3.1131 \(\int \frac{1}{x^{14} (a+b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=92 \[ \frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}} \]

[Out]

-(a + b*x^4)^(1/4)/(13*a*x^13) + (4*b*(a + b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*(a + b*x^4)^(1/4))/(195*a^3*x^
5) + (128*b^3*(a + b*x^4)^(1/4))/(195*a^4*x)

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Rubi [A]  time = 0.0286821, 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} \[ \frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^14*(a + b*x^4)^(3/4)),x]

[Out]

-(a + b*x^4)^(1/4)/(13*a*x^13) + (4*b*(a + b*x^4)^(1/4))/(39*a^2*x^9) - (32*b^2*(a + b*x^4)^(1/4))/(195*a^3*x^
5) + (128*b^3*(a + b*x^4)^(1/4))/(195*a^4*x)

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{1}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}-\frac{(12 b) \int \frac{1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx}{13 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}+\frac{\left (32 b^2\right ) \int \frac{1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx}{39 a^2}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}-\frac{\left (128 b^3\right ) \int \frac{1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx}{195 a^3}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}\\ \end{align*}

Mathematica [A]  time = 0.0189832, size = 53, normalized size = 0.58 \[ \frac{\sqrt [4]{a+b x^4} \left (20 a^2 b x^4-15 a^3-32 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^14*(a + b*x^4)^(3/4)),x]

[Out]

((a + b*x^4)^(1/4)*(-15*a^3 + 20*a^2*b*x^4 - 32*a*b^2*x^8 + 128*b^3*x^12))/(195*a^4*x^13)

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Maple [A]  time = 0.005, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}-20\,{a}^{2}b{x}^{4}+15\,{a}^{3}}{195\,{x}^{13}{a}^{4}}\sqrt [4]{b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^14/(b*x^4+a)^(3/4),x)

[Out]

-1/195*(b*x^4+a)^(1/4)*(-128*b^3*x^12+32*a*b^2*x^8-20*a^2*b*x^4+15*a^3)/x^13/a^4

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Maxima [A]  time = 0.992936, size = 93, normalized size = 1.01 \begin{align*} \frac{\frac{195 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x} - \frac{117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{65 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} - \frac{15 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{195 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^14/(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

1/195*(195*(b*x^4 + a)^(1/4)*b^3/x - 117*(b*x^4 + a)^(5/4)*b^2/x^5 + 65*(b*x^4 + a)^(9/4)*b/x^9 - 15*(b*x^4 +
a)^(13/4)/x^13)/a^4

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Fricas [A]  time = 1.4831, size = 122, normalized size = 1.33 \begin{align*} \frac{{\left (128 \, b^{3} x^{12} - 32 \, a b^{2} x^{8} + 20 \, a^{2} b x^{4} - 15 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{195 \, a^{4} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^14/(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/195*(128*b^3*x^12 - 32*a*b^2*x^8 + 20*a^2*b*x^4 - 15*a^3)*(b*x^4 + a)^(1/4)/(a^4*x^13)

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Sympy [B]  time = 6.76619, size = 692, normalized size = 7.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**14/(b*x**4+a)**(3/4),x)

[Out]

-45*a**6*b**(37/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16
*gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 75*a**5*b**(41/4)*x**4*(a/(
b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*
b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 51*a**4*b**(45/4)*x**8*(a/(b*x**4) + 1)**(1/4)*gam
ma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4)
+ 256*a**4*b**12*x**24*gamma(3/4)) + 231*a**3*b**(49/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b
**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**2
4*gamma(3/4)) + 924*a**2*b**(53/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4)
+ 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 1056*
a*b**(57/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*
gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 384*b**(61/4)*x**24*(a/(b*x*
*4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**1
1*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4))

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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{3}{4}} x^{14}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^14/(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^14), x)