3.1115 \(\int x^7 (a+b x^2+c x^4)^p \, dx\)

Optimal. Leaf size=257 \[ -\frac{b 2^{p-2} \left (6 a c-b^2 (p+3)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{8 c^3 (p+1) (p+2) (2 p+3)}+\frac{x^4 \left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+2)} \]

[Out]

(x^4*(a + b*x^2 + c*x^4)^(1 + p))/(4*c*(2 + p)) + ((b^2*(2 + p)*(3 + p) - 2*a*c*(3 + 2*p) - 2*b*c*(1 + p)*(3 +
p)*x^2)*(a + b*x^2 + c*x^4)^(1 + p))/(8*c^3*(1 + p)*(2 + p)*(3 + 2*p)) - (2^(-2 + p)*b*(6*a*c - b^2*(3 + 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)*Hypergeometric2F
1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(2*Sqrt[b^2 - 4*a*c])])/(c^3*Sqrt[b^2 - 4*a*c]*(1 + p)*(
3 + 2*p))

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Rubi [A]  time = 0.373316, antiderivative size = 257, 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, Rules used = {1114, 742, 779, 624} \[ -\frac{b 2^{p-2} \left (6 a c-b^2 (p+3)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{8 c^3 (p+1) (p+2) (2 p+3)}+\frac{x^4 \left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+2)} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2 + c*x^4)^p,x]

[Out]

(x^4*(a + b*x^2 + c*x^4)^(1 + p))/(4*c*(2 + p)) + ((b^2*(2 + p)*(3 + p) - 2*a*c*(3 + 2*p) - 2*b*c*(1 + p)*(3 +
p)*x^2)*(a + b*x^2 + c*x^4)^(1 + p))/(8*c^3*(1 + p)*(2 + p)*(3 + 2*p)) - (2^(-2 + p)*b*(6*a*c - b^2*(3 + 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)*Hypergeometric2F
1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(2*Sqrt[b^2 - 4*a*c])])/(c^3*Sqrt[b^2 - 4*a*c]*(1 + p)*(
3 + 2*p))

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 624

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*
x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p
+ 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps

\begin{align*} \int x^7 \left (a+b x^2+c x^4\right )^p \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac{x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac{\operatorname{Subst}\left (\int x (-2 a-b (3+p) x) \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{4 c (2+p)}\\ &=\frac{x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac{\left (b^2 (2+p) (3+p)-2 a c (3+2 p)-2 b c (1+p) (3+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{8 c^3 (1+p) (2+p) (3+2 p)}+\frac{\left (b \left (6 a c-b^2 (3+p)\right )\right ) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{8 c^3 (3+2 p)}\\ &=\frac{x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac{\left (b^2 (2+p) (3+p)-2 a c (3+2 p)-2 b c (1+p) (3+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{8 c^3 (1+p) (2+p) (3+2 p)}-\frac{2^{-2+p} b \left (6 a c-b^2 (3+p)\right ) \left (-\frac{b-\sqrt{b^2-4 a c}+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-1-p} \left (a+b x^2+c x^4\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+\sqrt{b^2-4 a c}+2 c x^2}{2 \sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} (1+p) (3+2 p)}\\ \end{align*}

Mathematica [C]  time = 0.242301, size = 162, normalized size = 0.63 \[ \frac{1}{8} x^8 \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (4;-p,-p;5;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^7*(a + b*x^2 + c*x^4)^p,x]

[Out]

(x^8*(a + b*x^2 + c*x^4)^p*AppellF1[4, -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])])/(8*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^
2)/(b + Sqrt[b^2 - 4*a*c]))^p)

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{x}^{7} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(c*x^4+b*x^2+a)^p,x)

[Out]

int(x^7*(c*x^4+b*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^p*x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^4 + b*x^2 + a)^p*x^7, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(c*x**4+b*x**2+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(c*x^4+b*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^p*x^7, x)