3.3151 \(\int \sqrt [3]{a+b x} \sqrt{c+d x} \sqrt [4]{e+f x} \, dx\)

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 = 0.500113, 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 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{4}{3};-\frac{1}{2},-\frac{1}{4};\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 b \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

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))

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Rubi in Sympy [A]  time = 63.5351, size = 104, normalized size = 0.83 \[ \frac{3 \left (a + b x\right )^{\frac{4}{3}} \sqrt{c + d x} \sqrt [4]{e + f x} \operatorname{appellf_{1}}{\left (\frac{4}{3},- \frac{1}{2},- \frac{1}{4},\frac{7}{3},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{4 b \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt [4]{\frac{b \left (- e - f x\right )}{a f - b e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/3)*(d*x+c)**(1/2)*(f*x+e)**(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)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(4*b*sqrt(b*(-c - d*x)/(a*d -
b*c))*(b*(-e - f*x)/(a*f - b*e))**(1/4))

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Mathematica [B]  time = 9.86585, size = 1078, normalized size = 8.62 \[ \frac{\sqrt{c+d x} \left (\frac{132 (a+b x) (e+f x) (4 a d f+b (3 d e+6 c f+13 d f x))}{b d f}-\frac{72 (c+d x) \left (-23 (b c-a d) (d e-c f) (3 b d e-7 b c f+4 a d f) 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 ) \left (9 b (d e-c f) F_1\left (\frac{11}{12};\frac{2}{3},\frac{7}{4};\frac{23}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+8 (a d-b c) f F_1\left (\frac{11}{12};\frac{5}{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 )-11 (c+d x) 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 ) \left (23 b f \left (-\left (d^2 (58 c+55 d x) e^2-2 c d f (38 c+33 d x) e+7 c^2 f^2 (12 c+11 d x)\right ) b^2+a d \left (e (3 e+44 f x) d^2+2 c f (15 e+44 f x) d+99 c^2 f^2\right ) b-2 a^2 d^2 f (-2 d e+35 c f+33 d f x)\right ) 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 )+11 \left (\left (5 d^2 e^2-6 c d f e+7 c^2 f^2\right ) b^2-4 a d f (d e+2 c f) b+6 a^2 d^2 f^2\right ) \left (9 b (d e-c f) F_1\left (\frac{23}{12};\frac{2}{3},\frac{7}{4};\frac{35}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+8 (a d-b c) f F_1\left (\frac{23}{12};\frac{5}{3},\frac{3}{4};\frac{35}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )\right )\right )}{d^3 \left (11 b f (c+d x) 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 (9 c f-9 d e) F_1\left (\frac{11}{12};\frac{2}{3},\frac{7}{4};\frac{23}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+8 (b c-a d) f F_1\left (\frac{11}{12};\frac{5}{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 ) \left (23 b f (c+d x) 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 )+b (9 c f-9 d e) F_1\left (\frac{23}{12};\frac{2}{3},\frac{7}{4};\frac{35}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+8 (b c-a d) f F_1\left (\frac{23}{12};\frac{5}{3},\frac{3}{4};\frac{35}{12};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )}\right )}{3575 (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]

(Sqrt[c + d*x]*((132*(a + b*x)*(e + f*x)*(4*a*d*f + b*(3*d*e + 6*c*f + 13*d*f*x)
))/(b*d*f) - (72*(c + d*x)*(-23*(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))]*(9*b*(d*e - c*f)*AppellF1[11/12, 2/3, 7/4, 23/12, (b*c - a*d)/(b
*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 8*(-(b*c) + a*d)*f*AppellF1[11/12,
5/3, 3/4, 23/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]) - 11*
(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))]*(23*b*f*(-2*a^2*d^2*f*(-2*d*e + 35*c*f + 33*d*f*x) - b^2*(7*
c^2*f^2*(12*c + 11*d*x) - 2*c*d*e*f*(38*c + 33*d*x) + d^2*e^2*(58*c + 55*d*x)) +
a*b*d*(99*c^2*f^2 + d^2*e*(3*e + 44*f*x) + 2*c*d*f*(15*e + 44*f*x)))*AppellF1[1
1/12, 2/3, 3/4, 23/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]
+ 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))*(9*b*(d*e - c*f)*AppellF1[23/12, 2/3, 7/4, 35/12, (b*c - a*d)/(b*c + b*
d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 8*(-(b*c) + a*d)*f*AppellF1[23/12, 5/3, 3/
4, 35/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]))))/(d^3*(11*
b*f*(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*(-9*d*e + 9*c*f)*AppellF1[11/12, 2/3, 7/4, 23/12, (b
*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 8*(b*c - a*d)*f*AppellF
1[11/12, 5/3, 3/4, 23/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x)
)])*(23*b*f*(c + d*x)*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))] + b*(-9*d*e + 9*c*f)*AppellF1[23/12, 2/3, 7/4, 3
5/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 8*(b*c - a*d)*f
*AppellF1[23/12, 5/3, 3/4, 35/12, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(
c + d*x))]))))/(3575*(a + b*x)^(2/3)*(e + f*x)^(3/4))

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \[ \int \sqrt [3]{bx+a}\sqrt{dx+c}\sqrt [4]{fx+e}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*sqrt(d*x + c)*(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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(1/3)*sqrt(d*x + c)*(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. \[ \text{Timed out} \]

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/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out