3.12.61 \(\int \frac {1}{x^{16} (a+b x^4)^{5/4}} \, dx\) [1161]

Optimal. Leaf size=114 \[ -\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac {64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac {512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac {2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}} \]

[Out]

-1/15/a/x^15/(b*x^4+a)^(1/4)+16/165*b/a^2/x^11/(b*x^4+a)^(1/4)-64/385*b^2/a^3/x^7/(b*x^4+a)^(1/4)+512/1155*b^3
/a^4/x^3/(b*x^4+a)^(1/4)+2048/1155*b^4*x/a^5/(b*x^4+a)^(1/4)

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Rubi [A]
time = 0.03, antiderivative size = 114, 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, 197} \begin {gather*} \frac {2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}+\frac {512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}-\frac {64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

-1/15*1/(a*x^15*(a + b*x^4)^(1/4)) + (16*b)/(165*a^2*x^11*(a + b*x^4)^(1/4)) - (64*b^2)/(385*a^3*x^7*(a + b*x^
4)^(1/4)) + (512*b^3)/(1155*a^4*x^3*(a + b*x^4)^(1/4)) + (2048*b^4*x)/(1155*a^5*(a + b*x^4)^(1/4))

Rule 197

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]

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^{16} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}-\frac {(16 b) \int \frac {1}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx}{15 a}\\ &=-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}+\frac {\left (64 b^2\right ) \int \frac {1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx}{55 a^2}\\ &=-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac {64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}-\frac {\left (512 b^3\right ) \int \frac {1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{385 a^3}\\ &=-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac {64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac {512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac {\left (2048 b^4\right ) \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{1155 a^4}\\ &=-\frac {1}{15 a x^{15} \sqrt [4]{a+b x^4}}+\frac {16 b}{165 a^2 x^{11} \sqrt [4]{a+b x^4}}-\frac {64 b^2}{385 a^3 x^7 \sqrt [4]{a+b x^4}}+\frac {512 b^3}{1155 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac {2048 b^4 x}{1155 a^5 \sqrt [4]{a+b x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.63, size = 64, normalized size = 0.56 \begin {gather*} \frac {-77 a^4+112 a^3 b x^4-192 a^2 b^2 x^8+512 a b^3 x^{12}+2048 b^4 x^{16}}{1155 a^5 x^{15} \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^16*(a + b*x^4)^(5/4)),x]

[Out]

(-77*a^4 + 112*a^3*b*x^4 - 192*a^2*b^2*x^8 + 512*a*b^3*x^12 + 2048*b^4*x^16)/(1155*a^5*x^15*(a + b*x^4)^(1/4))

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Maple [A]
time = 0.18, size = 61, normalized size = 0.54

method	result	size
gosper	\(-\frac {-2048 x^{16} b^{4}-512 a \,b^{3} x^{12}+192 a^{2} b^{2} x^{8}-112 a^{3} b \,x^{4}+77 a^{4}}{1155 x^{15} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{5}}\)	\(61\)
trager	\(-\frac {-2048 x^{16} b^{4}-512 a \,b^{3} x^{12}+192 a^{2} b^{2} x^{8}-112 a^{3} b \,x^{4}+77 a^{4}}{1155 x^{15} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{5}}\)	\(61\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-893 b^{3} x^{12}+381 a \,b^{2} x^{8}-189 a^{2} b \,x^{4}+77 a^{3}\right )}{1155 a^{5} x^{15}}+\frac {b^{4} x}{a^{5} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^16/(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)

[Out]

-1/1155*(-2048*b^4*x^16-512*a*b^3*x^12+192*a^2*b^2*x^8-112*a^3*b*x^4+77*a^4)/x^15/(b*x^4+a)^(1/4)/a^5

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Maxima [A]
time = 0.30, size = 87, normalized size = 0.76 \begin {gather*} \frac {b^{4} x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{5}} + \frac {\frac {1540 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} - \frac {990 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}} + \frac {420 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b}{x^{11}} - \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}}}{x^{15}}}{1155 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

b^4*x/((b*x^4 + a)^(1/4)*a^5) + 1/1155*(1540*(b*x^4 + a)^(3/4)*b^3/x^3 - 990*(b*x^4 + a)^(7/4)*b^2/x^7 + 420*(
b*x^4 + a)^(11/4)*b/x^11 - 77*(b*x^4 + a)^(15/4)/x^15)/a^5

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Fricas [A]
time = 0.39, size = 72, normalized size = 0.63 \begin {gather*} \frac {{\left (2048 \, b^{4} x^{16} + 512 \, a b^{3} x^{12} - 192 \, a^{2} b^{2} x^{8} + 112 \, a^{3} b x^{4} - 77 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, {\left (a^{5} b x^{19} + a^{6} x^{15}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^16/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/1155*(2048*b^4*x^16 + 512*a*b^3*x^12 - 192*a^2*b^2*x^8 + 112*a^3*b*x^4 - 77*a^4)*(b*x^4 + a)^(3/4)/(a^5*b*x^
19 + a^6*x^15)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (109) = 218\).
time = 2.08, size = 928, normalized size = 8.14 \begin {gather*} - \frac {231 a^{7} b^{\frac {67}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} - \frac {357 a^{6} b^{\frac {71}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} - \frac {261 a^{5} b^{\frac {75}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} + \frac {585 a^{4} b^{\frac {79}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} + \frac {9360 a^{3} b^{\frac {83}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} + \frac {22464 a^{2} b^{\frac {87}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} + \frac {19968 a b^{\frac {91}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} + \frac {6144 b^{\frac {95}{4}} x^{28} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {15}{4}\right )}{1024 a^{9} b^{16} x^{12} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{8} b^{17} x^{16} \Gamma \left (\frac {5}{4}\right ) + 6144 a^{7} b^{18} x^{20} \Gamma \left (\frac {5}{4}\right ) + 4096 a^{6} b^{19} x^{24} \Gamma \left (\frac {5}{4}\right ) + 1024 a^{5} b^{20} x^{28} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**16/(b*x**4+a)**(5/4),x)

[Out]

-231*a**7*b**(67/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x
**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*
gamma(5/4)) - 357*a**6*b**(71/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) +
4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024
*a**5*b**20*x**28*gamma(5/4)) - 261*a**5*b**(75/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*
x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24
*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 585*a**4*b**(79/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4
)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 40
96*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 9360*a**3*b**(83/4)*x**16*(a/(b*x**4) + 1
)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x
**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) + 22464*a**2*b**(87/4)*
x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**17*x**16*gamma(5/4
) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x**28*gamma(5/4)) +
19968*a*b**(91/4)*x**24*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4) + 4096*a**8*b**
17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 1024*a**5*b**20*x*
*28*gamma(5/4)) + 6144*b**(95/4)*x**28*(a/(b*x**4) + 1)**(3/4)*gamma(-15/4)/(1024*a**9*b**16*x**12*gamma(5/4)
+ 4096*a**8*b**17*x**16*gamma(5/4) + 6144*a**7*b**18*x**20*gamma(5/4) + 4096*a**6*b**19*x**24*gamma(5/4) + 102
4*a**5*b**20*x**28*gamma(5/4))

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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^16/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^16), x)

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Mupad [B]
time = 1.65, size = 93, normalized size = 0.82 \begin {gather*} \frac {b^4\,x}{a^5\,{\left (b\,x^4+a\right )}^{1/4}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{15\,a^2\,x^{15}}+\frac {9\,b\,{\left (b\,x^4+a\right )}^{3/4}}{55\,a^3\,x^{11}}+\frac {893\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{1155\,a^5\,x^3}-\frac {127\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{385\,a^4\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^16*(a + b*x^4)^(5/4)),x)

[Out]

(b^4*x)/(a^5*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(15*a^2*x^15) + (9*b*(a + b*x^4)^(3/4))/(55*a^3*x^11) + (8
93*b^3*(a + b*x^4)^(3/4))/(1155*a^5*x^3) - (127*b^2*(a + b*x^4)^(3/4))/(385*a^4*x^7)

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