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

Optimal. Leaf size=71 \[ \frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a e \sqrt{\frac{d x^4}{c}+1}} \]

[Out]

(2*(e*x)^(5/2)*Sqrt[c + d*x^4]*AppellF1[5/8, 1, -1/2, 13/8, -((b*x^4)/a), -((d*x
^4)/c)])/(5*a*e*Sqrt[1 + (d*x^4)/c])

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Rubi [A]  time = 0.381679, antiderivative size = 71, 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 \[ \frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a e \sqrt{\frac{d x^4}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*x)^(5/2)*Sqrt[c + d*x^4]*AppellF1[5/8, 1, -1/2, 13/8, -((b*x^4)/a), -((d*x
^4)/c)])/(5*a*e*Sqrt[1 + (d*x^4)/c])

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Rubi in Sympy [A]  time = 43.8723, size = 58, normalized size = 0.82 \[ \frac{2 \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{4}} \operatorname{appellf_{1}}{\left (\frac{5}{8},- \frac{1}{2},1,\frac{13}{8},- \frac{d x^{4}}{c},- \frac{b x^{4}}{a} \right )}}{5 a e \sqrt{1 + \frac{d x^{4}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e*x)**(5/2)*sqrt(c + d*x**4)*appellf1(5/8, -1/2, 1, 13/8, -d*x**4/c, -b*x**4/
a)/(5*a*e*sqrt(1 + d*x**4/c))

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Mathematica [B]  time = 0.260022, size = 170, normalized size = 2.39 \[ \frac{26 a c x (e x)^{3/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};-\frac{1}{2},1;\frac{13}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{5 \left (a+b x^4\right ) \left (4 x^4 \left (a d F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-2 b c F_1\left (\frac{13}{8};-\frac{1}{2},2;\frac{21}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )+13 a c F_1\left (\frac{5}{8};-\frac{1}{2},1;\frac{13}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(26*a*c*x*(e*x)^(3/2)*Sqrt[c + d*x^4]*AppellF1[5/8, -1/2, 1, 13/8, -((d*x^4)/c),
-((b*x^4)/a)])/(5*(a + b*x^4)*(13*a*c*AppellF1[5/8, -1/2, 1, 13/8, -((d*x^4)/c)
, -((b*x^4)/a)] + 4*x^4*(-2*b*c*AppellF1[13/8, -1/2, 2, 21/8, -((d*x^4)/c), -((b
*x^4)/a)] + a*d*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^4)/c), -((b*x^4)/a)])))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{4}+a} \left ( ex \right ) ^{{\frac{3}{2}}}\sqrt{d{x}^{4}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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