3.12.2 \(\int \frac {1}{x^{20} \sqrt [4]{a+b x^4}} \, dx\) [1102]

Optimal. Leaf size=116 \[ -\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {2048 b^4 \left (a+b x^4\right )^{3/4}}{21945 a^5 x^3} \]

[Out]

-1/19*(b*x^4+a)^(3/4)/a/x^19+16/285*b*(b*x^4+a)^(3/4)/a^2/x^15-64/1045*b^2*(b*x^4+a)^(3/4)/a^3/x^11+512/7315*b
^3*(b*x^4+a)^(3/4)/a^4/x^7-2048/21945*b^4*(b*x^4+a)^(3/4)/a^5/x^3

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Rubi [A]
time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {2048 b^4 \left (a+b x^4\right )^{3/4}}{21945 a^5 x^3}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^20*(a + b*x^4)^(1/4)),x]

[Out]

-1/19*(a + b*x^4)^(3/4)/(a*x^19) + (16*b*(a + b*x^4)^(3/4))/(285*a^2*x^15) - (64*b^2*(a + b*x^4)^(3/4))/(1045*
a^3*x^11) + (512*b^3*(a + b*x^4)^(3/4))/(7315*a^4*x^7) - (2048*b^4*(a + b*x^4)^(3/4))/(21945*a^5*x^3)

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^{20} \sqrt [4]{a+b x^4}} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}-\frac {(16 b) \int \frac {1}{x^{16} \sqrt [4]{a+b x^4}} \, dx}{19 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}+\frac {\left (64 b^2\right ) \int \frac {1}{x^{12} \sqrt [4]{a+b x^4}} \, dx}{95 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}-\frac {\left (512 b^3\right ) \int \frac {1}{x^8 \sqrt [4]{a+b x^4}} \, dx}{1045 a^3}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}+\frac {\left (2048 b^4\right ) \int \frac {1}{x^4 \sqrt [4]{a+b x^4}} \, dx}{7315 a^4}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{19 a x^{19}}+\frac {16 b \left (a+b x^4\right )^{3/4}}{285 a^2 x^{15}}-\frac {64 b^2 \left (a+b x^4\right )^{3/4}}{1045 a^3 x^{11}}+\frac {512 b^3 \left (a+b x^4\right )^{3/4}}{7315 a^4 x^7}-\frac {2048 b^4 \left (a+b x^4\right )^{3/4}}{21945 a^5 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 64, normalized size = 0.55 \begin {gather*} \frac {\left (a+b x^4\right )^{3/4} \left (-1155 a^4+1232 a^3 b x^4-1344 a^2 b^2 x^8+1536 a b^3 x^{12}-2048 b^4 x^{16}\right )}{21945 a^5 x^{19}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^20*(a + b*x^4)^(1/4)),x]

[Out]

((a + b*x^4)^(3/4)*(-1155*a^4 + 1232*a^3*b*x^4 - 1344*a^2*b^2*x^8 + 1536*a*b^3*x^12 - 2048*b^4*x^16))/(21945*a
^5*x^19)

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Maple [A]
time = 0.17, size = 61, normalized size = 0.53

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\)	\(61\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\)	\(61\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (2048 x^{16} b^{4}-1536 a \,b^{3} x^{12}+1344 a^{2} b^{2} x^{8}-1232 a^{3} b \,x^{4}+1155 a^{4}\right )}{21945 x^{19} a^{5}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^20/(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/21945*(b*x^4+a)^(3/4)*(2048*b^4*x^16-1536*a*b^3*x^12+1344*a^2*b^2*x^8-1232*a^3*b*x^4+1155*a^4)/x^19/a^5

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Maxima [A]
time = 0.29, size = 86, normalized size = 0.74 \begin {gather*} -\frac {\frac {7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{4}}{x^{3}} - \frac {12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} + \frac {11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} - \frac {5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} + \frac {1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{21945 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^20/(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/21945*(7315*(b*x^4 + a)^(3/4)*b^4/x^3 - 12540*(b*x^4 + a)^(7/4)*b^3/x^7 + 11970*(b*x^4 + a)^(11/4)*b^2/x^11
- 5852*(b*x^4 + a)^(15/4)*b/x^15 + 1155*(b*x^4 + a)^(19/4)/x^19)/a^5

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Fricas [A]
time = 0.38, size = 60, normalized size = 0.52 \begin {gather*} -\frac {{\left (2048 \, b^{4} x^{16} - 1536 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1232 \, a^{3} b x^{4} + 1155 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, a^{5} x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^20/(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/21945*(2048*b^4*x^16 - 1536*a*b^3*x^12 + 1344*a^2*b^2*x^8 - 1232*a^3*b*x^4 + 1155*a^4)*(b*x^4 + a)^(3/4)/(a
^5*x^19)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (109) = 218\).
time = 2.16, size = 1046, normalized size = 9.02 \begin {gather*} \frac {3465 a^{8} b^{\frac {67}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {10164 a^{7} b^{\frac {71}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {10038 a^{6} b^{\frac {75}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {3204 a^{5} b^{\frac {79}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {585 a^{4} b^{\frac {83}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {9360 a^{3} b^{\frac {87}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {22464 a^{2} b^{\frac {91}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {19968 a b^{\frac {95}{4}} x^{28} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} + \frac {6144 b^{\frac {99}{4}} x^{32} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{1024 a^{9} b^{16} x^{16} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{8} b^{17} x^{20} \Gamma \left (\frac {1}{4}\right ) + 6144 a^{7} b^{18} x^{24} \Gamma \left (\frac {1}{4}\right ) + 4096 a^{6} b^{19} x^{28} \Gamma \left (\frac {1}{4}\right ) + 1024 a^{5} b^{20} x^{32} \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**20/(b*x**4+a)**(1/4),x)

[Out]

3465*a**8*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x
**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*
gamma(1/4)) + 10164*a**7*b**(71/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4)
+ 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 10
24*a**5*b**20*x**32*gamma(1/4)) + 10038*a**6*b**(75/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b*
*16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x
**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 3204*a**5*b**(79/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(
-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4)
+ 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 585*a**4*b**(83/4)*x**16*(a/(b*x**4)
+ 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**
18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 9360*a**3*b**(87/
4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(
1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4))
+ 22464*a**2*b**(91/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*gamma(1/4) + 4096*a*
*8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1/4) + 1024*a**5*b*
*20*x**32*gamma(1/4)) + 19968*a*b**(95/4)*x**28*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9*b**16*x**16*ga
mma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**19*x**28*gamma(1
/4) + 1024*a**5*b**20*x**32*gamma(1/4)) + 6144*b**(99/4)*x**32*(a/(b*x**4) + 1)**(3/4)*gamma(-19/4)/(1024*a**9
*b**16*x**16*gamma(1/4) + 4096*a**8*b**17*x**20*gamma(1/4) + 6144*a**7*b**18*x**24*gamma(1/4) + 4096*a**6*b**1
9*x**28*gamma(1/4) + 1024*a**5*b**20*x**32*gamma(1/4))

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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^20/(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^20), x)

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Mupad [B]
time = 1.49, size = 96, normalized size = 0.83 \begin {gather*} \frac {16\,b\,{\left (b\,x^4+a\right )}^{3/4}}{285\,a^2\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,a\,x^{19}}-\frac {2048\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{21945\,a^5\,x^3}+\frac {512\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^7}-\frac {64\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^3\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^20*(a + b*x^4)^(1/4)),x)

[Out]

(16*b*(a + b*x^4)^(3/4))/(285*a^2*x^15) - (a + b*x^4)^(3/4)/(19*a*x^19) - (2048*b^4*(a + b*x^4)^(3/4))/(21945*
a^5*x^3) + (512*b^3*(a + b*x^4)^(3/4))/(7315*a^4*x^7) - (64*b^2*(a + b*x^4)^(3/4))/(1045*a^3*x^11)

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