3.3148 \(\int \frac{\sqrt{a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx\)

Optimal. Leaf size=100 \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac{3}{2};-\frac{1}{3},1;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

(2*(a + b*x)^(3/2)*(c + d*x)^(1/3)*AppellF1[3/2, -1/3, 1, 5/2, -((d*(a + b*x))/(
b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(3*(b*e - a*f)*((b*(c + d*x))/(b*c -
a*d))^(1/3))

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Rubi [A]  time = 0.248875, 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 \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac{3}{2};-\frac{1}{3},1;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(3/2)*(c + d*x)^(1/3)*AppellF1[3/2, -1/3, 1, 5/2, -((d*(a + b*x))/(
b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(3*(b*e - a*f)*((b*(c + d*x))/(b*c -
a*d))^(1/3))

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Rubi in Sympy [A]  time = 21.4712, size = 80, normalized size = 0.8 \[ - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \sqrt [3]{c + d x} \operatorname{appellf_{1}}{\left (\frac{3}{2},- \frac{1}{3},1,\frac{5}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{3 \sqrt [3]{\frac{b \left (- c - d x\right )}{a d - b c}} \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b*x)**(3/2)*(c + d*x)**(1/3)*appellf1(3/2, -1/3, 1, 5/2, d*(a + b*x)/(a*
d - b*c), f*(a + b*x)/(a*f - b*e))/(3*(b*(-c - d*x)/(a*d - b*c))**(1/3)*(a*f - b
*e))

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Mathematica [B]  time = 4.85267, size = 901, normalized size = 9.01 \[ \frac{6 \sqrt{a+b x} \left (\frac{7 (c+d x)}{f}-\frac{d (a+b x) \left (78 (b c-a d) (b e-a f) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right ) \left (3 d (b e-a f) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+2 (b c-a d) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )-7 (a+b x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right ) \left (13 d f \left (3 c e b^2+(a (32 d e-17 c f)+7 b (5 d e-2 c f) x) b-3 a d f (6 a+7 b x)\right ) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )-14 (5 b d e-2 b c f-3 a d f) \left (3 d (b e-a f) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+2 (b c-a d) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )\right )\right )}{b^2 (e+f x) \left (7 d f (a+b x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(6 a d f-6 b d e) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+4 (a d-b c) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right ) \left (13 d f (a+b x) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(6 a d f-6 b d e) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+4 (a d-b c) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )}\right )}{35 (c+d x)^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e}\sqrt{bx+a}\sqrt [3]{dx+c}}\, dx \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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