3.1173 \(\int \frac{1}{(a+b x^4)^{13/4}} \, dx\)

Optimal. Leaf size=58 \[ \frac{32 x}{45 a^3 \sqrt [4]{a+b x^4}}+\frac{8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac{x}{9 a \left (a+b x^4\right )^{9/4}} \]

[Out]

x/(9*a*(a + b*x^4)^(9/4)) + (8*x)/(45*a^2*(a + b*x^4)^(5/4)) + (32*x)/(45*a^3*(a + b*x^4)^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 0.0098094, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{32 x}{45 a^3 \sqrt [4]{a+b x^4}}+\frac{8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac{x}{9 a \left (a+b x^4\right )^{9/4}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(-13/4),x]

[Out]

x/(9*a*(a + b*x^4)^(9/4)) + (8*x)/(45*a^2*(a + b*x^4)^(5/4)) + (32*x)/(45*a^3*(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 )^{13/4}} \, dx &=\frac{x}{9 a \left (a+b x^4\right )^{9/4}}+\frac{8 \int \frac{1}{\left (a+b x^4\right )^{9/4}} \, dx}{9 a}\\ &=\frac{x}{9 a \left (a+b x^4\right )^{9/4}}+\frac{8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac{32 \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{45 a^2}\\ &=\frac{x}{9 a \left (a+b x^4\right )^{9/4}}+\frac{8 x}{45 a^2 \left (a+b x^4\right )^{5/4}}+\frac{32 x}{45 a^3 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0129195, size = 40, normalized size = 0.69 \[ \frac{x \left (45 a^2+72 a b x^4+32 b^2 x^8\right )}{45 a^3 \left (a+b x^4\right )^{9/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(-13/4),x]

[Out]

(x*(45*a^2 + 72*a*b*x^4 + 32*b^2*x^8))/(45*a^3*(a + b*x^4)^(9/4))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 37, normalized size = 0.6 \begin{align*}{\frac{x \left ( 32\,{b}^{2}{x}^{8}+72\,ab{x}^{4}+45\,{a}^{2} \right ) }{45\,{a}^{3}} \left ( b{x}^{4}+a \right ) ^{-{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)^(13/4),x)

[Out]

1/45*x*(32*b^2*x^8+72*a*b*x^4+45*a^2)/(b*x^4+a)^(9/4)/a^3

________________________________________________________________________________________

Maxima [A]  time = 1.22938, size = 68, normalized size = 1.17 \begin{align*} \frac{{\left (5 \, b^{2} - \frac{18 \,{\left (b x^{4} + a\right )} b}{x^{4}} + \frac{45 \,{\left (b x^{4} + a\right )}^{2}}{x^{8}}\right )} x^{9}}{45 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(13/4),x, algorithm="maxima")

[Out]

1/45*(5*b^2 - 18*(b*x^4 + a)*b/x^4 + 45*(b*x^4 + a)^2/x^8)*x^9/((b*x^4 + a)^(9/4)*a^3)

________________________________________________________________________________________

Fricas [A]  time = 1.50299, size = 151, normalized size = 2.6 \begin{align*} \frac{{\left (32 \, b^{2} x^{9} + 72 \, a b x^{5} + 45 \, a^{2} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{45 \,{\left (a^{3} b^{3} x^{12} + 3 \, a^{4} b^{2} x^{8} + 3 \, a^{5} b x^{4} + a^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(13/4),x, algorithm="fricas")

[Out]

1/45*(32*b^2*x^9 + 72*a*b*x^5 + 45*a^2*x)*(b*x^4 + a)^(3/4)/(a^3*b^3*x^12 + 3*a^4*b^2*x^8 + 3*a^5*b*x^4 + a^6)

________________________________________________________________________________________

Sympy [B]  time = 6.86493, size = 515, normalized size = 8.88 \begin{align*} \frac{45 a^{5} x \Gamma \left (\frac{1}{4}\right )}{64 a^{\frac{33}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{29}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 64 a^{\frac{21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right )} + \frac{117 a^{4} b x^{5} \Gamma \left (\frac{1}{4}\right )}{64 a^{\frac{33}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{29}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 64 a^{\frac{21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right )} + \frac{104 a^{3} b^{2} x^{9} \Gamma \left (\frac{1}{4}\right )}{64 a^{\frac{33}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{29}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 64 a^{\frac{21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right )} + \frac{32 a^{2} b^{3} x^{13} \Gamma \left (\frac{1}{4}\right )}{64 a^{\frac{33}{4}} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{29}{4}} b x^{4} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 192 a^{\frac{25}{4}} b^{2} x^{8} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right ) + 64 a^{\frac{21}{4}} b^{3} x^{12} \sqrt [4]{1 + \frac{b x^{4}}{a}} \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)**(13/4),x)

[Out]

45*a**5*x*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(1
/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*x
**4/a)**(1/4)*gamma(13/4)) + 117*a**4*b*x**5*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*
a**(29/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4)
+ 64*a**(21/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(13/4)) + 104*a**3*b**2*x**9*gamma(1/4)/(64*a**(33/4)*(1
+ b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(25/4)*b**2*
x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(13/4)) + 32*a**2*
b**3*x**13*gamma(1/4)/(64*a**(33/4)*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 192*a**(29/4)*b*x**4*(1 + b*x**4/a)**(
1/4)*gamma(13/4) + 192*a**(25/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(13/4) + 64*a**(21/4)*b**3*x**12*(1 + b*
x**4/a)**(1/4)*gamma(13/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{13}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)^(13/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-13/4), x)