3.1040 \(\int x^{10} (a+b x^4)^{3/4} \, dx\)

Optimal. Leaf size=150 \[ -\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}+\frac{3 a^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{80 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b} \]

[Out]

(3*a^3*x^3)/(80*b^2*(a + b*x^4)^(1/4)) - (a^2*x^3*(a + b*x^4)^(3/4))/(40*b^2) + (3*a*x^7*(a + b*x^4)^(3/4))/(1
40*b) + (x^11*(a + b*x^4)^(3/4))/14 + (3*a^(7/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a
]]/2, 2])/(80*b^(5/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0709193, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {279, 321, 310, 281, 335, 275, 196} \[ -\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}+\frac{3 a^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{80 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b} \]

Antiderivative was successfully verified.

[In]

Int[x^10*(a + b*x^4)^(3/4),x]

[Out]

(3*a^3*x^3)/(80*b^2*(a + b*x^4)^(1/4)) - (a^2*x^3*(a + b*x^4)^(3/4))/(40*b^2) + (3*a*x^7*(a + b*x^4)^(3/4))/(1
40*b) + (x^11*(a + b*x^4)^(3/4))/14 + (3*a^(7/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a
]]/2, 2])/(80*b^(5/2)*(a + b*x^4)^(1/4))

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 310

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4)^(1/4)), x] - Dist[a/2, Int[x^2/(a
+ b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 281

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Dist[(x*(1 + a/(b*x^4))^(1/4))/(b*(a + b*x^4)^(1/4)), Int
[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 196

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(5/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int x^{10} \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{1}{14} (3 a) \int \frac{x^{10}}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^2\right ) \int \frac{x^6}{\sqrt [4]{a+b x^4}} \, dx}{20 b}\\ &=-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{\left (3 a^3\right ) \int \frac{x^2}{\sqrt [4]{a+b x^4}} \, dx}{40 b^2}\\ &=\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^4\right ) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{80 b^2}\\ &=\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}-\frac{\left (3 a^4 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{80 b^3 \sqrt [4]{a+b x^4}}\\ &=\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{\left (3 a^4 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{80 b^3 \sqrt [4]{a+b x^4}}\\ &=\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{\left (3 a^4 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{160 b^3 \sqrt [4]{a+b x^4}}\\ &=\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{3 a^{7/2} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{80 b^{5/2} \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0506913, size = 96, normalized size = 0.64 \[ \frac{x^3 \left (a+b x^4\right )^{3/4} \left (\left (\frac{b x^4}{a}+1\right )^{3/4} \left (-7 a^2+3 a b x^4+10 b^2 x^8\right )+7 a^2 \, _2F_1\left (-\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )\right )}{140 b^2 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10*(a + b*x^4)^(3/4),x]

[Out]

(x^3*(a + b*x^4)^(3/4)*((1 + (b*x^4)/a)^(3/4)*(-7*a^2 + 3*a*b*x^4 + 10*b^2*x^8) + 7*a^2*Hypergeometric2F1[-3/4
, 3/4, 7/4, -((b*x^4)/a)]))/(140*b^2*(1 + (b*x^4)/a)^(3/4))

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{x}^{10} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(b*x^4+a)^(3/4),x)

[Out]

int(x^10*(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)*x^10, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)*x^10, x)

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Sympy [C]  time = 4.3101, size = 39, normalized size = 0.26 \begin{align*} \frac{a^{\frac{3}{4}} x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10*(b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x**11*gamma(11/4)*hyper((-3/4, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(15/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)*x^10, x)