3.11.8 \(\int \frac {\sqrt [4]{a+b x^4}}{x^{18}} \, dx\) [1008]

Optimal. Leaf size=92 \[ -\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5} \]

[Out]

-1/17*(b*x^4+a)^(5/4)/a/x^17+12/221*b*(b*x^4+a)^(5/4)/a^2/x^13-32/663*b^2*(b*x^4+a)^(5/4)/a^3/x^9+128/3315*b^3
*(b*x^4+a)^(5/4)/a^4/x^5

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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 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(1/4)/x^18,x]

[Out]

-1/17*(a + b*x^4)^(5/4)/(a*x^17) + (12*b*(a + b*x^4)^(5/4))/(221*a^2*x^13) - (32*b^2*(a + b*x^4)^(5/4))/(663*a
^3*x^9) + (128*b^3*(a + b*x^4)^(5/4))/(3315*a^4*x^5)

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 {\sqrt [4]{a+b x^4}}{x^{18}} \, dx &=-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}-\frac {(12 b) \int \frac {\sqrt [4]{a+b x^4}}{x^{14}} \, dx}{17 a}\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}+\frac {\left (96 b^2\right ) \int \frac {\sqrt [4]{a+b x^4}}{x^{10}} \, dx}{221 a^2}\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}-\frac {\left (128 b^3\right ) \int \frac {\sqrt [4]{a+b x^4}}{x^6} \, dx}{663 a^3}\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{17 a x^{17}}+\frac {12 b \left (a+b x^4\right )^{5/4}}{221 a^2 x^{13}}-\frac {32 b^2 \left (a+b x^4\right )^{5/4}}{663 a^3 x^9}+\frac {128 b^3 \left (a+b x^4\right )^{5/4}}{3315 a^4 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^4\right )^{5/4} \left (-195 a^3+180 a^2 b x^4-160 a b^2 x^8+128 b^3 x^{12}\right )}{3315 a^4 x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(1/4)/x^18,x]

[Out]

((a + b*x^4)^(5/4)*(-195*a^3 + 180*a^2*b*x^4 - 160*a*b^2*x^8 + 128*b^3*x^12))/(3315*a^4*x^17)

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Maple [A]
time = 0.17, size = 50, normalized size = 0.54

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-128 b^{3} x^{12}+160 a \,b^{2} x^{8}-180 a^{2} b \,x^{4}+195 a^{3}\right )}{3315 x^{17} a^{4}}\)	\(50\)
trager	\(-\frac {\left (-128 x^{16} b^{4}+32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}+15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\)	\(61\)
risch	\(-\frac {\left (-128 x^{16} b^{4}+32 a \,b^{3} x^{12}-20 a^{2} b^{2} x^{8}+15 a^{3} b \,x^{4}+195 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{3315 x^{17} a^{4}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(1/4)/x^18,x,method=_RETURNVERBOSE)

[Out]

-1/3315*(b*x^4+a)^(5/4)*(-128*b^3*x^12+160*a*b^2*x^8-180*a^2*b*x^4+195*a^3)/x^17/a^4

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Maxima [A]
time = 0.29, size = 69, normalized size = 0.75 \begin {gather*} \frac {\frac {663 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}{x^{5}} - \frac {1105 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{2}}{x^{9}} + \frac {765 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b}{x^{13}} - \frac {195 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}}}{x^{17}}}{3315 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^18,x, algorithm="maxima")

[Out]

1/3315*(663*(b*x^4 + a)^(5/4)*b^3/x^5 - 1105*(b*x^4 + a)^(9/4)*b^2/x^9 + 765*(b*x^4 + a)^(13/4)*b/x^13 - 195*(
b*x^4 + a)^(17/4)/x^17)/a^4

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Fricas [A]
time = 0.37, size = 60, normalized size = 0.65 \begin {gather*} \frac {{\left (128 \, b^{4} x^{16} - 32 \, a b^{3} x^{12} + 20 \, a^{2} b^{2} x^{8} - 15 \, a^{3} b x^{4} - 195 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, a^{4} x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(1/4)/x^18,x, algorithm="fricas")

[Out]

1/3315*(128*b^4*x^16 - 32*a*b^3*x^12 + 20*a^2*b^2*x^8 - 15*a^3*b*x^4 - 195*a^4)*(b*x^4 + a)^(1/4)/(a^4*x^17)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (85) = 170\).
time = 1.60, size = 847, normalized size = 9.21 \begin {gather*} - \frac {585 a^{7} b^{\frac {37}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {1800 a^{6} b^{\frac {41}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {1830 a^{5} b^{\frac {45}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} - \frac {636 a^{4} b^{\frac {49}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {231 a^{3} b^{\frac {53}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {924 a^{2} b^{\frac {57}{4}} x^{20} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {1056 a b^{\frac {61}{4}} x^{24} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} + \frac {384 b^{\frac {65}{4}} x^{28} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {17}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {1}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {1}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(1/4)/x**18,x)

[Out]

-585*a**7*b**(37/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**
20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 1800*a**6*b**(41/4)*x*
*4*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) +
768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 1830*a**5*b**(45/4)*x**8*(a/(b*x**4) +
1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x*
*24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) - 636*a**4*b**(49/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-
17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) +
256*a**4*b**12*x**28*gamma(-1/4)) + 231*a**3*b**(53/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b
**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x
**28*gamma(-1/4)) + 924*a**2*b**(57/4)*x**20*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-
1/4) + 768*a**6*b**10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4))
+ 1056*a*b**(61/4)*x**24*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**
10*x**20*gamma(-1/4) + 768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*gamma(-1/4)) + 384*b**(65/4)*x*
*28*(a/(b*x**4) + 1)**(1/4)*gamma(-17/4)/(256*a**7*b**9*x**16*gamma(-1/4) + 768*a**6*b**10*x**20*gamma(-1/4) +
768*a**5*b**11*x**24*gamma(-1/4) + 256*a**4*b**12*x**28*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((b*x^4+a)^(1/4)/x^18,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)/x^18, x)

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Mupad [B]
time = 1.70, size = 93, normalized size = 1.01 \begin {gather*} \frac {128\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{3315\,a^4\,x}-\frac {b\,{\left (b\,x^4+a\right )}^{1/4}}{221\,a\,x^{13}}-\frac {{\left (b\,x^4+a\right )}^{1/4}}{17\,x^{17}}-\frac {32\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{3315\,a^3\,x^5}+\frac {4\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{663\,a^2\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(1/4)/x^18,x)

[Out]

(128*b^4*(a + b*x^4)^(1/4))/(3315*a^4*x) - (b*(a + b*x^4)^(1/4))/(221*a*x^13) - (a + b*x^4)^(1/4)/(17*x^17) -
(32*b^3*(a + b*x^4)^(1/4))/(3315*a^3*x^5) + (4*b^2*(a + b*x^4)^(1/4))/(663*a^2*x^9)

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