3.1037 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx\)

Optimal. Leaf size=92 \[ \frac{128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \]

[Out]

-(a + b*x^4)^(7/4)/(19*a*x^19) + (4*b*(a + b*x^4)^(7/4))/(95*a^2*x^15) - (32*b^2
*(a + b*x^4)^(7/4))/(1045*a^3*x^11) + (128*b^3*(a + b*x^4)^(7/4))/(7315*a^4*x^7)

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Rubi [A]  time = 0.0904368, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^20,x]

[Out]

-(a + b*x^4)^(7/4)/(19*a*x^19) + (4*b*(a + b*x^4)^(7/4))/(95*a^2*x^15) - (32*b^2
*(a + b*x^4)^(7/4))/(1045*a^3*x^11) + (128*b^3*(a + b*x^4)^(7/4))/(7315*a^4*x^7)

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Rubi in Sympy [A]  time = 9.77648, size = 85, normalized size = 0.92 \[ - \frac{\left (a + b x^{4}\right )^{\frac{7}{4}}}{19 a x^{19}} + \frac{4 b \left (a + b x^{4}\right )^{\frac{7}{4}}}{95 a^{2} x^{15}} - \frac{32 b^{2} \left (a + b x^{4}\right )^{\frac{7}{4}}}{1045 a^{3} x^{11}} + \frac{128 b^{3} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7315 a^{4} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**20,x)

[Out]

-(a + b*x**4)**(7/4)/(19*a*x**19) + 4*b*(a + b*x**4)**(7/4)/(95*a**2*x**15) - 32
*b**2*(a + b*x**4)**(7/4)/(1045*a**3*x**11) + 128*b**3*(a + b*x**4)**(7/4)/(7315
*a**4*x**7)

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Mathematica [A]  time = 0.0401515, size = 64, normalized size = 0.7 \[ -\frac{\left (a+b x^4\right )^{3/4} \left (385 a^4+77 a^3 b x^4-84 a^2 b^2 x^8+96 a b^3 x^{12}-128 b^4 x^{16}\right )}{7315 a^4 x^{19}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^20,x]

[Out]

-((a + b*x^4)^(3/4)*(385*a^4 + 77*a^3*b*x^4 - 84*a^2*b^2*x^8 + 96*a*b^3*x^12 - 1
28*b^4*x^16))/(7315*a^4*x^19)

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Maple [A]  time = 0.01, size = 50, normalized size = 0.5 \[ -{\frac{-128\,{b}^{3}{x}^{12}+224\,a{b}^{2}{x}^{8}-308\,{a}^{2}b{x}^{4}+385\,{a}^{3}}{7315\,{x}^{19}{a}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^20,x)

[Out]

-1/7315*(b*x^4+a)^(7/4)*(-128*b^3*x^12+224*a*b^2*x^8-308*a^2*b*x^4+385*a^3)/x^19
/a^4

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Maxima [A]  time = 1.43583, size = 93, normalized size = 1.01 \[ \frac{\frac{1045 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b^{3}}{x^{7}} - \frac{1995 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} b^{2}}{x^{11}} + \frac{1463 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} b}{x^{15}} - \frac{385 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{x^{19}}}{7315 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^20,x, algorithm="maxima")

[Out]

1/7315*(1045*(b*x^4 + a)^(7/4)*b^3/x^7 - 1995*(b*x^4 + a)^(11/4)*b^2/x^11 + 1463
*(b*x^4 + a)^(15/4)*b/x^15 - 385*(b*x^4 + a)^(19/4)/x^19)/a^4

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Fricas [A]  time = 0.32024, size = 81, normalized size = 0.88 \[ \frac{{\left (128 \, b^{4} x^{16} - 96 \, a b^{3} x^{12} + 84 \, a^{2} b^{2} x^{8} - 77 \, a^{3} b x^{4} - 385 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{7315 \, a^{4} x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^20,x, algorithm="fricas")

[Out]

1/7315*(128*b^4*x^16 - 96*a*b^3*x^12 + 84*a^2*b^2*x^8 - 77*a^3*b*x^4 - 385*a^4)*
(b*x^4 + a)^(3/4)/(a^4*x^19)

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Sympy [A]  time = 65.7367, size = 847, normalized size = 9.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**20,x)

[Out]

-1155*a**7*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*g
amma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4)
+ 256*a**4*b**12*x**28*gamma(-3/4)) - 3696*a**6*b**(43/4)*x**4*(a/(b*x**4) + 1)
**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gam
ma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4))
- 3906*a**5*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x
**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma
(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) - 1380*a**4*b**(51/4)*x**12*(a/(b*x**
4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x*
*20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(
-3/4)) + 45*a**3*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*
b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24
*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) + 540*a**2*b**(59/4)*x**20*(a/(
b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**
10*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*g
amma(-3/4)) + 864*a*b**(63/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(256*a*
*7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10*x**20*gamma(-3/4) + 768*a**5*b**11*x*
*24*gamma(-3/4) + 256*a**4*b**12*x**28*gamma(-3/4)) + 384*b**(67/4)*x**28*(a/(b*
x**4) + 1)**(3/4)*gamma(-19/4)/(256*a**7*b**9*x**16*gamma(-3/4) + 768*a**6*b**10
*x**20*gamma(-3/4) + 768*a**5*b**11*x**24*gamma(-3/4) + 256*a**4*b**12*x**28*gam
ma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{20}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^20,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^20, x)