a)/(3*gamma(4/3)) + 2*a**(7/3)*c*d*x**4*gamma(4/3)*hyper((-1/3, 4/3), ( 
7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(7/3)*d**2*x**7*gamma 
(7/3)*hyper((-1/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3) 
) + 2*a**(4/3)*b*c**2*x**4*gamma(4/3)*hyper((-1/3, 4/3), (7/3,), b*x**3*ex 
p_polar(I*pi)/a)/(3*gamma(7/3)) + 4*a**(4/3)*b*c*d*x**7*gamma(7/3)*hyper(( 
-1/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + 2*a**(4/3 
)*b*d**2*x**10*gamma(10/3)*hyper((-1/3, 10/3), (13/3,), b*x**3*exp_polar(I 
*pi)/a)/(3*gamma(13/3)) + a**(1/3)*b**2*c**2*x**7*gamma(7/3)*hyper((-1/3, 
7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + 2*a**(1/3)*b**2 
*c*d*x**10*gamma(10/3)*hyper((-1/3, 10/3), (13/3,), b*x**3*exp_polar(I*pi) 
/a)/(3*gamma(13/3)) + a**(1/3)*b**2*d**2*x**13*gamma(13/3)*hyper((-1/3, 13 
/3), (16/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(16/3))
 

Maxima [F]

\[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \] Input:

integrate((b*x^3+a)^(7/3)*(d*x^3+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(7/3)*(d*x^3 + c)^2, x)
 

Giac [F]

\[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \] Input:

integrate((b*x^3+a)^(7/3)*(d*x^3+c)^2,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(7/3)*(d*x^3 + c)^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\int {\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^2 \,d x \] Input:

int((a + b*x^3)^(7/3)*(c + d*x^3)^2,x)
 

Output:

int((a + b*x^3)^(7/3)*(c + d*x^3)^2, x)
 

Reduce [F]

\[ \int \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 \, dx=\frac {-28 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} d^{2} x +196 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b c d x +14 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b \,d^{2} x^{4}+2002 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} c^{2} x +1442 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} c d \,x^{4}+430 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} d^{2} x^{7}+1309 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{3} c^{2} x^{4}+1610 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{3} c d \,x^{7}+580 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{3} d^{2} x^{10}+385 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{4} c^{2} x^{7}+560 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{4} c d \,x^{10}+220 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{4} d^{2} x^{13}+28 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) a^{5} d^{2}-196 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) a^{4} b c d +1078 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x \right ) a^{3} b^{2} c^{2}}{3080 b^{2}} \] Input:

int((b*x^3+a)^(7/3)*(d*x^3+c)^2,x)
 

Output:

( - 28*(a + b*x**3)**(1/3)*a**4*d**2*x + 196*(a + b*x**3)**(1/3)*a**3*b*c* 
d*x + 14*(a + b*x**3)**(1/3)*a**3*b*d**2*x**4 + 2002*(a + b*x**3)**(1/3)*a 
**2*b**2*c**2*x + 1442*(a + b*x**3)**(1/3)*a**2*b**2*c*d*x**4 + 430*(a + b 
*x**3)**(1/3)*a**2*b**2*d**2*x**7 + 1309*(a + b*x**3)**(1/3)*a*b**3*c**2*x 
**4 + 1610*(a + b*x**3)**(1/3)*a*b**3*c*d*x**7 + 580*(a + b*x**3)**(1/3)*a 
*b**3*d**2*x**10 + 385*(a + b*x**3)**(1/3)*b**4*c**2*x**7 + 560*(a + b*x** 
3)**(1/3)*b**4*c*d*x**10 + 220*(a + b*x**3)**(1/3)*b**4*d**2*x**13 + 28*in 
t((a + b*x**3)**(1/3)/(a + b*x**3),x)*a**5*d**2 - 196*int((a + b*x**3)**(1 
/3)/(a + b*x**3),x)*a**4*b*c*d + 1078*int((a + b*x**3)**(1/3)/(a + b*x**3) 
,x)*a**3*b**2*c**2)/(3080*b**2)