\(\int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx\) [3179]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (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 = {145, 144, 143} \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\frac {3 (a+b x)^{4/3} \sqrt {c+d x} \sqrt [4]{e+f x} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{4},\frac {7}{3},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 b \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \]

[In]

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

[Out]

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

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

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

Rule 145

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^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[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rubi steps \begin{align*} \text {integral}& = \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}}} \\ & = \frac {\left (\sqrt {c+d x} \sqrt [4]{e+f x}\right ) \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}}} \\ & = \frac {3 (a+b x)^{4/3} \sqrt {c+d x} \sqrt [4]{e+f x} F_1\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}}} \\ \end{align*}

Mathematica [B] (verified)

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

Time = 22.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.54 \[ \int \sqrt [3]{a+b x} \sqrt {c+d x} \sqrt [4]{e+f x} \, dx=\frac {12 \sqrt {c+d x} \left (11 d^2 (a+b x) (e+f x) (4 a d f+b (3 d e+6 c f+13 d f x))-6 \left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (\frac {d (e+f x)}{f (c+d x)}\right )^{3/4} \left (11 \left (6 a^2 d^2 f^2-4 a b d f (d e+2 c f)+b^2 \left (5 d^2 e^2-6 c d e f+7 c^2 f^2\right )\right ) (c+d x) \operatorname {AppellF1}\left (-\frac {1}{12},\frac {2}{3},\frac {3}{4},\frac {11}{12},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )+(b c-a d) (d e-c f) (3 b d e-7 b c f+4 a d f) \operatorname {AppellF1}\left (\frac {11}{12},\frac {2}{3},\frac {3}{4},\frac {23}{12},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )\right )\right )}{3575 b d^3 f (a+b x)^{2/3} (e+f x)^{3/4}} \]

[In]

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

[Out]

(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))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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