3.1113 \(\int \frac{(d x)^m}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) \sqrt{a+b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.467935, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.2231, size = 138, normalized size = 0.86 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},\frac{3}{2},\frac{3}{2},\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} d \left (m + 1\right ) \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*sqrt(a + b*x**2 + c*x**4)*appellf1(m/2 + 1/2, 3/2, 3/2, m/2 + 3/2
, -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*(m + 1)*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 = 2.83284, size = 426, normalized size = 2.66 \[ \frac{2 a^2 (m+3) x (d x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) F_1\left (\frac{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) \left (a+b x^2+c x^4\right )^{5/2} \left (2 a (m+3) F_1\left (\frac{m+1}{2};\frac{3}{2},\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-3 x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+3}{2};\frac{3}{2},\frac{5}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+3}{2};\frac{5}{2},\frac{3}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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