3.1116 \(\int x^5 \left (a+b x^2+c x^4\right )^p \, dx\)

Optimal. Leaf size=223 \[ \frac{2^{p-1} \left (2 a c-b^2 (p+2)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{4 c^2 (p+1) (2 p+3)}+\frac{x^2 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (2 p+3)} \]

[Out]

-(b*(2 + p)*(a + b*x^2 + c*x^4)^(1 + p))/(4*c^2*(1 + p)*(3 + 2*p)) + (x^2*(a + b
*x^2 + c*x^4)^(1 + p))/(2*c*(3 + 2*p)) + (2^(-1 + p)*(2*a*c - b^2*(2 + p))*(-((b
- Sqrt[b^2 - 4*a*c] + 2*c*x^2)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^2 + c*x^4)
^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(
2*Sqrt[b^2 - 4*a*c])])/(c^2*Sqrt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))

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Rubi [A]  time = 0.429162, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2^{p-1} \left (2 a c-b^2 (p+2)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{4 c^2 (p+1) (2 p+3)}+\frac{x^2 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (2 p+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2 + c*x^4)^p,x]

[Out]

-(b*(2 + p)*(a + b*x^2 + c*x^4)^(1 + p))/(4*c^2*(1 + p)*(3 + 2*p)) + (x^2*(a + b
*x^2 + c*x^4)^(1 + p))/(2*c*(3 + 2*p)) + (2^(-1 + p)*(2*a*c - b^2*(2 + p))*(-((b
- Sqrt[b^2 - 4*a*c] + 2*c*x^2)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^2 + c*x^4)
^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(
2*Sqrt[b^2 - 4*a*c])])/(c^2*Sqrt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))

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Rubi in Sympy [A]  time = 40.7271, size = 192, normalized size = 0.86 \[ - \frac{b \left (p + 2\right ) \left (a + b x^{2} + c x^{4}\right )^{p + 1}}{4 c^{2} \left (p + 1\right ) \left (2 p + 3\right )} + \frac{x^{2} \left (a + b x^{2} + c x^{4}\right )^{p + 1}}{2 c \left (2 p + 3\right )} + \frac{\left (\frac{- \frac{b}{2} - c x^{2} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (2 a c - b^{2} \left (p + 2\right )\right ) \left (a + b x^{2} + c x^{4}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x^{2} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{2} \left (p + 1\right ) \left (2 p + 3\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(c*x**4+b*x**2+a)**p,x)

[Out]

-b*(p + 2)*(a + b*x**2 + c*x**4)**(p + 1)/(4*c**2*(p + 1)*(2*p + 3)) + x**2*(a +
b*x**2 + c*x**4)**(p + 1)/(2*c*(2*p + 3)) + ((-b/2 - c*x**2 + sqrt(-4*a*c + b**
2)/2)/sqrt(-4*a*c + b**2))**(-p - 1)*(2*a*c - b**2*(p + 2))*(a + b*x**2 + c*x**4
)**(p + 1)*hyper((-p, p + 1), (p + 2,), (b/2 + c*x**2 + sqrt(-4*a*c + b**2)/2)/s
qrt(-4*a*c + b**2))/(4*c**2*(p + 1)*(2*p + 3)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 3.64026, size = 395, normalized size = 1.77 \[ \frac{x^6 \left (\sqrt{b^2-4 a c}+b\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (x^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 \left (a+x^2 \left (b+c x^2\right )\right )^{p-1} F_1\left (3;-p,-p;4;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{3 \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (p x^2 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (4;1-p,-p;5;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (4;-p,1-p;5;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-8 a F_1\left (3;-p,-p;4;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5*(a + b*x^2 + c*x^4)^p,x]

[Out]

((b + Sqrt[b^2 - 4*a*c])*x^6*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(2*a + (b - Sqrt[
b^2 - 4*a*c])*x^2)^2*(a + x^2*(b + c*x^2))^(-1 + p)*AppellF1[3, -p, -p, 4, (-2*c
*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(3*(-b + Sqr
t[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(-8*a*AppellF1[3, -p, -p, 4, (
-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + p*x^2*(
(-b + Sqrt[b^2 - 4*a*c])*AppellF1[4, 1 - p, -p, 5, (-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[4,
-p, 1 - p, 5, (-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 = 0.054, size = 0, normalized size = 0. \[ \int{x}^{5} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(c*x^4+b*x^2+a)^p,x)

[Out]

int(x^5*(c*x^4+b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p*x^5,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^p*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p*x^5,x, algorithm="fricas")

[Out]

integral((c*x^4 + b*x^2 + a)^p*x^5, 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(x**5*(c*x**4+b*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p*x^5,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^p*x^5, x)