3.1100 \(\int \frac{1}{(d x)^{3/2} \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{3}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d \sqrt{d x} \sqrt{a+b x^2+c x^4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.447976, antiderivative size = 145, 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 \[ -\frac{2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{4};\frac{1}{2},\frac{1}{2};\frac{3}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d \sqrt{d x} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.4651, size = 131, normalized size = 0.9 \[ - \frac{2 \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (- \frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a d \sqrt{d x} \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 1.23925, size = 710, normalized size = 4.9 \[ \frac{2 x \left (\frac{49 a b x^2 \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{4 c \left (7 a F_1\left (\frac{3}{4};\frac{1}{2},\frac{1}{2};\frac{7}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{7}{4};\frac{1}{2},\frac{3}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{7}{4};\frac{3}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}+\frac{99 a x^4 \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{44 a F_1\left (\frac{7}{4};\frac{1}{2},\frac{1}{2};\frac{11}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-4 x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{11}{4};\frac{1}{2},\frac{3}{2};\frac{15}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{11}{4};\frac{3}{2},\frac{1}{2};\frac{15}{4};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}-21 \left (a+b x^2+c x^4\right )^2\right )}{21 a (d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*(-21*(a + b*x^2 + c*x^4)^2 + (49*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[3/4, 1/2, 1/2, 7/4, (-2*c*x^2)/(b + S
qrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(4*c*(7*a*AppellF1[3/4,
1/2, 1/2, 7/4, (-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[7/4, 1/2, 3/2, 11/4, (-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[7/4, 3/2, 1/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2
)/(-b + Sqrt[b^2 - 4*a*c])]))) + (99*a*x^4*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x^2)*AppellF1[7/4, 1/2, 1/2, 11/4, (-2*c*x^2)/(b + Sqr
t[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(44*a*AppellF1[7/4, 1/2, 1
/2, 11/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])
] - 4*x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[11/4, 1/2, 3/2, 15/4, (-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[11/4, 3/2, 1/2, 15/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)
/(-b + Sqrt[b^2 - 4*a*c])]))))/(21*a*(d*x)^(3/2)*(a + b*x^2 + c*x^4)^(3/2))

_______________________________________________________________________________________

Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{1 \left ( dx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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