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

Optimal. Leaf size=303 \[ \frac{2 d (f x)^{5/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{5}{4};\frac{3}{2},\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 a f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{9}{4};\frac{3}{2},\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 a f^3 \sqrt{a+b x^2+c x^4}} \]

[Out]

(2*d*(f*x)^(5/2)*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[5/4, 3/2, 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])])/(5*a*f*Sqrt[a + b*x^2 + c*x^4]) +
(2*e*(f*x)^(9/2)*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[9/4, 3/2, 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])])/(9*a*f^3*Sqrt[a + b*x^2 + c*x^4
])

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Rubi [A]  time = 1.02874, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{2 d (f x)^{5/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{5}{4};\frac{3}{2},\frac{3}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 a f \sqrt{a+b x^2+c x^4}}+\frac{2 e (f x)^{9/2} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{9}{4};\frac{3}{2},\frac{3}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 a f^3 \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(f*x)^(5/2)*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[5/4, 3/2, 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])])/(5*a*f*Sqrt[a + b*x^2 + c*x^4]) +
(2*e*(f*x)^(9/2)*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[9/4, 3/2, 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])])/(9*a*f^3*Sqrt[a + b*x^2 + c*x^4
])

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Rubi in Sympy [A]  time = 102.03, size = 269, normalized size = 0.89 \[ \frac{2 d \left (f x\right )^{\frac{5}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{5}{4},\frac{3}{2},\frac{3}{2},\frac{9}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{5 a^{2} f \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{2 e \left (f x\right )^{\frac{9}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{9}{4},\frac{3}{2},\frac{3}{2},\frac{13}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{9 a^{2} f^{3} \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*(f*x)**(5/2)*sqrt(a + b*x**2 + c*x**4)*appellf1(5/4, 3/2, 3/2, 9/4, -2*c*x**
2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(5*a**2*f*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)) + 2*e*(f*x)**(9/2)*sqrt(a + b*x**2 + c*x**4)*appellf1(9/4, 3/2, 3/2, 13/4
, -2*c*x**2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(9*a
**2*f**3*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 = 3.76852, size = 1404, normalized size = 4.63 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(f*Sqrt[f*x]*(5*(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)*(a + b*x^2 + c*x^4) + (25
*a^2*e*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*Appe
llF1[1/4, 1/2, 1/2, 5/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqr
t[b^2 - 4*a*c])])/(2*c*(5*a*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])] - 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
*d*(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])])/(4*c*(5*a*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])] - x^2*((b + Sqrt[b^2 - 4*a*c])*Appel
lF1[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*d*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])])/(18*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])] - 2*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])])) - (9*a*b*e*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])])/(4*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])])))))/(5*(b^2 - 4*a*c)*
(a + b*x^2 + c*x^4)^(3/2))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{(e{x}^{2}+d) \left ( fx \right ) ^{{\frac{3}{2}}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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