3.1171 \(\int \frac{1}{(a+b x^4)^{9/4}} \, dx\)

Optimal. Leaf size=39 \[ \frac{4 x}{5 a^2 \sqrt [4]{a+b x^4}}+\frac{x}{5 a \left (a+b x^4\right )^{5/4}} \]

[Out]

x/(5*a*(a + b*x^4)^(5/4)) + (4*x)/(5*a^2*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0051588, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{4 x}{5 a^2 \sqrt [4]{a+b x^4}}+\frac{x}{5 a \left (a+b x^4\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-9/4),x]

[Out]

x/(5*a*(a + b*x^4)^(5/4)) + (4*x)/(5*a^2*(a + b*x^4)^(1/4))

Rule 192

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{9/4}} \, dx &=\frac{x}{5 a \left (a+b x^4\right )^{5/4}}+\frac{4 \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{5 a}\\ &=\frac{x}{5 a \left (a+b x^4\right )^{5/4}}+\frac{4 x}{5 a^2 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0105574, size = 29, normalized size = 0.74 \[ \frac{x \left (5 a+4 b x^4\right )}{5 a^2 \left (a+b x^4\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-9/4),x]

[Out]

(x*(5*a + 4*b*x^4))/(5*a^2*(a + b*x^4)^(5/4))

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Maple [A]  time = 0.003, size = 26, normalized size = 0.7 \begin{align*}{\frac{x \left ( 4\,b{x}^{4}+5\,a \right ) }{5\,{a}^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(9/4),x)

[Out]

1/5*x*(4*b*x^4+5*a)/(b*x^4+a)^(5/4)/a^2

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Maxima [A]  time = 1.01096, size = 42, normalized size = 1.08 \begin{align*} -\frac{{\left (b - \frac{5 \,{\left (b x^{4} + a\right )}}{x^{4}}\right )} x^{5}}{5 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4),x, algorithm="maxima")

[Out]

-1/5*(b - 5*(b*x^4 + a)/x^4)*x^5/((b*x^4 + a)^(5/4)*a^2)

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Fricas [A]  time = 1.43547, size = 101, normalized size = 2.59 \begin{align*} \frac{{\left (4 \, b x^{5} + 5 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{5 \,{\left (a^{2} b^{2} x^{8} + 2 \, a^{3} b x^{4} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4),x, algorithm="fricas")

[Out]

1/5*(4*b*x^5 + 5*a*x)*(b*x^4 + a)^(3/4)/(a^2*b^2*x^8 + 2*a^3*b*x^4 + a^4)

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Sympy [B]  time = 1.58255, size = 126, normalized size = 3.23 \begin{align*} \frac{5 a x \Gamma \left (\frac{1}{4}\right )}{16 a^{\frac{13}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{9}{4}\right ) + 16 a^{\frac{9}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{9}{4}\right )} + \frac{4 b x^{5} \Gamma \left (\frac{1}{4}\right )}{16 a^{\frac{13}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{9}{4}\right ) + 16 a^{\frac{9}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(9/4),x)

[Out]

5*a*x*gamma(1/4)/(16*a**(13/4)*(1 + b*x**4/a)**(1/4)*gamma(9/4) + 16*a**(9/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gam
ma(9/4)) + 4*b*x**5*gamma(1/4)/(16*a**(13/4)*(1 + b*x**4/a)**(1/4)*gamma(9/4) + 16*a**(9/4)*b*x**4*(1 + b*x**4
/a)**(1/4)*gamma(9/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(9/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-9/4), x)