3.635 \(\int \frac{\sqrt{c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(-2*Sqrt[c + d*x^4]*AppellF1[-1/8, 1, -1/2, 7/8, -((b*x^4)/a), -((d*x^4)/c)])/(a
*e*Sqrt[e*x]*Sqrt[1 + (d*x^4)/c])

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Rubi [A]  time = 0.390963, 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 \[ -\frac{2 \sqrt{c+d x^4} F_1\left (-\frac{1}{8};1,-\frac{1}{2};\frac{7}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a e \sqrt{e x} \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]

[Out]

(-2*Sqrt[c + d*x^4]*AppellF1[-1/8, 1, -1/2, 7/8, -((b*x^4)/a), -((d*x^4)/c)])/(a
*e*Sqrt[e*x]*Sqrt[1 + (d*x^4)/c])

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Rubi in Sympy [A]  time = 45.4652, size = 60, normalized size = 0.87 \[ - \frac{2 \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (- \frac{1}{8},- \frac{1}{2},1,\frac{7}{8},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{a e \sqrt{e x} \sqrt{1 + \frac{d x^{4}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(c + d*x**4)*appellf1(-1/8, -1/2, 1, 7/8, -d*x**4/c, -b*x**4/a)/(a*e*sqrt
(e*x)*sqrt(1 + d*x**4/c))

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Mathematica [B]  time = 1.00336, size = 348, normalized size = 5.04 \[ \frac{2 x \left (\frac{75 c x^4 (b c-4 a d) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (4 x^4 \left (2 b c F_1\left (\frac{15}{8};\frac{1}{2},2;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{15}{8};\frac{3}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-15 a c F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{161 b c d x^8 F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{\left (a+b x^4\right ) \left (4 x^4 \left (2 b c F_1\left (\frac{23}{8};\frac{1}{2},2;\frac{31}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+a d F_1\left (\frac{23}{8};\frac{3}{2},1;\frac{31}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )-23 a c F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )}-\frac{35 \left (c+d x^4\right )}{a}\right )}{35 (e x)^{3/2} \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]

[Out]

(2*x*((-35*(c + d*x^4))/a + (75*c*(b*c - 4*a*d)*x^4*AppellF1[7/8, 1/2, 1, 15/8,
-((d*x^4)/c), -((b*x^4)/a)])/((a + b*x^4)*(-15*a*c*AppellF1[7/8, 1/2, 1, 15/8, -
((d*x^4)/c), -((b*x^4)/a)] + 4*x^4*(2*b*c*AppellF1[15/8, 1/2, 2, 23/8, -((d*x^4)
/c), -((b*x^4)/a)] + a*d*AppellF1[15/8, 3/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)
]))) - (161*b*c*d*x^8*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)])/
((a + b*x^4)*(-23*a*c*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)] +
4*x^4*(2*b*c*AppellF1[23/8, 1/2, 2, 31/8, -((d*x^4)/c), -((b*x^4)/a)] + a*d*App
ellF1[23/8, 3/2, 1, 31/8, -((d*x^4)/c), -((b*x^4)/a)])))))/(35*(e*x)^(3/2)*Sqrt[
c + d*x^4])

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a}\sqrt{d{x}^{4}+c} \left ( ex \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)

[Out]

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{4} + c}}{{\left (b e x^{5} + a e x\right )} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^4 + c)/((b*e*x^5 + a*e*x)*sqrt(e*x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{4}}}{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{4}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**4)/((e*x)**(3/2)*(a + b*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)