3.1039 \(\int \frac{(a+b x^4)^{3/4}}{x^{20}} \, dx\)

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 = 0.0277969, 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, Rules used = {271, 264} \[ \frac{128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \]

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)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx &=-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}}-\frac{(12 b) \int \frac{\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx}{19 a}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}+\frac{\left (32 b^2\right ) \int \frac{\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{95 a^2}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}-\frac{\left (128 b^3\right ) \int \frac{\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{1045 a^3}\\ &=-\frac{\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac{4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac{32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac{128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0133863, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^4\right )^{7/4} \left (308 a^2 b x^4-385 a^3-224 a b^2 x^8+128 b^3 x^{12}\right )}{7315 a^4 x^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^20,x]

[Out]

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

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Maple [A]  time = 0.006, size = 50, normalized size = 0.5 \begin{align*} -{\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}}}} \end{align*}

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 = 0.972672, size = 93, normalized size = 1.01 \begin{align*} \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}} \end{align*}

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 - 3
85*(b*x^4 + a)^(19/4)/x^19)/a^4

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Fricas [A]  time = 1.76956, size = 149, normalized size = 1.62 \begin{align*} \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}} \end{align*}

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 [B]  time = 22.6397, size = 847, normalized size = 9.21 \begin{align*} \text{result too large to display} \end{align*}

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*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)) - 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*gamma(-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*gamma(-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*gamma(-3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{20}}\,{d x} \end{align*}

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)