3.1174 \(\int \frac{1}{(a+b x^4)^{17/4}} \, dx\)
Optimal. Leaf size=77 \[ \frac{128 x}{195 a^4 \sqrt [4]{a+b x^4}}+\frac{32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{x}{13 a \left (a+b x^4\right )^{13/4}} \]
[Out]
x/(13*a*(a + b*x^4)^(13/4)) + (4*x)/(39*a^2*(a + b*x^4)^(9/4)) + (32*x)/(195*a^3*(a + b*x^4)^(5/4)) + (128*x)/
(195*a^4*(a + b*x^4)^(1/4))
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Rubi [A] time = 0.0159634, antiderivative size = 77, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used =
{192, 191} \[ \frac{128 x}{195 a^4 \sqrt [4]{a+b x^4}}+\frac{32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{x}{13 a \left (a+b x^4\right )^{13/4}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^4)^(-17/4),x]
[Out]
x/(13*a*(a + b*x^4)^(13/4)) + (4*x)/(39*a^2*(a + b*x^4)^(9/4)) + (32*x)/(195*a^3*(a + b*x^4)^(5/4)) + (128*x)/
(195*a^4*(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 )^{17/4}} \, dx &=\frac{x}{13 a \left (a+b x^4\right )^{13/4}}+\frac{12 \int \frac{1}{\left (a+b x^4\right )^{13/4}} \, dx}{13 a}\\ &=\frac{x}{13 a \left (a+b x^4\right )^{13/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{32 \int \frac{1}{\left (a+b x^4\right )^{9/4}} \, dx}{39 a^2}\\ &=\frac{x}{13 a \left (a+b x^4\right )^{13/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac{128 \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{195 a^3}\\ &=\frac{x}{13 a \left (a+b x^4\right )^{13/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac{128 x}{195 a^4 \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [A] time = 0.0154447, size = 51, normalized size = 0.66 \[ \frac{x \left (468 a^2 b x^4+195 a^3+416 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 \left (a+b x^4\right )^{13/4}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^4)^(-17/4),x]
[Out]
(x*(195*a^3 + 468*a^2*b*x^4 + 416*a*b^2*x^8 + 128*b^3*x^12))/(195*a^4*(a + b*x^4)^(13/4))
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Maple [A] time = 0.003, size = 48, normalized size = 0.6 \begin{align*}{\frac{x \left ( 128\,{b}^{3}{x}^{12}+416\,a{b}^{2}{x}^{8}+468\,{a}^{2}b{x}^{4}+195\,{a}^{3} \right ) }{195\,{a}^{4}} \left ( b{x}^{4}+a \right ) ^{-{\frac{13}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x^4+a)^(17/4),x)
[Out]
1/195*x*(128*b^3*x^12+416*a*b^2*x^8+468*a^2*b*x^4+195*a^3)/(b*x^4+a)^(13/4)/a^4
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Maxima [A] time = 1.38891, size = 90, normalized size = 1.17 \begin{align*} -\frac{{\left (15 \, b^{3} - \frac{65 \,{\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac{117 \,{\left (b x^{4} + a\right )}^{2} b}{x^{8}} - \frac{195 \,{\left (b x^{4} + a\right )}^{3}}{x^{12}}\right )} x^{13}}{195 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^4+a)^(17/4),x, algorithm="maxima")
[Out]
-1/195*(15*b^3 - 65*(b*x^4 + a)*b^2/x^4 + 117*(b*x^4 + a)^2*b/x^8 - 195*(b*x^4 + a)^3/x^12)*x^13/((b*x^4 + a)^
(13/4)*a^4)
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Fricas [A] time = 1.48307, size = 205, normalized size = 2.66 \begin{align*} \frac{{\left (128 \, b^{3} x^{13} + 416 \, a b^{2} x^{9} + 468 \, a^{2} b x^{5} + 195 \, a^{3} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{195 \,{\left (a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{12} + 6 \, a^{6} b^{2} x^{8} + 4 \, a^{7} b x^{4} + a^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^4+a)^(17/4),x, algorithm="fricas")
[Out]
1/195*(128*b^3*x^13 + 416*a*b^2*x^9 + 468*a^2*b*x^5 + 195*a^3*x)*(b*x^4 + a)^(3/4)/(a^4*b^4*x^16 + 4*a^5*b^3*x
^12 + 6*a^6*b^2*x^8 + 4*a^7*b*x^4 + a^8)
________________________________________________________________________________________
Sympy [B] time = 34.9791, size = 1550, normalized size = 20.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x**4+a)**(17/4),x)
[Out]
585*a**14*x*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(
1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4
)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) +
3159*a**13*b*x**5*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*
x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*
x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a
**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(1
7/4)) + 7215*a**12*b**2*x**9*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x*
*4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(6
1/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4
) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/
4)*gamma(17/4)) + 8925*a**11*b**3*x**13*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**
(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*gamma(17/4) +
5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**4/a)**(1/4)
*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**24*(1 + b*x
**4/a)**(1/4)*gamma(17/4)) + 6300*a**10*b**4*x**17*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*gamma(17/4)
+ 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)**(1/4)*ga
mma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*(1 + b*x**
4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4)*b**6*x**
24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) + 2400*a**9*b**5*x**21*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)**(1/4)*g
amma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b**4*x**16*
(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256*a**(49/4
)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4)) + 384*a**8*b**6*x**25*gamma(1/4)/(256*a**(73/4)*(1 + b*x**4/a)
**(1/4)*gamma(17/4) + 1536*a**(69/4)*b*x**4*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(65/4)*b**2*x**8*(1 +
b*x**4/a)**(1/4)*gamma(17/4) + 5120*a**(61/4)*b**3*x**12*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 3840*a**(57/4)*b*
*4*x**16*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 1536*a**(53/4)*b**5*x**20*(1 + b*x**4/a)**(1/4)*gamma(17/4) + 256
*a**(49/4)*b**6*x**24*(1 + b*x**4/a)**(1/4)*gamma(17/4))
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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{17}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x^4+a)^(17/4),x, algorithm="giac")
[Out]
integrate((b*x^4 + a)^(-17/4), x)