3.1172 \(\int \frac{1}{\left (a+b x^4\right )^{17/4}} \, dx\)

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 = 0.0446786, 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 \[ \frac{128 x}{195 a^4 \sqrt [4]{a+b x^4}}+\frac{32 x}{195 a^3 \left (a+b x^4\right )^{5/4}}+\frac{4 x}{39 a^2 \left (a+b x^4\right )^{9/4}}+\frac{x}{13 a \left (a+b x^4\right )^{13/4}} \]

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))

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Rubi in Sympy [A]  time = 4.94269, size = 70, normalized size = 0.91 \[ \frac{x}{13 a \left (a + b x^{4}\right )^{\frac{13}{4}}} + \frac{4 x}{39 a^{2} \left (a + b x^{4}\right )^{\frac{9}{4}}} + \frac{32 x}{195 a^{3} \left (a + b x^{4}\right )^{\frac{5}{4}}} + \frac{128 x}{195 a^{4} \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(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))

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Mathematica [A]  time = 0.034681, size = 51, normalized size = 0.66 \[ \frac{x \left (195 a^3+468 a^2 b x^4+416 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 \left (a+b x^4\right )^{13/4}} \]

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 = 0.008, size = 48, normalized size = 0.6 \[{\frac{x \left ( 128\,{b}^{3}{x}^{12}+416\,a{b}^{2}{x}^{8}+468\,{a}^{2}b{x}^{4}+195\,{a}^{3} \right ) }{195\,{a}^{4}} \left ( b{x}^{4}+a \right ) ^{-{\frac{13}{4}}}} \]

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 = 1.43924, size = 90, normalized size = 1.17 \[ -\frac{{\left (15 \, b^{3} - \frac{65 \,{\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac{117 \,{\left (b x^{4} + a\right )}^{2} b}{x^{8}} - \frac{195 \,{\left (b x^{4} + a\right )}^{3}}{x^{12}}\right )} x^{13}}{195 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 = 0.244398, size = 123, normalized size = 1.6 \[ \frac{{\left (128 \, b^{3} x^{13} + 416 \, a b^{2} x^{9} + 468 \, a^{2} b x^{5} + 195 \, a^{3} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{195 \,{\left (a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{12} + 6 \, a^{6} b^{2} x^{8} + 4 \, a^{7} b x^{4} + a^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)

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Sympy [A]  time = 114.815, size = 1550, normalized size = 20.13 \[ \text{result too large to display} \]

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) + 153
6*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**(7
3/4)*(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)*gam
ma(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)) + 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)*gam
ma(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)) +
2400*a**9*b**5*x**21*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)) + 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)*gam
ma(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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{17}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(-17/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-17/4), x)