Integrand size = 54, antiderivative size = 144 \[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\frac {\sqrt [3]{d+e x} (c d+b e+3 c e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {(c d+b e+3 c e x)^3}{(2 c d-b e)^3}\right )}{3 c e \sqrt [3]{1+\frac {(c d+b e+3 c e x)^3}{(2 c d-b e)^3}}} \] Output:
1/3*(e*x+d)^(1/3)*(3*c*e*x+b*e+c*d)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c *d*e+c^2*d^2)^(1/3)*hypergeom([-1/3, 1/3],[4/3],-(3*c*e*x+b*e+c*d)^3/(-b*e +2*c*d)^3)/c/e/(1+(3*c*e*x+b*e+c*d)^3/(-b*e+2*c*d)^3)^(1/3)
Leaf count is larger than twice the leaf count of optimal. \(463\) vs. \(2(144)=288\).
Time = 11.77 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.22 \[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\frac {\sqrt [3]{d+e x} \left (b^3 e^3+6 b^2 c e^3 x+12 b c^2 e^3 x^2+c^3 \left (d^3+3 d^2 e x+3 d e^2 x^2+9 e^3 x^3\right )+(2 c d-b e)^3 \sqrt [3]{\frac {-3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}-6 c^2 e^2 x}{6 c^2 d e-3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}}} \left (\frac {3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}+6 c^2 e^2 x}{-6 c^2 d e+3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {12 \sqrt {3} c^2 e \sqrt {-c^2 e^2 (-2 c d+b e)^2} (d+e x)}{\left (-6 c^2 d e+3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}\right ) \left (-3 b c e^2+\sqrt {3} \sqrt {-c^2 e^2 (-2 c d+b e)^2}-6 c^2 e^2 x\right )}\right )\right )}{6 c e \left (b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )\right )^{2/3}} \] Input:
Integrate[(d + e*x)^(1/3)*(c^2*d^2 - b*c*d*e + b^2*e^2 + 3*b*c*e^2*x + 3*c ^2*e^2*x^2)^(1/3),x]
Output:
((d + e*x)^(1/3)*(b^3*e^3 + 6*b^2*c*e^3*x + 12*b*c^2*e^3*x^2 + c^3*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + 9*e^3*x^3) + (2*c*d - b*e)^3*((-3*b*c*e^2 + Sqrt [3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] - 6*c^2*e^2*x)/(6*c^2*d*e - 3*b*c*e^ 2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)]))^(1/3)*((3*b*c*e^2 + Sqrt[3 ]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)/(-6*c^2*d*e + 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)]))^(2/3)*Hypergeometric2F1[1/3 , 2/3, 4/3, (-12*Sqrt[3]*c^2*e*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)]*(d + e*x) )/((-6*c^2*d*e + 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])*(- 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] - 6*c^2*e^2*x))]))/( 6*c*e*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(2/3))
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.58 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.77, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1179, 150}
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.
| \(\displaystyle \int \sqrt [3]{d+e x} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {\sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2} \int \sqrt [3]{d+e x} \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (3 c-\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)}} \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (3 c+\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)}}d(d+e x)}{e \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (3 c-\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)}} \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (\sqrt {3} \sqrt {-c^2}+3 c\right ) (2 c d-b e)}}}\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {3 (d+e x)^{4/3} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{3},-\frac {1}{3},\frac {7}{3},\frac {6 c^2 (d+e x)}{\left (3 c-\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)},\frac {6 c^2 (d+e x)}{\left (3 c+\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)}\right )}{4 e \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (3 c-\sqrt {3} \sqrt {-c^2}\right ) (2 c d-b e)}} \sqrt [3]{1-\frac {6 c^2 (d+e x)}{\left (\sqrt {3} \sqrt {-c^2}+3 c\right ) (2 c d-b e)}}}\) |
Input:
Int[(d + e*x)^(1/3)*(c^2*d^2 - b*c*d*e + b^2*e^2 + 3*b*c*e^2*x + 3*c^2*e^2 *x^2)^(1/3),x]
Output:
(3*(d + e*x)^(4/3)*(c^2*d^2 - b*c*d*e + b^2*e^2 + 3*b*c*e^2*x + 3*c^2*e^2* x^2)^(1/3)*AppellF1[4/3, -1/3, -1/3, 7/3, (6*c^2*(d + e*x))/((3*c - Sqrt[3 ]*Sqrt[-c^2])*(2*c*d - b*e)), (6*c^2*(d + e*x))/((3*c + Sqrt[3]*Sqrt[-c^2] )*(2*c*d - b*e))])/(4*e*(1 - (6*c^2*(d + e*x))/((3*c - Sqrt[3]*Sqrt[-c^2]) *(2*c*d - b*e)))^(1/3)*(1 - (6*c^2*(d + e*x))/((3*c + Sqrt[3]*Sqrt[-c^2])* (2*c*d - b*e)))^(1/3))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) ^p) Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d - e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m , p}, x]
\[\int \left (e x +d \right )^{\frac {1}{3}} \left (3 c^{2} e^{2} x^{2}+3 e^{2} x b c +b^{2} e^{2}-b c d e +c^{2} d^{2}\right )^{\frac {1}{3}}d x\]
Input:
int((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2)^(1/3 ),x)
Output:
int((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2)^(1/3 ),x)
\[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\int { {\left (3 \, c^{2} e^{2} x^{2} + 3 \, b c e^{2} x + c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2 )^(1/3),x, algorithm="fricas")
Output:
integral((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3) *(e*x + d)^(1/3), x)
\[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\int \sqrt [3]{d + e x} \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}}\, dx \] Input:
integrate((e*x+d)**(1/3)*(3*c**2*e**2*x**2+3*b*c*e**2*x+b**2*e**2-b*c*d*e+ c**2*d**2)**(1/3),x)
Output:
Integral((d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2)**(1/3), x)
\[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\int { {\left (3 \, c^{2} e^{2} x^{2} + 3 \, b c e^{2} x + c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2 )^(1/3),x, algorithm="maxima")
Output:
integrate((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3 )*(e*x + d)^(1/3), x)
\[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\int { {\left (3 \, c^{2} e^{2} x^{2} + 3 \, b c e^{2} x + c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}^{\frac {1}{3}} \,d x } \] Input:
integrate((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2 )^(1/3),x, algorithm="giac")
Output:
integrate((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3 )*(e*x + d)^(1/3), x)
Timed out. \[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\int {\left (d+e\,x\right )}^{1/3}\,{\left (b^2\,e^2-b\,c\,d\,e+3\,b\,c\,e^2\,x+c^2\,d^2+3\,c^2\,e^2\,x^2\right )}^{1/3} \,d x \] Input:
int((d + e*x)^(1/3)*(b^2*e^2 + c^2*d^2 + 3*c^2*e^2*x^2 + 3*b*c*e^2*x - b*c *d*e)^(1/3),x)
Output:
int((d + e*x)^(1/3)*(b^2*e^2 + c^2*d^2 + 3*c^2*e^2*x^2 + 3*b*c*e^2*x - b*c *d*e)^(1/3), x)
\[ \int \sqrt [3]{d+e x} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2} \, dx=\text {too large to display} \] Input:
int((e*x+d)^(1/3)*(3*c^2*e^2*x^2+3*b*c*e^2*x+b^2*e^2-b*c*d*e+c^2*d^2)^(1/3 ),x)
Output:
(9*(d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c* *2*e**2*x**2)**(1/3)*b**2*d*e**2 + 3*(d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2)**(1/3)*b**2*e**3*x - 9*(d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e** 2*x**2)**(1/3)*b*c*d**2*e + 6*(d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b* c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2)**(1/3)*b*c*d*e**2*x + 9*(d + e*x) **(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2 )**(1/3)*c**2*d**3 + 3*(d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2* x + c**2*d**2 + 3*c**2*e**2*x**2)**(1/3)*c**2*d**2*e*x + 3*int(((d + e*x)* *(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2) **(1/3)*x**2)/(b**4*d*e**4 + b**4*e**5*x + b**3*c*d**2*e**3 + 4*b**3*c*d*e **4*x + 3*b**3*c*e**5*x**2 + 6*b**2*c**2*d**2*e**3*x + 9*b**2*c**2*d*e**4* x**2 + 3*b**2*c**2*e**5*x**3 + b*c**3*d**4*e + 4*b*c**3*d**3*e**2*x + 9*b* c**3*d**2*e**3*x**2 + 6*b*c**3*d*e**4*x**3 + c**4*d**5 + c**4*d**4*e*x + 3 *c**4*d**3*e**2*x**2 + 3*c**4*d**2*e**3*x**3),x)*b**5*c*e**8 - 12*int(((d + e*x)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e** 2*x**2)**(1/3)*x**2)/(b**4*d*e**4 + b**4*e**5*x + b**3*c*d**2*e**3 + 4*b** 3*c*d*e**4*x + 3*b**3*c*e**5*x**2 + 6*b**2*c**2*d**2*e**3*x + 9*b**2*c**2* d*e**4*x**2 + 3*b**2*c**2*e**5*x**3 + b*c**3*d**4*e + 4*b*c**3*d**3*e**2*x + 9*b*c**3*d**2*e**3*x**2 + 6*b*c**3*d*e**4*x**3 + c**4*d**5 + c**4*d*...