3.1071 \(\int x^{12} (a+b x^4)^{5/4} \, dx\)

Optimal. Leaf size=171 \[ \frac{5 a^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{672 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4} \]

[Out]

(5*a^4*x*(a + b*x^4)^(1/4))/(672*b^3) - (a^3*x^5*(a + b*x^4)^(1/4))/(336*b^2) + (a^2*x^9*(a + b*x^4)^(1/4))/(5
04*b) + (5*a*x^13*(a + b*x^4)^(1/4))/252 + (x^13*(a + b*x^4)^(5/4))/18 + (5*a^(9/2)*(1 + a/(b*x^4))^(3/4)*x^3*
EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(672*b^(5/2)*(a + b*x^4)^(3/4))

________________________________________________________________________________________

Rubi [A]  time = 0.087997, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 321, 237, 335, 275, 231} \[ -\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}+\frac{5 a^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{672 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^12*(a + b*x^4)^(5/4),x]

[Out]

(5*a^4*x*(a + b*x^4)^(1/4))/(672*b^3) - (a^3*x^5*(a + b*x^4)^(1/4))/(336*b^2) + (a^2*x^9*(a + b*x^4)^(1/4))/(5
04*b) + (5*a*x^13*(a + b*x^4)^(1/4))/252 + (x^13*(a + b*x^4)^(5/4))/18 + (5*a^(9/2)*(1 + a/(b*x^4))^(3/4)*x^3*
EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(672*b^(5/2)*(a + b*x^4)^(3/4))

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int x^{12} \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{1}{18} (5 a) \int x^{12} \sqrt [4]{a+b x^4} \, dx\\ &=\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{1}{252} \left (5 a^2\right ) \int \frac{x^{12}}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}-\frac{a^3 \int \frac{x^8}{\left (a+b x^4\right )^{3/4}} \, dx}{56 b}\\ &=-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{\left (5 a^4\right ) \int \frac{x^4}{\left (a+b x^4\right )^{3/4}} \, dx}{336 b^2}\\ &=\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}-\frac{\left (5 a^5\right ) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{672 b^3}\\ &=\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}-\frac{\left (5 a^5 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{672 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{\left (5 a^5 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{672 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{\left (5 a^5 \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{1344 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{5 a^{9/2} \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{672 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0888975, size = 89, normalized size = 0.52 \[ \frac{x \sqrt [4]{a+b x^4} \left (\left (a+b x^4\right )^2 \left (9 a^2-18 a b x^4+28 b^2 x^8\right )-\frac{9 a^4 \, _2F_1\left (-\frac{5}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}}\right )}{504 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12*(a + b*x^4)^(5/4),x]

[Out]

(x*(a + b*x^4)^(1/4)*((a + b*x^4)^2*(9*a^2 - 18*a*b*x^4 + 28*b^2*x^8) - (9*a^4*Hypergeometric2F1[-5/4, 1/4, 5/
4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(1/4)))/(504*b^3)

________________________________________________________________________________________

Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x}^{12} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12*(b*x^4+a)^(5/4),x)

[Out]

int(x^12*(b*x^4+a)^(5/4),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{12}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)*x^12, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{16} + a x^{12}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

integral((b*x^16 + a*x^12)*(b*x^4 + a)^(1/4), x)

________________________________________________________________________________________

Sympy [C]  time = 11.11, size = 39, normalized size = 0.23 \begin{align*} \frac{a^{\frac{5}{4}} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12*(b*x**4+a)**(5/4),x)

[Out]

a**(5/4)*x**13*gamma(13/4)*hyper((-5/4, 13/4), (17/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(17/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{12}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12*(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)*x^12, x)