3.1156 \(\int \frac{x^4}{(a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}-\frac{x}{b \sqrt [4]{a+b x^4}} \]

[Out]

-(x/(b*(a + b*x^4)^(1/4))) + ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4)) + ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)]/(2*b^(5/4))

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Rubi [A]  time = 0.0219148, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {288, 240, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}-\frac{x}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^4)^(5/4),x]

[Out]

-(x/(b*(a + b*x^4)^(1/4))) + ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4)) + ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)]/(2*b^(5/4))

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 240

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

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

Rubi steps

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

Mathematica [C]  time = 0.0086163, size = 54, normalized size = 0.73 \[ \frac{x^5 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^4}{a}\right )}{5 a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^4)^(5/4),x]

[Out]

(x^5*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -((b*x^4)/a)])/(5*a*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \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^4/(b*x^4+a)^(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.59409, size = 494, normalized size = 6.68 \begin{align*} \frac{4 \,{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \arctan \left (\frac{b \frac{1}{b^{5}}^{\frac{1}{4}} x \sqrt{\frac{b^{3} \sqrt{\frac{1}{b^{5}}} x^{2} + \sqrt{b x^{4} + a}}{x^{2}}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} b \frac{1}{b^{5}}^{\frac{1}{4}}}{x}\right ) +{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \log \left (\frac{b^{4} \frac{1}{b^{5}}^{\frac{3}{4}} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) -{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \log \left (-\frac{b^{4} \frac{1}{b^{5}}^{\frac{3}{4}} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{4 \,{\left (b^{2} x^{4} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(b^2*x^4 + a*b)*(b^(-5))^(1/4)*arctan((b*(b^(-5))^(1/4)*x*sqrt((b^3*sqrt(b^(-5))*x^2 + sqrt(b*x^4 + a))
/x^2) - (b*x^4 + a)^(1/4)*b*(b^(-5))^(1/4))/x) + (b^2*x^4 + a*b)*(b^(-5))^(1/4)*log((b^4*(b^(-5))^(3/4)*x + (b
*x^4 + a)^(1/4))/x) - (b^2*x^4 + a*b)*(b^(-5))^(1/4)*log(-(b^4*(b^(-5))^(3/4)*x - (b*x^4 + a)^(1/4))/x) - 4*(b
*x^4 + a)^(3/4)*x)/(b^2*x^4 + a*b)

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Sympy [C]  time = 1.49545, size = 37, normalized size = 0.5 \begin{align*} \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*gamma(5/4)*hyper((5/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5/4)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(b*x^4 + a)^(5/4), x)