\(\int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx\) [1674]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\frac {3 (a+b x)^{4/3} \sqrt {c+d x} \sqrt [4]{e+f x} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{4},\frac {7}{3},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 b \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \] Output:

3/4*(b*x+a)^(4/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4)*AppellF1(4/3,-1/2,-1/4,7/3,- 
d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/b/(b*(d*x+c)/(-a*d+b*c))^(1/2) 
/(b*(f*x+e)/(-a*f+b*e))^(1/4)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(318\) vs. \(2(125)=250\).

Time = 21.51 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.54 \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\frac {12 \sqrt {c+d x} \left (11 d^2 (a+b x) (e+f x) (4 a d f+b (3 d e+6 c f+13 d f x))-6 \left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (\frac {d (e+f x)}{f (c+d x)}\right )^{3/4} \left (11 \left (6 a^2 d^2 f^2-4 a b d f (d e+2 c f)+b^2 \left (5 d^2 e^2-6 c d e f+7 c^2 f^2\right )\right ) (c+d x) \operatorname {AppellF1}\left (-\frac {1}{12},\frac {2}{3},\frac {3}{4},\frac {11}{12},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )+(b c-a d) (d e-c f) (3 b d e-7 b c f+4 a d f) \operatorname {AppellF1}\left (\frac {11}{12},\frac {2}{3},\frac {3}{4},\frac {23}{12},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )\right )\right )}{3575 b d^3 f (a+b x)^{2/3} (e+f x)^{3/4}} \] Input:

Integrate[(a + b*x)^(1/3)*Sqrt[c + d*x]*(e + f*x)^(1/4),x]
 

Output:

(12*Sqrt[c + d*x]*(11*d^2*(a + b*x)*(e + f*x)*(4*a*d*f + b*(3*d*e + 6*c*f 
+ 13*d*f*x)) - 6*((d*(a + b*x))/(b*(c + d*x)))^(2/3)*((d*(e + f*x))/(f*(c 
+ d*x)))^(3/4)*(11*(6*a^2*d^2*f^2 - 4*a*b*d*f*(d*e + 2*c*f) + b^2*(5*d^2*e 
^2 - 6*c*d*e*f + 7*c^2*f^2))*(c + d*x)*AppellF1[-1/12, 2/3, 3/4, 11/12, (b 
*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + (b*c - a*d)*(d*e 
- c*f)*(3*b*d*e - 7*b*c*f + 4*a*d*f)*AppellF1[11/12, 2/3, 3/4, 23/12, (b*c 
 - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))])))/(3575*b*d^3*f*(a + 
 b*x)^(2/3)*(e + f*x)^(3/4))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {157, 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.

\(\displaystyle \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx\)

\(\Big \downarrow \) 157

\(\displaystyle \frac {\sqrt {c+d x} \int \sqrt [3]{a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt [4]{e+f x}dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {\sqrt {c+d x} \sqrt [4]{e+f x} \int \sqrt [3]{a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt [4]{\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}dx}{\sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {3 (a+b x)^{4/3} \sqrt {c+d x} \sqrt [4]{e+f x} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{4},\frac {7}{3},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 b \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}}\)

Input:

Int[(a + b*x)^(1/3)*Sqrt[c + d*x]*(e + f*x)^(1/4),x]
 

Output:

(3*(a + b*x)^(4/3)*Sqrt[c + d*x]*(e + f*x)^(1/4)*AppellF1[4/3, -1/2, -1/4, 
 7/3, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(4*b*Sq 
rt[(b*(c + d*x))/(b*c - a*d)]*((b*(e + f*x))/(b*e - a*f))^(1/4))
 

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 157

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n 
]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c 
 - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + 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] &&  !SimplerQ[c + d*x, a + b*x] &&  !Si 
mplerQ[e + f*x, a + b*x]
 

Maple [F]

\[\int \left (b x +a \right )^{\frac {1}{3}} \sqrt {x d +c}\, \left (f x +e \right )^{\frac {1}{4}}d x\]

Input:

int((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x)
 

Output:

int((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\int \sqrt [3]{a + b x} \sqrt {c + d x} \sqrt [4]{e + f x}\, dx \] Input:

integrate((b*x+a)**(1/3)*(d*x+c)**(1/2)*(f*x+e)**(1/4),x)
 

Output:

Integral((a + b*x)**(1/3)*sqrt(c + d*x)*(e + f*x)**(1/4), x)
 

Maxima [F]

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

integrate((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/3)*sqrt(d*x + c)*(f*x + e)^(1/4), x)
 

Giac [F]

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

integrate((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/3)*sqrt(d*x + c)*(f*x + e)^(1/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\int {\left (e+f\,x\right )}^{1/4}\,{\left (a+b\,x\right )}^{1/3}\,\sqrt {c+d\,x} \,d x \] Input:

int((e + f*x)^(1/4)*(a + b*x)^(1/3)*(c + d*x)^(1/2),x)
 

Output:

int((e + f*x)^(1/4)*(a + b*x)^(1/3)*(c + d*x)^(1/2), x)
 

Reduce [F]

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

int((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x)
 

Output:

int((b*x+a)^(1/3)*(d*x+c)^(1/2)*(f*x+e)^(1/4),x)