3.1104 \(\int \frac{1}{(d x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

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

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

_______________________________________________________________________________________

Rubi [A]  time = 0.445524, 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 \[ -\frac{2 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (-\frac{1}{4};\frac{3}{2},\frac{3}{2};\frac{3}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{a d \sqrt{d x} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.7389, size = 133, normalized size = 0.9 \[ - \frac{2 \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (- \frac{1}{4},\frac{3}{2},\frac{3}{2},\frac{3}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} d \sqrt{d x} \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

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)**(3/2),x)

[Out]

-2*sqrt(a + b*x**2 + c*x**4)*appellf1(-1/4, 3/2, 3/2, 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**2*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 = 3.84863, size = 1600, normalized size = 10.81 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((7*x^2*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2)*(a + b*x^2 + c*x^4))/(a^2*(
-b^2 + 4*a*c)) - (14*(a + b*x^2 + c*x^4)^2)/a^2 + (49*b^3*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((b^2 -
4*a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*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])]))) - (147*a*b*c*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 + Sqr
t[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c)*(-b + Sqrt[
b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*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 + Sqr
t[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*b^2*c*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 + Sqrt[b^2 - 4*a*c]), (
2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b +
Sqrt[b^2 - 4*a*c])*(-11*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])] + 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])]))) - (33
0*a*c^2*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b
+ Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c)*(-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4
*a*c])*(-11*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])] + 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])])))))/(7*(d*x)^(3/2)*(
a + b*x^2 + c*x^4)^(3/2))

_______________________________________________________________________________________

Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{1 \left ( dx \right ) ^{-{\frac{3}{2}}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c d x^{5} + b d x^{3} + a d x\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{d x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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