3.3149 \(\int \frac{\sqrt [3]{a+b x} \sqrt{c+d x}}{e+f x} \, dx\)

Optimal. Leaf size=100 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} F_1\left (\frac{4}{3};-\frac{1}{2},1;\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

(3*(a + b*x)^(4/3)*Sqrt[c + d*x]*AppellF1[4/3, -1/2, 1, 7/3, -((d*(a + b*x))/(b*
c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(4*(b*e - a*f)*Sqrt[(b*(c + d*x))/(b*c
- a*d)])

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Rubi [A]  time = 0.240913, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} F_1\left (\frac{4}{3};-\frac{1}{2},1;\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a + b*x)^(4/3)*Sqrt[c + d*x]*AppellF1[4/3, -1/2, 1, 7/3, -((d*(a + b*x))/(b*
c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(4*(b*e - a*f)*Sqrt[(b*(c + d*x))/(b*c
- a*d)])

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Rubi in Sympy [A]  time = 21.4789, size = 80, normalized size = 0.8 \[ - \frac{3 \left (a + b x\right )^{\frac{4}{3}} \sqrt{c + d x} \operatorname{appellf_{1}}{\left (\frac{4}{3},- \frac{1}{2},1,\frac{7}{3},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{4 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \left (a f - b e\right )} \]

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),x)

[Out]

-3*(a + b*x)**(4/3)*sqrt(c + d*x)*appellf1(4/3, -1/2, 1, 7/3, d*(a + b*x)/(a*d -
b*c), f*(a + b*x)/(a*f - b*e))/(4*sqrt(b*(-c - d*x)/(a*d - b*c))*(a*f - b*e))

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Mathematica [B]  time = 3.03309, size = 895, normalized size = 8.95 \[ \frac{6 \sqrt{c+d x} \left (\frac{7 (a+b x)}{f}+\frac{b (c+d x) \left (-78 (b c-a d) (d e-c f) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right ) \left (b (3 c f-3 d e) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 (b c-a d) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )-7 (c+d x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right ) \left (13 b f \left (a d (-3 d e+17 c f+14 d f x)+b \left (18 f c^2-32 d e c+21 d f x c-35 d^2 e x\right )\right ) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+14 (5 b d e-3 b c f-2 a d f) \left (3 b (d e-c f) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 (a d-b c) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )\right )\right )}{d^2 (e+f x) \left (7 b f (c+d x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (6 c f-6 d e) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+4 (b c-a d) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right ) \left (13 b f (c+d x) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (6 c f-6 d e) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+4 (b c-a d) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )}\right )}{35 (a+b x)^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(6*Sqrt[c + d*x]*((7*(a + b*x))/f + (b*(c + d*x)*(-78*(b*c - a*d)*(d*e - c*f)*Ap
pellF1[7/6, 2/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x)
)]*(b*(-3*d*e + 3*c*f)*AppellF1[7/6, 2/3, 2, 13/6, (b*c - a*d)/(b*c + b*d*x), (-
(d*e) + c*f)/(f*(c + d*x))] + 2*(b*c - a*d)*f*AppellF1[7/6, 5/3, 1, 13/6, (b*c -
a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]) - 7*(c + d*x)*AppellF1[1/6,
2/3, 1, 7/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]*(13*b*f*(a
*d*(-3*d*e + 17*c*f + 14*d*f*x) + b*(-32*c*d*e + 18*c^2*f - 35*d^2*e*x + 21*c*d*
f*x))*AppellF1[7/6, 2/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(
c + d*x))] + 14*(5*b*d*e - 3*b*c*f - 2*a*d*f)*(3*b*(d*e - c*f)*AppellF1[13/6, 2/
3, 2, 19/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 2*(-(b*c)
+ a*d)*f*AppellF1[13/6, 5/3, 1, 19/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)
/(f*(c + d*x))]))))/(d^2*(e + f*x)*(7*b*f*(c + d*x)*AppellF1[1/6, 2/3, 1, 7/6, (
b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + b*(-6*d*e + 6*c*f)*App
ellF1[7/6, 2/3, 2, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))
] + 4*(b*c - a*d)*f*AppellF1[7/6, 5/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*
e) + c*f)/(f*(c + d*x))])*(13*b*f*(c + d*x)*AppellF1[7/6, 2/3, 1, 13/6, (b*c - a
*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + b*(-6*d*e + 6*c*f)*AppellF1[1
3/6, 2/3, 2, 19/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 4*
(b*c - a*d)*f*AppellF1[13/6, 5/3, 1, 19/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) +
c*f)/(f*(c + d*x))]))))/(35*(a + b*x)^(2/3))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e}\sqrt [3]{bx+a}\sqrt{dx+c}}\, dx \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{a + b x} \sqrt{c + d x}}{e + f x}\, dx \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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