\(\int \sqrt {a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx\) [3178]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \sqrt {a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},-\frac {1}{4},\frac {5}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \]

[Out]

2/3*(b*x+a)^(3/2)*(d*x+c)^(1/3)*(f*x+e)^(1/4)*AppellF1(3/2,-1/3,-1/4,5/2,-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))^(1/3)/(b*(f*x+e)/(-a*f+b*e))^(1/4)

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} \[ \int \sqrt {a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},-\frac {1}{4},\frac {5}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \]

[In]

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

[Out]

(2*(a + b*x)^(3/2)*(c + d*x)^(1/3)*(e + f*x)^(1/4)*AppellF1[3/2, -1/3, -1/4, 5/2, -((d*(a + b*x))/(b*c - a*d))
, -((f*(a + b*x))/(b*e - a*f))])/(3*b*((b*(c + d*x))/(b*c - a*d))^(1/3)*((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 [3]{c+d x} \int \sqrt {a+b x} \sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt [4]{e+f x} \, dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \\ & = \frac {\left (\sqrt [3]{c+d x} \sqrt [4]{e+f x}\right ) \int \sqrt {a+b x} \sqrt [3]{\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 [3]{\frac {b (c+d x)}{b c-a d}} \sqrt [4]{\frac {b (e+f x)}{b e-a f}}} \\ & = \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac {3}{2};-\frac {1}{3},-\frac {1}{4};\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\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. \(473\) vs. \(2(125)=250\).

Time = 22.60 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.78 \[ \int \sqrt {a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx=\frac {12 \left (253 b^2 d f (a+b x) (c+d x) (e+f x) (6 a d f+b (3 d e+4 c f+13 d f x))-276 \left (21 a^3 d^3 f^3-9 a^2 b d^2 f^2 (3 d e+4 c f)+a b^2 d f \left (20 d^2 e^2+14 c d e f+29 c^2 f^2\right )-b^3 \left (5 d^3 e^3+5 c d^2 e^2 f+2 c^2 d e f^2+9 c^3 f^3\right )\right ) (a+b x) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{3/4} \operatorname {AppellF1}\left (\frac {11}{12},\frac {2}{3},\frac {3}{4},\frac {23}{12},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )-66 \left (7 a^2 d^2 f^2-2 a b d f (3 d e+4 c f)+b^2 \left (5 d^2 e^2-4 c d e f+6 c^2 f^2\right )\right ) \left (23 b^2 (c+d x) (e+f x)-6 (b c-a d) (b e-a f) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{3/4} \operatorname {AppellF1}\left (\frac {23}{12},\frac {2}{3},\frac {3}{4},\frac {35}{12},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )\right )\right )}{82225 b^3 d^2 f^2 \sqrt {a+b x} (c+d x)^{2/3} (e+f x)^{3/4}} \]

[In]

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

[Out]

(12*(253*b^2*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(6*a*d*f + b*(3*d*e + 4*c*f + 13*d*f*x)) - 276*(21*a^3*d^3*f^3
- 9*a^2*b*d^2*f^2*(3*d*e + 4*c*f) + a*b^2*d*f*(20*d^2*e^2 + 14*c*d*e*f + 29*c^2*f^2) - b^3*(5*d^3*e^3 + 5*c*d^
2*e^2*f + 2*c^2*d*e*f^2 + 9*c^3*f^3))*(a + b*x)*((b*(c + d*x))/(d*(a + b*x)))^(2/3)*((b*(e + f*x))/(f*(a + b*x
)))^(3/4)*AppellF1[11/12, 2/3, 3/4, 23/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] - 66*(7
*a^2*d^2*f^2 - 2*a*b*d*f*(3*d*e + 4*c*f) + b^2*(5*d^2*e^2 - 4*c*d*e*f + 6*c^2*f^2))*(23*b^2*(c + d*x)*(e + f*x
) - 6*(b*c - a*d)*(b*e - a*f)*((b*(c + d*x))/(d*(a + b*x)))^(2/3)*((b*(e + f*x))/(f*(a + b*x)))^(3/4)*AppellF1
[23/12, 2/3, 3/4, 35/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))])))/(82225*b^3*d^2*f^2*Sqr
t[a + b*x]*(c + d*x)^(2/3)*(e + f*x)^(3/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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