3.1103 \(\int \frac{1}{\sqrt{d x} \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{2 \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{4};\frac{3}{2},\frac{3}{2};\frac{5}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{a d \sqrt{a+b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.441563, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{d x} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{4};\frac{3}{2},\frac{3}{2};\frac{5}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{a d \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.9585, size = 129, normalized size = 0.87 \[ \frac{2 \sqrt{d x} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{1}{4},\frac{3}{2},\frac{3}{2},\frac{5}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} d \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d*x)*sqrt(a + b*x**2 + c*x**4)*appellf1(1/4, 3/2, 3/2, 5/4, -2*c*x**2/(b
- sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(a**2*d*sqrt(2*c*x*
*2/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqrt(-4*a*c + b**2)) + 1))

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Mathematica [B]  time = 2.03135, size = 1058, normalized size = 7.15 \[ \frac{x \left (\frac{300 \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a^2}{10 a F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-2 x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{3}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}-\frac{25 b^2 \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a}{c \left (5 a F_1\left (\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{5}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{3}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}-\frac{9 b x^2 \left (2 c x^2+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^2+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) a}{x^2 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{1}{2},\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{9}{4};\frac{3}{2},\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-9 a F_1\left (\frac{5}{4};\frac{1}{2},\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}-20 \left (b^2+c x^2 b-2 a c\right ) \left (c x^4+b x^2+a\right )\right )}{20 a \left (4 a c-b^2\right ) \sqrt{d x} \left (c x^4+b x^2+a\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(-20*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^4) + (300*a^2*(b - Sqrt[b^2 - 4
*a*c] + 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[1/4, 1/2, 1/2, 5/4,
(-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(10*a*A
ppellF1[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b +
Sqrt[b^2 - 4*a*c])] - 2*x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[5/4, 1/2, 3/2, 9/4
, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b -
Sqrt[b^2 - 4*a*c])*AppellF1[5/4, 3/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*
c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])) - (25*a*b^2*(b - Sqrt[b^2 - 4*a*c] +
2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[1/4, 1/2, 1/2, 5/4, (-2*c*x^
2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(c*(5*a*AppellF
1[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b
^2 - 4*a*c])] - x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[5/4, 1/2, 3/2, 9/4, (-2*c*
x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^
2 - 4*a*c])*AppellF1[5/4, 3/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*
c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))) - (9*a*b*x^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^
2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[5/4, 1/2, 1/2, 9/4, (-2*c*x^2)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(-9*a*AppellF1[5/4, 1
/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a
*c])] + x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[9/4, 1/2, 3/2, 13/4, (-2*c*x^2)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a
*c])*AppellF1[9/4, 3/2, 1/2, 13/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)
/(-b + Sqrt[b^2 - 4*a*c])]))))/(20*a*(-b^2 + 4*a*c)*Sqrt[d*x]*(a + b*x^2 + c*x^4
)^(3/2))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{dx}}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*sqrt(d*x)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{d x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*sqrt(d*x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(d*x)*(a + b*x**2 + c*x**4)**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*sqrt(d*x)),x, algorithm="giac")

[Out]

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