3.3178 \(\int \sqrt{a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{3}{2};-\frac{1}{3},-\frac{1}{4};\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

[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))

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Rubi [A]  time = 0.0730139, 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} \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{3}{2};-\frac{1}{3},-\frac{1}{4};\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully verified.

[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 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{a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx &=\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]  time = 3.60359, size = 473, normalized size = 3.78 \[ \frac{12 \left (-276 (a+b x) \left (\frac{b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac{b (e+f x)}{f (a+b x)}\right )^{3/4} \left (-9 a^2 b d^2 f^2 (4 c f+3 d e)+21 a^3 d^3 f^3+a b^2 d f \left (29 c^2 f^2+14 c d e f+20 d^2 e^2\right )+b^3 \left (-\left (2 c^2 d e f^2+9 c^3 f^3+5 c d^2 e^2 f+5 d^3 e^3\right )\right )\right ) F_1\left (\frac{11}{12};\frac{2}{3},\frac{3}{4};\frac{23}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )-66 \left (7 a^2 d^2 f^2-2 a b d f (4 c f+3 d e)+b^2 \left (6 c^2 f^2-4 c d e f+5 d^2 e^2\right )\right ) \left (23 b^2 (c+d x) (e+f x)-6 (b c-a d) (b e-a f) \left (\frac{b (c+d x)}{d (a+b x)}\right )^{2/3} \left (\frac{b (e+f x)}{f (a+b x)}\right )^{3/4} F_1\left (\frac{23}{12};\frac{2}{3},\frac{3}{4};\frac{35}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )+253 b^2 d f (a+b x) (c+d x) (e+f x) (6 a d f+b (4 c f+3 d e+13 d f x))\right )}{82225 b^3 d^2 f^2 \sqrt{a+b x} (c+d x)^{2/3} (e+f x)^{3/4}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \sqrt{bx+a}\sqrt [3]{dx+c}\sqrt [4]{fx+e}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)*(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}{\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)^(1/2)*(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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{3}}{\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)^(1/2)*(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)