3.404 \(\int \left (c+e x^2\right )^3 \left (a+c x^2+b x^4\right )^p \, dx\)

Optimal. Leaf size=498 \[ \frac{e x^3 \left (-3 b e \left (a e (4 p+5)+c^2 \left (8 p^2+26 p+21\right )\right )+3 b^2 c^2 \left (16 p^2+48 p+35\right )+c^2 e^2 \left (4 p^2+16 p+15\right )\right ) \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )}{3 b^2 (4 p+5) (4 p+7)}+\frac{c x \left (-3 a b e^2 (4 p+7)+a e^3 (2 p+5)+b^2 c^2 \left (16 p^2+48 p+35\right )\right ) \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )}{b^2 (4 p+5) (4 p+7)}+\frac{c e^2 x (12 b p+21 b-2 e p-5 e) \left (a+b x^4+c x^2\right )^{p+1}}{b^2 (4 p+5) (4 p+7)}+\frac{e^3 x^3 \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+7)} \]

[Out]

(c*e^2*(21*b - 5*e + 12*b*p - 2*e*p)*x*(a + c*x^2 + b*x^4)^(1 + p))/(b^2*(5 + 4*
p)*(7 + 4*p)) + (e^3*x^3*(a + c*x^2 + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c*(a*e^3*
(5 + 2*p) - 3*a*b*e^2*(7 + 4*p) + b^2*c^2*(35 + 48*p + 16*p^2))*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])])/(b^2*(5 + 4*p)*(7 + 4*p)*(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*(c^2*e^2*(15
+ 16*p + 4*p^2) + 3*b^2*c^2*(35 + 48*p + 16*p^2) - 3*b*e*(a*e*(5 + 4*p) + c^2*(2
1 + 26*p + 8*p^2)))*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*b^2*(5 +
4*p)*(7 + 4*p)*(1 + (2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*(1 + (2*b*x^2)/(c + Sq
rt[-4*a*b + c^2]))^p)

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Rubi [A]  time = 1.59861, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{e x^3 \left (-3 b e \left (a e (4 p+5)+c^2 \left (8 p^2+26 p+21\right )\right )+3 b^2 c^2 \left (16 p^2+48 p+35\right )+c^2 e^2 \left (4 p^2+16 p+15\right )\right ) \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )}{3 b^2 (4 p+5) (4 p+7)}+\frac{c x \left (-3 a b e^2 (4 p+7)+a e^3 (2 p+5)+b^2 c^2 \left (16 p^2+48 p+35\right )\right ) \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )}{b^2 (4 p+5) (4 p+7)}+\frac{c e^2 x (12 b p+21 b-2 e p-5 e) \left (a+b x^4+c x^2\right )^{p+1}}{b^2 (4 p+5) (4 p+7)}+\frac{e^3 x^3 \left (a+b x^4+c x^2\right )^{p+1}}{b (4 p+7)} \]

Antiderivative was successfully verified.

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

[Out]

(c*e^2*(21*b - 5*e + 12*b*p - 2*e*p)*x*(a + c*x^2 + b*x^4)^(1 + p))/(b^2*(5 + 4*
p)*(7 + 4*p)) + (e^3*x^3*(a + c*x^2 + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c*(a*e^3*
(5 + 2*p) - 3*a*b*e^2*(7 + 4*p) + b^2*c^2*(35 + 48*p + 16*p^2))*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])])/(b^2*(5 + 4*p)*(7 + 4*p)*(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*(c^2*e^2*(15
+ 16*p + 4*p^2) + 3*b^2*c^2*(35 + 48*p + 16*p^2) - 3*b*e*(a*e*(5 + 4*p) + c^2*(2
1 + 26*p + 8*p^2)))*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*b^2*(5 +
4*p)*(7 + 4*p)*(1 + (2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*(1 + (2*b*x^2)/(c + Sq
rt[-4*a*b + c^2]))^p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c + e*x**2)**3*(a + b*x**4 + c*x**2)**p, x)

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Mathematica [B]  time = 18.3258, size = 1871, normalized size = 3.76 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*4^(-1 - p)*c^3*(c + Sqrt[-4*a*b + c^2])*x*((c - Sqrt[-4*a*b + c^2] + 2*b*x^2)
/b)^(1 + p)*((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)^2*(a + c*x^2 + b*x^4)^(-1 + 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])])/((-c + S
qrt[-4*a*b + c^2])*((c - Sqrt[-4*a*b + c^2])/(2*b) + x^2)^p*((c + Sqrt[-4*a*b +
c^2])/(2*b) + x^2)^p*(-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*2^(-2 - p)*b*c^2*(c + Sqrt[-4*a*b + c^2])*e*x^3*((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)^2*(a + c*x^2 + b*x^4
)^(-1 + 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])])/((-c + Sqrt[-4*a*b + c^2])*((c - Sqrt[-4*a*b +
c^2])/(2*b) + x^2)^p*(c + Sqrt[-4*a*b + c^2] + 2*b*x^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])]))) + (21*2^(-2 - p)*b*c*(c + Sqrt[-4*a*b + c^2]
)*e^2*x^5*((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)^2*(a + c*x^2 + b*x^4)^(-1 + p)*AppellF1[5/2, -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])])/(5*(-c + Sq
rt[-4*a*b + c^2])*((c - Sqrt[-4*a*b + c^2])/(2*b) + x^2)^p*(c + Sqrt[-4*a*b + c^
2] + 2*b*x^2)*(-7*a*AppellF1[5/2, -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])] + p*x^2*((-c + Sqrt[-4*a*b + c^2])*Appe
llF1[7/2, 1 - p, -p, 9/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + S
qrt[-4*a*b + c^2])] - (c + Sqrt[-4*a*b + c^2])*AppellF1[7/2, -p, 1 - p, 9/2, (-2
*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])]))) + (9*2
^(-2 - p)*b*(c + Sqrt[-4*a*b + c^2])*e^3*x^7*((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)^2*(a + c*x^2 + b*x^4)^(-1 + p
)*AppellF1[7/2, -p, -p, 9/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)/(-c
+ Sqrt[-4*a*b + c^2])])/(7*(-c + Sqrt[-4*a*b + c^2])*((c - Sqrt[-4*a*b + c^2])/(
2*b) + x^2)^p*(c + Sqrt[-4*a*b + c^2] + 2*b*x^2)*(-9*a*AppellF1[7/2, -p, -p, 9/2
, (-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[9/2, 1 - p, -p, 11/2, (-2*b*x^2)/(c + Sq
rt[-4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])] - (c + Sqrt[-4*a*b + c^2
])*AppellF1[9/2, -p, 1 - p, 11/2, (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2]), (2*b*x^2)
/(-c + Sqrt[-4*a*b + c^2])])))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) ^{3} \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x^2+c)^3*(b*x^4+c*x^2+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^2 + c)^3*(b*x^4 + c*x^2 + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{6} + 3 \, c e^{2} x^{4} + 3 \, c^{2} e x^{2} + c^{3}\right )}{\left (b x^{4} + c x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^6 + 3*c*e^2*x^4 + 3*c^2*e*x^2 + c^3)*(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)**3*(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. \[ \int{\left (e x^{2} + c\right )}^{3}{\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^2 + c)^3*(b*x^4 + c*x^2 + a)^p, x)