3.3179 \(\int \sqrt [3]{a+b x} \sqrt{c+d x} \sqrt [4]{e+f x} \, dx\)

Optimal. Leaf size=125 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

[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))

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Rubi [A]  time = 0.0727979, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {140, 139, 138} \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully verified.

[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 140

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]

Rule 139

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 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), 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])

Rubi steps

\begin{align*} \int \sqrt [3]{a+b x} \sqrt{c+d x} \sqrt [4]{e+f x} \, dx &=\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]  time = 2.43141, size = 318, normalized size = 2.54 \[ \frac{12 \sqrt{c+d x} \left (11 d^2 (a+b x) (e+f x) (4 a d f+b (6 c f+3 d e+13 d f x))-6 \left (\frac{d (a+b x)}{b (c+d x)}\right )^{2/3} \left (\frac{d (e+f x)}{f (c+d x)}\right )^{3/4} \left (11 (c+d x) \left (6 a^2 d^2 f^2-4 a b d f (2 c f+d e)+b^2 \left (7 c^2 f^2-6 c d e f+5 d^2 e^2\right )\right ) F_1\left (-\frac{1}{12};\frac{2}{3},\frac{3}{4};\frac{11}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+(b c-a d) (d e-c f) (4 a d f-7 b c f+3 b d e) F_1\left (\frac{11}{12};\frac{2}{3},\frac{3}{4};\frac{23}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )\right )}{3575 b d^3 f (a+b x)^{2/3} (e+f x)^{3/4}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{bx+a}\sqrt{dx+c}\sqrt [4]{fx+e}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{\frac{1}{3}} \sqrt{d x + c}{\left (f x + e\right )}^{\frac{1}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out