3.634 \(\int \frac{\sqrt{c+d x^4}}{\sqrt{e x} \left (a+b x^4\right )} \, dx\)

Optimal. Leaf size=69 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(2*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[1/8, 1, -1/2, 9/8, -((b*x^4)/a), -((d*x^4)
/c)])/(a*e*Sqrt[1 + (d*x^4)/c])

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Rubi [A]  time = 0.223926, antiderivative size = 69, 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 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^4} F_1\left (\frac{1}{8};1,-\frac{1}{2};\frac{9}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[1/8, 1, -1/2, 9/8, -((b*x^4)/a), -((d*x^4)
/c)])/(a*e*Sqrt[1 + (d*x^4)/c])

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Rubi in Sympy [A]  time = 50.2075, size = 56, normalized size = 0.81 \[ \frac{2 \sqrt{e x} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{1}{8},- \frac{1}{2},1,\frac{9}{8},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a e \sqrt{1 + \frac{d x^{4}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(e*x)*sqrt(c + d*x**4)*appellf1(1/8, -1/2, 1, 9/8, -d*x**4/c, -b*x**4/a)/(
a*e*sqrt(1 + d*x**4/c))

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Mathematica [B]  time = 0.263671, size = 168, normalized size = 2.43 \[ \frac{18 a c x \sqrt{c+d x^4} F_1\left (\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\sqrt{e x} \left (a+b x^4\right ) \left (4 x^4 \left (a d F_1\left (\frac{9}{8};\frac{1}{2},1;\frac{17}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{9}{8};-\frac{1}{2},2;\frac{17}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+9 a c F_1\left (\frac{1}{8};-\frac{1}{2},1;\frac{9}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(18*a*c*x*Sqrt[c + d*x^4]*AppellF1[1/8, -1/2, 1, 9/8, -((d*x^4)/c), -((b*x^4)/a)
])/(Sqrt[e*x]*(a + b*x^4)*(9*a*c*AppellF1[1/8, -1/2, 1, 9/8, -((d*x^4)/c), -((b*
x^4)/a)] + 4*x^4*(-2*b*c*AppellF1[9/8, -1/2, 2, 17/8, -((d*x^4)/c), -((b*x^4)/a)
] + a*d*AppellF1[9/8, 1/2, 1, 17/8, -((d*x^4)/c), -((b*x^4)/a)])))

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c}{\frac{1}{\sqrt{ex}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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