3.12.31 \(\int \frac {1}{x^{14} (a+b x^4)^{3/4}} \, dx\) [1131]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^14*(a + b*x^4)^(3/4)),x]

[Out]

-1/13*(a + b*x^4)^(1/4)/(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 270

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]

Rule 277

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]

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*}

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Mathematica [A]
time = 0.33, size = 53, normalized size = 0.58 \begin {gather*} \frac {\sqrt [4]{a+b x^4} \left (-15 a^3+20 a^2 b x^4-32 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 x^{13}} \end {gather*}

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 = 0.17, size = 50, normalized size = 0.54

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-128 b^{3} x^{12}+32 a \,b^{2} x^{8}-20 a^{2} b \,x^{4}+15 a^{3}\right )}{195 x^{13} a^{4}}\)	\(50\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-128 b^{3} x^{12}+32 a \,b^{2} x^{8}-20 a^{2} b \,x^{4}+15 a^{3}\right )}{195 x^{13} a^{4}}\)	\(50\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-128 b^{3} x^{12}+32 a \,b^{2} x^{8}-20 a^{2} b \,x^{4}+15 a^{3}\right )}{195 x^{13} a^{4}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 69, normalized size = 0.75 \begin {gather*} \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 {gather*}

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 = 0.38, size = 49, normalized size = 0.53 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (83) = 166\).
time = 1.38, size = 692, normalized size = 7.52 \begin {gather*} - \frac {45 a^{6} b^{\frac {37}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} - \frac {75 a^{5} b^{\frac {41}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} - \frac {51 a^{4} b^{\frac {45}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} + \frac {231 a^{3} b^{\frac {49}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} + \frac {924 a^{2} b^{\frac {53}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} + \frac {1056 a b^{\frac {57}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} + \frac {384 b^{\frac {61}{4}} x^{24} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {13}{4}\right )}{256 a^{7} b^{9} x^{12} \Gamma \left (\frac {3}{4}\right ) + 768 a^{6} b^{10} x^{16} \Gamma \left (\frac {3}{4}\right ) + 768 a^{5} b^{11} x^{20} \Gamma \left (\frac {3}{4}\right ) + 256 a^{4} b^{12} x^{24} \Gamma \left (\frac {3}{4}\right )} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 1.36, size = 76, normalized size = 0.83 \begin {gather*} \frac {4\,b\,{\left (b\,x^4+a\right )}^{1/4}}{39\,a^2\,x^9}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{13\,a\,x^{13}}+\frac {128\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{195\,a^4\,x}-\frac {32\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{195\,a^3\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^14*(a + b*x^4)^(3/4)),x)

[Out]

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

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