3.1027 \(\int x (a+b x^4)^{3/4} \, dx\)

Optimal. Leaf size=98 \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]

[Out]

(3*a*x^2)/(5*(a + b*x^4)^(1/4)) + (x^2*(a + b*x^4)^(3/4))/5 - (3*a^(3/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTa
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a + b*x^4)^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 0.0496406, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {275, 195, 229, 227, 196} \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^4)^(3/4),x]

[Out]

(3*a*x^2)/(5*(a + b*x^4)^(1/4)) + (x^2*(a + b*x^4)^(3/4))/5 - (3*a^(3/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTa
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a + b*x^4)^(1/4))

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 229

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

Rule 227

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

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 \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/4} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{1}{10} (3 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{\left (3 a \sqrt [4]{1+\frac{b x^4}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx,x,x^2\right )}{10 \sqrt [4]{a+b x^4}}\\ &=\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}-\frac{\left (3 a \sqrt [4]{1+\frac{b x^4}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{10 \sqrt [4]{a+b x^4}}\\ &=\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}-\frac{3 a^{3/2} \sqrt [4]{1+\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0099279, size = 51, normalized size = 0.52 \[ \frac{x^2 \left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )}{2 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^4)^(3/4),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int x \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*(b*x^4+a)^(3/4),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^4 + a)^(3/4)*x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^4 + a)^(3/4)*x, x)

________________________________________________________________________________________

Sympy [C]  time = 1.31051, size = 29, normalized size = 0.3 \begin{align*} \frac{a^{\frac{3}{4}} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/4)*x**2*hyper((-3/4, 1/2), (3/2,), b*x**4*exp_polar(I*pi)/a)/2

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError