3.1160 \(\int \frac{1}{x^{12} (a+b x^4)^{5/4}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(11*a*x^11*(a + b*x^4)^(1/4)) + (12*b)/(77*a^2*x^7*(a + b*x^4)^(1/4)) - (32*b^2)/(77*a^3*x^3*(a + b*x^4)^(1
/4)) - (128*b^3*x)/(77*a^4*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0260121, antiderivative size = 90, 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, 191} \[ -\frac{128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

-1/(11*a*x^11*(a + b*x^4)^(1/4)) + (12*b)/(77*a^2*x^7*(a + b*x^4)^(1/4)) - (32*b^2)/(77*a^3*x^3*(a + b*x^4)^(1
/4)) - (128*b^3*x)/(77*a^4*(a + b*x^4)^(1/4))

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 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}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}}-\frac{(12 b) \int \frac{1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx}{11 a}\\ &=-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}+\frac{\left (96 b^2\right ) \int \frac{1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{77 a^2}\\ &=-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}-\frac{\left (128 b^3\right ) \int \frac{1}{\left (a+b x^4\right )^{5/4}} \, dx}{77 a^3}\\ &=-\frac{1}{11 a x^{11} \sqrt [4]{a+b x^4}}+\frac{12 b}{77 a^2 x^7 \sqrt [4]{a+b x^4}}-\frac{32 b^2}{77 a^3 x^3 \sqrt [4]{a+b x^4}}-\frac{128 b^3 x}{77 a^4 \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0088963, size = 53, normalized size = 0.59 \[ -\frac{-12 a^2 b x^4+7 a^3+32 a b^2 x^8+128 b^3 x^{12}}{77 a^4 x^{11} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^12*(a + b*x^4)^(5/4)),x]

[Out]

-(7*a^3 - 12*a^2*b*x^4 + 32*a*b^2*x^8 + 128*b^3*x^12)/(77*a^4*x^11*(a + b*x^4)^(1/4))

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Maple [A]  time = 0.005, size = 50, normalized size = 0.6 \begin{align*} -{\frac{128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}-12\,{a}^{2}b{x}^{4}+7\,{a}^{3}}{77\,{x}^{11}{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^12/(b*x^4+a)^(5/4),x)

[Out]

-1/77*(128*b^3*x^12+32*a*b^2*x^8-12*a^2*b*x^4+7*a^3)/x^11/(b*x^4+a)^(1/4)/a^4

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Maxima [A]  time = 1.00044, size = 96, normalized size = 1.07 \begin{align*} -\frac{b^{3} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}} - \frac{\frac{77 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} b^{2}}{x^{3}} - \frac{33 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} b}{x^{7}} + \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{x^{11}}}{77 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

-b^3*x/((b*x^4 + a)^(1/4)*a^4) - 1/77*(77*(b*x^4 + a)^(3/4)*b^2/x^3 - 33*(b*x^4 + a)^(7/4)*b/x^7 + 7*(b*x^4 +
a)^(11/4)/x^11)/a^4

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Fricas [A]  time = 1.51716, size = 138, normalized size = 1.53 \begin{align*} -\frac{{\left (128 \, b^{3} x^{12} + 32 \, a b^{2} x^{8} - 12 \, a^{2} b x^{4} + 7 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \,{\left (a^{4} b x^{15} + a^{5} x^{11}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

-1/77*(128*b^3*x^12 + 32*a*b^2*x^8 - 12*a^2*b*x^4 + 7*a^3)*(b*x^4 + a)^(3/4)/(a^4*b*x^15 + a^5*x^11)

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Sympy [B]  time = 5.31332, size = 592, normalized size = 6.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**12/(b*x**4+a)**(5/4),x)

[Out]

21*a**5*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*g
amma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 6*a**4*b**(43/4)*x**4*(a/(b*x
**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**1
1*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 45*a**3*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-
11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256
*a**4*b**12*x**20*gamma(5/4)) + 540*a**2*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x
**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamm
a(5/4)) + 864*a*b**(55/4)*x**16*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6
*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 384*b**(59/4)*x
**20*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 7
68*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4))

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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{5}{4}} x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^12/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^12), x)