3.3027 \(\int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt [4]{e+f x}} \, dx\)

Optimal. Leaf size=266 \[ \frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}} \]

[Out]

(2*(d*e - c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[-((Sqrt[b]*Sq
rt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e
- c*f)^(1/4)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]*Sqrt[c + d*x]) - (2*(d*e -
c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*
f])/(Sqrt[d]*Sqrt[b*e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4
)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]*Sqrt[c + d*x])

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Rubi [A]  time = 1.21152, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (-\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}}-\frac{2 \sqrt [4]{d e-c f} \sqrt{-\frac{f (c+d x)}{d e-c f}} \Pi \left (\frac{\sqrt{b} \sqrt{d e-c f}}{\sqrt{d} \sqrt{b e-a f}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right )\right |-1\right )}{\sqrt{b} \sqrt [4]{d} \sqrt{c+d x} \sqrt{b e-a f}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*Sqrt[c + d*x]*(e + f*x)^(1/4)),x]

[Out]

(2*(d*e - c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[-((Sqrt[b]*Sq
rt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e
- c*f)^(1/4)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]*Sqrt[c + d*x]) - (2*(d*e -
c*f)^(1/4)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*
f])/(Sqrt[d]*Sqrt[b*e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4
)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]*Sqrt[c + d*x])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.889217, size = 270, normalized size = 1.02 \[ -\frac{28 d f (a+b x) F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{4};\frac{7}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{3 b \sqrt{c+d x} \sqrt [4]{e+f x} \left (7 d f (a+b x) F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{4};\frac{7}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(a d f-b d e) F_1\left (\frac{7}{4};\frac{1}{2},\frac{5}{4};\frac{11}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+2 f (a d-b c) F_1\left (\frac{7}{4};\frac{3}{2},\frac{1}{4};\frac{11}{4};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x)*Sqrt[c + d*x]*(e + f*x)^(1/4)),x]

[Out]

(-28*d*f*(a + b*x)*AppellF1[3/4, 1/2, 1/4, 7/4, (-(b*c) + a*d)/(d*(a + b*x)), (-
(b*e) + a*f)/(f*(a + b*x))])/(3*b*Sqrt[c + d*x]*(e + f*x)^(1/4)*(7*d*f*(a + b*x)
*AppellF1[3/4, 1/2, 1/4, 7/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a
+ b*x))] + (-(b*d*e) + a*d*f)*AppellF1[7/4, 1/2, 5/4, 11/4, (-(b*c) + a*d)/(d*(
a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] + 2*(-(b*c) + a*d)*f*AppellF1[7/4, 3/2,
1/4, 11/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))]))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt [4]{fx+e}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(f*x+e)^(1/4)/(d*x+c)^(1/2),x)

[Out]

int(1/(b*x+a)/(f*x+e)^(1/4)/(d*x+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x + a)*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(1/((b*x + a)*sqrt(d*x + c)*(f*x + e)^(1/4)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x)*sqrt(c + d*x)*(e + f*x)**(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x + a)*sqrt(d*x + c)*(f*x + e)^(1/4)), x)