3.406 \(\int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx\)

Optimal. Leaf size=274 \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.511617, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 67.9417, size = 233, normalized size = 0.85 \[ c x \left (\frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (\frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (a + b x^{4} + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},- p,- p,\frac{3}{2},- \frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}},- \frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} \right )} + \frac{e x^{3} \left (\frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (\frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (a + b x^{4} + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{3}{2},- p,- p,\frac{5}{2},- \frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}},- \frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+c)*(b*x**4+c*x**2+a)**p,x)

[Out]

c*x*(2*b*x**2/(c - sqrt(-4*a*b + c**2)) + 1)**(-p)*(2*b*x**2/(c + sqrt(-4*a*b +
c**2)) + 1)**(-p)*(a + b*x**4 + c*x**2)**p*appellf1(1/2, -p, -p, 3/2, -2*b*x**2/
(c - sqrt(-4*a*b + c**2)), -2*b*x**2/(c + sqrt(-4*a*b + c**2))) + e*x**3*(2*b*x*
*2/(c - sqrt(-4*a*b + c**2)) + 1)**(-p)*(2*b*x**2/(c + sqrt(-4*a*b + c**2)) + 1)
**(-p)*(a + b*x**4 + c*x**2)**p*appellf1(3/2, -p, -p, 5/2, -2*b*x**2/(c - sqrt(-
4*a*b + c**2)), -2*b*x**2/(c + sqrt(-4*a*b + c**2)))/3

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Mathematica [B]  time = 1.25916, size = 706, normalized size = 2.58 \[ \frac{1}{3} 2^{-p-3} x \left (\sqrt{c^2-4 a b}+c\right ) \left (x^2 \left (\sqrt{c^2-4 a b}-c\right )-2 a\right ) \left (\frac{c-\sqrt{c^2-4 a b}}{2 b}+x^2\right )^{-p} \left (\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{b}\right )^{p+1} \left (a+b x^4+c x^2\right )^{p-1} \left (\frac{5 e x^2 F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}{p x^2 \left (\left (\sqrt{c^2-4 a b}-c\right ) F_1\left (\frac{5}{2};1-p,-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )-\left (\sqrt{c^2-4 a b}+c\right ) F_1\left (\frac{5}{2};-p,1-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )\right )-5 a F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}-\frac{9 c F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}{p x^2 \left (\left (c-\sqrt{c^2-4 a b}\right ) F_1\left (\frac{3}{2};1-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )+\left (\sqrt{c^2-4 a b}+c\right ) F_1\left (\frac{3}{2};-p,1-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )\right )+3 a F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]

[Out]

(2^(-3 - p)*(c + Sqrt[-4*a*b + c^2])*x*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)/b)^(1
+ p)*(-2*a + (-c + Sqrt[-4*a*b + c^2])*x^2)*(a + c*x^2 + b*x^4)^(-1 + p)*((-9*c
*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c +
Sqrt[-4*a*b + c^2])])/(3*a*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[-4*a
*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + p*x^2*((c - Sqrt[-4*a*b + c^2
])*AppellF1[3/2, 1 - p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/
(-c + Sqrt[-4*a*b + c^2])] + (c + Sqrt[-4*a*b + c^2])*AppellF1[3/2, -p, 1 - p, 5
/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])]))
+ (5*e*x^2*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*
x^2)/(-c + Sqrt[-4*a*b + c^2])])/(-5*a*AppellF1[3/2, -p, -p, 5/2, (-2*b*x^2)/(c
+ Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] + p*x^2*((-c + Sqrt[
-4*a*b + c^2])*AppellF1[5/2, 1 - p, -p, 7/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2])
, (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] - (c + Sqrt[-4*a*b + c^2])*AppellF1[5/2,
-p, 1 - p, 7/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b
+ c^2])]))))/(3*((c - Sqrt[-4*a*b + c^2])/(2*b) + x^2)^p)

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+c)*(b*x^4+c*x^2+a)^p,x)

[Out]

int((e*x^2+c)*(b*x^4+c*x^2+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^2 + c)*(b*x^4 + c*x^2 + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^2 + c)*(b*x^4 + c*x^2 + a)^p, 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((e*x**2+c)*(b*x**4+c*x**2+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^2 + c)*(b*x^4 + c*x^2 + a)^p, x)