3.3150 \(\int \sqrt{a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x} \, dx\)

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 = 0.487757, 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 \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} \sqrt [4]{e+f x} F_1\left (\frac{3}{2};-\frac{1}{3},-\frac{1}{4};\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \sqrt [4]{\frac{b (e+f x)}{b e-a f}}} \]

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))

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Rubi in Sympy [A]  time = 63.3689, size = 104, normalized size = 0.83 \[ \frac{2 \left (a + b x\right )^{\frac{3}{2}} \sqrt [3]{c + d x} \sqrt [4]{e + f x} \operatorname{appellf_{1}}{\left (\frac{3}{2},- \frac{1}{3},- \frac{1}{4},\frac{5}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{3 b \sqrt [3]{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt [4]{\frac{b \left (- e - f x\right )}{a f - b e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [B]  time = 13.8262, size = 1077, normalized size = 8.62 \[ \left (\frac{12 (3 b d e+4 b c f+6 a d f)}{325 b d f}+\frac{12 x}{25}\right ) \sqrt{a+b x} \sqrt [3]{c+d x} \sqrt [4]{e+f x}-\frac{72 (a+b x)^{3/2} \left (\frac{1058 \left (\left (5 d^3 e^3+5 c d^2 f e^2+2 c^2 d f^2 e+9 c^3 f^3\right ) b^3-a d f \left (20 d^2 e^2+14 c d f e+29 c^2 f^2\right ) b^2+9 a^2 d^2 f^2 (3 d e+4 c f) b-21 a^3 d^3 f^3\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 )}{(a+b x) \left (\frac{9 d (b e-a f) F_1\left (\frac{23}{12};\frac{2}{3},\frac{7}{4};\frac{35}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+8 (b c-a d) f F_1\left (\frac{23}{12};\frac{5}{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 )}{a+b x}-23 d f 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 )\right )}+\frac{11 \left (\left (5 d^2 e^2-4 c d f e+6 c^2 f^2\right ) b^2-2 a d f (3 d e+4 c f) b+7 a^2 d^2 f^2\right ) \left (35 d f \left (\frac{b c \left (\frac{17 b e}{a+b x}+f \left (23-\frac{17 a}{a+b x}\right )\right )}{a+b x}+d \left (f \left (\frac{17 a^2}{(a+b x)^2}-\frac{46 a}{a+b x}+23\right )+\frac{b e \left (23-\frac{17 a}{a+b x}\right )}{a+b x}\right )\right ) 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 )-\frac{23 \left (\frac{b c}{a+b x}+d-\frac{a d}{a+b x}\right ) \left (\frac{b e}{a+b x}+f-\frac{a f}{a+b x}\right ) \left (9 d (b e-a f) F_1\left (\frac{35}{12};\frac{2}{3},\frac{7}{4};\frac{47}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+8 (b c-a d) f F_1\left (\frac{35}{12};\frac{5}{3},\frac{3}{4};\frac{47}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )}{a+b x}\right )}{d f \left (35 d f 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 )+\frac{(9 a d f-9 b d e) F_1\left (\frac{35}{12};\frac{2}{3},\frac{7}{4};\frac{47}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+8 (a d-b c) f F_1\left (\frac{35}{12};\frac{5}{3},\frac{3}{4};\frac{47}{12};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{a+b x}\right )}\right )}{82225 b^3 d f \left (c+\frac{(a+b x) \left (d-\frac{a d}{a+b x}\right )}{b}\right )^{2/3} \left (e+\frac{(a+b x) \left (f-\frac{a f}{a+b x}\right )}{b}\right )^{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*(3*b*d*e + 4*b*c*f + 6*a*d*f))/(325*b*d*f) + (12*x)/25)*Sqrt[a + b*x]*(c +
d*x)^(1/3)*(e + f*x)^(1/4) - (72*(a + b*x)^(3/2)*((1058*(-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))*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))])/((a +
b*x)*(-23*d*f*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))] + (9*d*(b*e - a*f)*AppellF1[23/12, 2/3, 7/4, 35/12,
(-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] + 8*(b*c - a*d)*f*Ap
pellF1[23/12, 5/3, 3/4, 35/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(
a + b*x))])/(a + b*x))) + (11*(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))*(35*d*f*((b*c*((17*b*e)/(a + b*x) + f*(23 -
(17*a)/(a + b*x))))/(a + b*x) + d*(f*(23 + (17*a^2)/(a + b*x)^2 - (46*a)/(a + b*
x)) + (b*e*(23 - (17*a)/(a + b*x)))/(a + b*x)))*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))] - (23*(d + (b*c)/(a
+ b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x))*(9*d*(b*e - a
*f)*AppellF1[35/12, 2/3, 7/4, 47/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f
)/(f*(a + b*x))] + 8*(b*c - a*d)*f*AppellF1[35/12, 5/3, 3/4, 47/12, (-(b*c) + a*
d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))]))/(a + b*x)))/(d*f*(35*d*f*Appel
lF1[23/12, 2/3, 3/4, 35/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a +
b*x))] + ((-9*b*d*e + 9*a*d*f)*AppellF1[35/12, 2/3, 7/4, 47/12, (-(b*c) + a*d)/
(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] + 8*(-(b*c) + a*d)*f*AppellF1[35/12
, 5/3, 3/4, 47/12, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))])/
(a + b*x)))))/(82225*b^3*d*f*(c + ((a + b*x)*(d - (a*d)/(a + b*x)))/b)^(2/3)*(e
+ ((a + b*x)*(f - (a*f)/(a + b*x)))/b)^(3/4))

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Maple [F]  time = 0.145, size = 0, normalized size = 0. \[ \int \sqrt{bx+a}\sqrt [3]{dx+c}\sqrt [4]{fx+e}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(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)