\(\int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx\) [76]

Optimal result

Integrand size = 29, antiderivative size = 85 \[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=-\frac {2 \sqrt {2} \sqrt {1-d x} \sqrt [3]{e+f x} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-d x),\frac {f (1-d x)}{d e+f}\right )}{d \sqrt [3]{\frac {d (e+f x)}{d e+f}}} \] Output:

-2*2^(1/2)*(-d*x+1)^(1/2)*(f*x+e)^(1/3)*AppellF1(1/2,-1/3,-1/2,3/2,f*(-d*x 
+1)/(d*e+f),-1/2*d*x+1/2)/d/(d*(f*x+e)/(d*e+f))^(1/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(306\) vs. \(2(85)=170\).

Time = 21.93 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.60 \[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\frac {3 \sqrt [3]{e+f x} \sqrt {1-d^2 x^2} \left (-4-\frac {\sqrt {\frac {\sqrt {d^4 f^2}-d^3 f x}{d^3 e+\sqrt {d^4 f^2}}} \sqrt {\frac {\sqrt {d^4 f^2}+d^3 f x}{-d^3 e+\sqrt {d^4 f^2}}} \left (\left (-4 d^2 e^2+4 f^2\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {d^3 (e+f x)}{d^3 e-\sqrt {d^4 f^2}},\frac {d^3 (e+f x)}{d^3 e+\sqrt {d^4 f^2}}\right )+d (d e+4 f) (e+f x) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {d^3 (e+f x)}{d^3 e-\sqrt {d^4 f^2}},\frac {d^3 (e+f x)}{d^3 e+\sqrt {d^4 f^2}}\right )\right )}{f^2 \left (-1+d^2 x^2\right )}\right )}{16 d} \] Input:

Integrate[((1 + d*x)*(e + f*x)^(1/3))/Sqrt[1 - d^2*x^2],x]
 

Output:

(3*(e + f*x)^(1/3)*Sqrt[1 - d^2*x^2]*(-4 - (Sqrt[(Sqrt[d^4*f^2] - d^3*f*x) 
/(d^3*e + Sqrt[d^4*f^2])]*Sqrt[(Sqrt[d^4*f^2] + d^3*f*x)/(-(d^3*e) + Sqrt[ 
d^4*f^2])]*((-4*d^2*e^2 + 4*f^2)*AppellF1[1/3, 1/2, 1/2, 4/3, (d^3*(e + f* 
x))/(d^3*e - Sqrt[d^4*f^2]), (d^3*(e + f*x))/(d^3*e + Sqrt[d^4*f^2])] + d* 
(d*e + 4*f)*(e + f*x)*AppellF1[4/3, 1/2, 1/2, 7/3, (d^3*(e + f*x))/(d^3*e 
- Sqrt[d^4*f^2]), (d^3*(e + f*x))/(d^3*e + Sqrt[d^4*f^2])]))/(f^2*(-1 + d^ 
2*x^2))))/(16*d)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {717, 156, 155}

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.

\(\displaystyle \int \frac {(d x+1) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx\)

\(\Big \downarrow \) 717

\(\displaystyle \int \frac {\sqrt {d x+1} \sqrt [3]{e+f x}}{\sqrt {1-d x}}dx\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {\sqrt [3]{e+f x} \int \frac {\sqrt {d x+1} \sqrt [3]{\frac {d e}{d e+f}+\frac {d f x}{d e+f}}}{\sqrt {1-d x}}dx}{\sqrt [3]{\frac {d (e+f x)}{d e+f}}}\)

\(\Big \downarrow \) 155

\(\displaystyle -\frac {2 \sqrt {2} \sqrt {1-d x} \sqrt [3]{e+f x} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-d x),\frac {f (1-d x)}{d e+f}\right )}{d \sqrt [3]{\frac {d (e+f x)}{d e+f}}}\)

Input:

Int[((1 + d*x)*(e + f*x)^(1/3))/Sqrt[1 - d^2*x^2],x]
 

Output:

(-2*Sqrt[2]*Sqrt[1 - d*x]*(e + f*x)^(1/3)*AppellF1[1/2, -1/2, -1/3, 3/2, ( 
1 - d*x)/2, (f*(1 - d*x))/(d*e + f)])/(d*((d*(e + f*x))/(d*e + f))^(1/3))
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 717

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, 
 x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a 
, 0] && GtQ[d, 0]
 

Maple [F]

\[\int \frac {\left (d x +1\right ) \left (f x +e \right )^{\frac {1}{3}}}{\sqrt {-d^{2} x^{2}+1}}d x\]

Input:

int((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x)
 

Output:

int((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{\frac {1}{3}}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:

integrate((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-d^2*x^2 + 1)*(f*x + e)^(1/3)/(d*x - 1), x)
 

Sympy [F]

\[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {\sqrt [3]{e + f x} \left (d x + 1\right )}{\sqrt {- \left (d x - 1\right ) \left (d x + 1\right )}}\, dx \] Input:

integrate((d*x+1)*(f*x+e)**(1/3)/(-d**2*x**2+1)**(1/2),x)
 

Output:

Integral((e + f*x)**(1/3)*(d*x + 1)/sqrt(-(d*x - 1)*(d*x + 1)), x)
 

Maxima [F]

\[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{\frac {1}{3}}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:

integrate((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate((d*x + 1)*(f*x + e)^(1/3)/sqrt(-d^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{\frac {1}{3}}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:

integrate((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate((d*x + 1)*(f*x + e)^(1/3)/sqrt(-d^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {{\left (e+f\,x\right )}^{1/3}\,\left (d\,x+1\right )}{\sqrt {1-d^2\,x^2}} \,d x \] Input:

int(((e + f*x)^(1/3)*(d*x + 1))/(1 - d^2*x^2)^(1/2),x)
 

Output:

int(((e + f*x)^(1/3)*(d*x + 1))/(1 - d^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(1+d x) \sqrt [3]{e+f x}}{\sqrt {1-d^2 x^2}} \, dx=\frac {-3 \left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}\, d e -3 \left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}\, f +\left (\int \frac {\left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}\, x^{2}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-f x -e}d x \right ) d^{3} e f +4 \left (\int \frac {\left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}\, x^{2}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-f x -e}d x \right ) d^{2} f^{2}-3 \left (\int \frac {\left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-f x -e}d x \right ) d^{2} e^{2}-\left (\int \frac {\left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-f x -e}d x \right ) d e f -\left (\int \frac {\left (f x +e \right )^{\frac {1}{3}} \sqrt {-d^{2} x^{2}+1}}{d^{2} f \,x^{3}+d^{2} e \,x^{2}-f x -e}d x \right ) f^{2}}{3 d^{2} e} \] Input:

int((d*x+1)*(f*x+e)^(1/3)/(-d^2*x^2+1)^(1/2),x)
 

Output:

( - 3*(e + f*x)**(1/3)*sqrt( - d**2*x**2 + 1)*d*e - 3*(e + f*x)**(1/3)*sqr 
t( - d**2*x**2 + 1)*f + int(((e + f*x)**(1/3)*sqrt( - d**2*x**2 + 1)*x**2) 
/(d**2*e*x**2 + d**2*f*x**3 - e - f*x),x)*d**3*e*f + 4*int(((e + f*x)**(1/ 
3)*sqrt( - d**2*x**2 + 1)*x**2)/(d**2*e*x**2 + d**2*f*x**3 - e - f*x),x)*d 
**2*f**2 - 3*int(((e + f*x)**(1/3)*sqrt( - d**2*x**2 + 1))/(d**2*e*x**2 + 
d**2*f*x**3 - e - f*x),x)*d**2*e**2 - int(((e + f*x)**(1/3)*sqrt( - d**2*x 
**2 + 1))/(d**2*e*x**2 + d**2*f*x**3 - e - f*x),x)*d*e*f - int(((e + f*x)* 
*(1/3)*sqrt( - d**2*x**2 + 1))/(d**2*e*x**2 + d**2*f*x**3 - e - f*x),x)*f* 
*2)/(3*d**2*e)