3.1005 \(\int \frac{\sqrt [4]{a+b x^4}}{x^{14}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}} \]

[Out]

-(a + b*x^4)^(5/4)/(13*a*x^13) + (8*b*(a + b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2
*(a + b*x^4)^(5/4))/(585*a^3*x^5)

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Rubi [A]  time = 0.0629874, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{32 b^2 \left (a+b x^4\right )^{5/4}}{585 a^3 x^5}+\frac{8 b \left (a+b x^4\right )^{5/4}}{117 a^2 x^9}-\frac{\left (a+b x^4\right )^{5/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^14,x]

[Out]

-(a + b*x^4)^(5/4)/(13*a*x^13) + (8*b*(a + b*x^4)^(5/4))/(117*a^2*x^9) - (32*b^2
*(a + b*x^4)^(5/4))/(585*a^3*x^5)

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Rubi in Sympy [A]  time = 6.66644, size = 61, normalized size = 0.9 \[ - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{13 a x^{13}} + \frac{8 b \left (a + b x^{4}\right )^{\frac{5}{4}}}{117 a^{2} x^{9}} - \frac{32 b^{2} \left (a + b x^{4}\right )^{\frac{5}{4}}}{585 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x**14,x)

[Out]

-(a + b*x**4)**(5/4)/(13*a*x**13) + 8*b*(a + b*x**4)**(5/4)/(117*a**2*x**9) - 32
*b**2*(a + b*x**4)**(5/4)/(585*a**3*x**5)

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Mathematica [A]  time = 0.0296195, size = 53, normalized size = 0.78 \[ -\frac{\sqrt [4]{a+b x^4} \left (45 a^3+5 a^2 b x^4-8 a b^2 x^8+32 b^3 x^{12}\right )}{585 a^3 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^14,x]

[Out]

-((a + b*x^4)^(1/4)*(45*a^3 + 5*a^2*b*x^4 - 8*a*b^2*x^8 + 32*b^3*x^12))/(585*a^3
*x^13)

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Maple [A]  time = 0.009, size = 39, normalized size = 0.6 \[ -{\frac{32\,{b}^{2}{x}^{8}-40\,ab{x}^{4}+45\,{a}^{2}}{585\,{x}^{13}{a}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^14,x)

[Out]

-1/585*(b*x^4+a)^(5/4)*(32*b^2*x^8-40*a*b*x^4+45*a^2)/x^13/a^3

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Maxima [A]  time = 1.43925, size = 70, normalized size = 1.03 \[ -\frac{\frac{117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} - \frac{130 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} + \frac{45 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{585 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^14,x, algorithm="maxima")

[Out]

-1/585*(117*(b*x^4 + a)^(5/4)*b^2/x^5 - 130*(b*x^4 + a)^(9/4)*b/x^9 + 45*(b*x^4
+ a)^(13/4)/x^13)/a^3

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Fricas [A]  time = 0.323688, size = 66, normalized size = 0.97 \[ -\frac{{\left (32 \, b^{3} x^{12} - 8 \, a b^{2} x^{8} + 5 \, a^{2} b x^{4} + 45 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{585 \, a^{3} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^14,x, algorithm="fricas")

[Out]

-1/585*(32*b^3*x^12 - 8*a*b^2*x^8 + 5*a^2*b*x^4 + 45*a^3)*(b*x^4 + a)^(1/4)/(a^3
*x^13)

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Sympy [A]  time = 15.7905, size = 520, normalized size = 7.65 \[ \frac{45 a^{5} b^{\frac{17}{4}} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{95 a^{4} b^{\frac{21}{4}} x^{4} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{47 a^{3} b^{\frac{25}{4}} x^{8} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{21 a^{2} b^{\frac{29}{4}} x^{12} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{56 a b^{\frac{33}{4}} x^{16} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} + \frac{32 b^{\frac{37}{4}} x^{20} \sqrt [4]{\frac{a}{b x^{4}} + 1} \Gamma \left (- \frac{13}{4}\right )}{64 a^{5} b^{4} x^{12} \Gamma \left (- \frac{1}{4}\right ) + 128 a^{4} b^{5} x^{16} \Gamma \left (- \frac{1}{4}\right ) + 64 a^{3} b^{6} x^{20} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x**14,x)

[Out]

45*a**5*b**(17/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gamma
(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 95*
a**4*b**(21/4)*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*gam
ma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) + 4
7*a**3*b**(25/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*g
amma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)) +
21*a**2*b**(29/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**1
2*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)
) + 56*a*b**(33/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**1
2*gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4)
) + 32*b**(37/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(64*a**5*b**4*x**12*
gamma(-1/4) + 128*a**4*b**5*x**16*gamma(-1/4) + 64*a**3*b**6*x**20*gamma(-1/4))

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GIAC/XCAS [A]  time = 0.222613, size = 143, normalized size = 2.1 \[ -\frac{\frac{117 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{2}}{x} - \frac{130 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{9}} + \frac{45 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{13}}}{585 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^14,x, algorithm="giac")

[Out]

-1/585*(117*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^2/x - 130*(b^2*x^8 + 2*a*b*x^4 + a^2
)*(b*x^4 + a)^(1/4)*b/x^9 + 45*(b^3*x^12 + 3*a*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x
^4 + a)^(1/4)/x^13)/a^3