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

Optimal. Leaf size=297 \[ \frac{2 d (f x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{1}{2},-\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{9}{4};-\frac{1}{2},-\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

(2*d*(f*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*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])])/(5*f*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])]) +
(2*e*(f*x)^(9/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[9/4, -1/2, -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*f^3*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])
])

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Rubi [A]  time = 1.12305, antiderivative size = 297, 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 \[ \frac{2 d (f x)^{5/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{5}{4};-\frac{1}{2},-\frac{1}{2};\frac{9}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{5 f \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{2 e (f x)^{9/2} \sqrt{a+b x^2+c x^4} F_1\left (\frac{9}{4};-\frac{1}{2},-\frac{1}{2};\frac{13}{4};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{9 f^3 \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(f*x)^(5/2)*Sqrt[a + b*x^2 + c*x^4]*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])])/(5*f*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])]) +
(2*e*(f*x)^(9/2)*Sqrt[a + b*x^2 + c*x^4]*AppellF1[9/4, -1/2, -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*f^3*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])
])

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Rubi in Sympy [A]  time = 80.8413, size = 269, normalized size = 0.91 \[ \frac{2 d \left (f x\right )^{\frac{5}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{5}{4},- \frac{1}{2},- \frac{1}{2},\frac{9}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{5 f \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{2 e \left (f x\right )^{\frac{9}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{9}{4},- \frac{1}{2},- \frac{1}{2},\frac{13}{4},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{9 f^{3} \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

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)**(1/2),x)

[Out]

2*d*(f*x)**(5/2)*sqrt(a + b*x**2 + c*x**4)*appellf1(5/4, -1/2, -1/2, 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*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, -1/2, -1/2, 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*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 = 7.46563, size = 2835, normalized size = 9.55 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

((f*x)^(3/2)*Sqrt[a + b*x^2 + c*x^4]*((4*(13*b*c*d - 7*b^2*e + 18*a*c*e)*Sqrt[x]
)/(585*c^2) + (2*(13*c*d + 2*b*e)*x^(5/2))/(117*c) + (2*e*x^(9/2))/13))/x^(3/2)
+ (4*a^3*b*d*(f*x)^(3/2)*(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])])/(9*c*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 -
4*a*c])*x*(a + b*x^2 + c*x^4)^(3/2)*(-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])
]))) - (28*a^3*b^2*e*(f*x)^(3/2)*(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])])/(117*c^2*(b - Sqrt[b^2 - 4*a*c])*(b +
Sqrt[b^2 - 4*a*c])*x*(a + b*x^2 + c*x^4)^(3/2)*(-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])*AppellF
1[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])]))) + (8*a^4*e*(f*x)^(3/2)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(b + S
qrt[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])])/(13*c*(b - Sqrt[b^2 - 4*a*c])*
(b + Sqrt[b^2 - 4*a*c])*x*(a + b*x^2 + c*x^4)^(3/2)*(-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])*App
ellF1[5/4, 3/2, 1/2, 9/4, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sq
rt[b^2 - 4*a*c])]))) - (8*a^3*d*x*(f*x)^(3/2)*(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 + S
qrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(5*(b - Sqrt[b^2 - 4*a*c
])*(b + Sqrt[b^2 - 4*a*c])*(a + b*x^2 + c*x^4)^(3/2)*(-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 + Sqr
t[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*A
ppellF1[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])]))) + (12*a^2*b^2*d*x*(f*x)^(3/2)*(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])])/(25*c*(b - Sqrt[
b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(a + b*x^2 + c*x^4)^(3/2)*(-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])]))) - (84*a^2*b^3*e*x*(f*x)^(3/2)*(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])])/(325*
c^2*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(a + b*x^2 + c*x^4)^(3/2)*(-
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])]))) + (316*a^3*b*e*x*(f*x)^(3/2)*(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])])/(325*c*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(a + b*x^2 + c*x^
4)^(3/2)*(-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])])))

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int \left ( fx \right ) ^{{\frac{3}{2}}} \left ( e{x}^{2}+d \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}\, dx \]

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)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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