Integrand size = 16, antiderivative size = 71 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=-\frac {\left (a-b x^4\right )^{3/2}}{14 a x^{14}}-\frac {2 b \left (a-b x^4\right )^{3/2}}{35 a^2 x^{10}}-\frac {4 b^2 \left (a-b x^4\right )^{3/2}}{105 a^3 x^6} \] Output:
-1/14*(-b*x^4+a)^(3/2)/a/x^14-2/35*b*(-b*x^4+a)^(3/2)/a^2/x^10-4/105*b^2*( -b*x^4+a)^(3/2)/a^3/x^6
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=\frac {\sqrt {a-b x^4} \left (-15 a^3+3 a^2 b x^4+4 a b^2 x^8+8 b^3 x^{12}\right )}{210 a^3 x^{14}} \] Input:
Integrate[Sqrt[a - b*x^4]/x^15,x]
Output:
(Sqrt[a - b*x^4]*(-15*a^3 + 3*a^2*b*x^4 + 4*a*b^2*x^8 + 8*b^3*x^12))/(210* a^3*x^14)
Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {803, 803, 796}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \int \frac {\sqrt {a-b x^4}}{x^{11}}dx}{7 a}-\frac {\left (a-b x^4\right )^{3/2}}{14 a x^{14}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \left (\frac {2 b \int \frac {\sqrt {a-b x^4}}{x^7}dx}{5 a}-\frac {\left (a-b x^4\right )^{3/2}}{10 a x^{10}}\right )}{7 a}-\frac {\left (a-b x^4\right )^{3/2}}{14 a x^{14}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {4 b \left (-\frac {b \left (a-b x^4\right )^{3/2}}{15 a^2 x^6}-\frac {\left (a-b x^4\right )^{3/2}}{10 a x^{10}}\right )}{7 a}-\frac {\left (a-b x^4\right )^{3/2}}{14 a x^{14}}\) |
Input:
Int[Sqrt[a - b*x^4]/x^15,x]
Output:
-1/14*(a - b*x^4)^(3/2)/(a*x^14) + (4*b*(-1/10*(a - b*x^4)^(3/2)/(a*x^10) - (b*(a - b*x^4)^(3/2))/(15*a^2*x^6)))/(7*a)
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]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[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] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 1.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{8}+12 a b \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(40\) |
default | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{8}+12 a b \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(40\) |
elliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{8}+12 a b \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(40\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{8}+12 a b \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(40\) |
orering | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {3}{2}} \left (8 b^{2} x^{8}+12 a b \,x^{4}+15 a^{2}\right )}{210 x^{14} a^{3}}\) | \(40\) |
trager | \(-\frac {\left (-8 b^{3} x^{12}-4 a \,b^{2} x^{8}-3 a^{2} b \,x^{4}+15 a^{3}\right ) \sqrt {-b \,x^{4}+a}}{210 x^{14} a^{3}}\) | \(51\) |
risch | \(-\frac {\left (-8 b^{3} x^{12}-4 a \,b^{2} x^{8}-3 a^{2} b \,x^{4}+15 a^{3}\right ) \sqrt {-b \,x^{4}+a}}{210 x^{14} a^{3}}\) | \(51\) |
Input:
int((-b*x^4+a)^(1/2)/x^15,x,method=_RETURNVERBOSE)
Output:
-1/210*(-b*x^4+a)^(3/2)*(8*b^2*x^8+12*a*b*x^4+15*a^2)/x^14/a^3
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=\frac {{\left (8 \, b^{3} x^{12} + 4 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} - 15 \, a^{3}\right )} \sqrt {-b x^{4} + a}}{210 \, a^{3} x^{14}} \] Input:
integrate((-b*x^4+a)^(1/2)/x^15,x, algorithm="fricas")
Output:
1/210*(8*b^3*x^12 + 4*a*b^2*x^8 + 3*a^2*b*x^4 - 15*a^3)*sqrt(-b*x^4 + a)/( a^3*x^14)
Result contains complex when optimal does not.
Time = 3.80 (sec) , antiderivative size = 733, normalized size of antiderivative = 10.32 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=\begin {cases} - \frac {15 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {33 a^{4} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} - \frac {17 a^{3} b^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {3 a^{2} b^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} - \frac {12 a b^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {8 b^{\frac {19}{2}} x^{20} \sqrt {\frac {a}{b x^{4}} - 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {15 i a^{5} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {33 i a^{4} b^{\frac {11}{2}} x^{4} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} - \frac {17 i a^{3} b^{\frac {13}{2}} x^{8} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {3 i a^{2} b^{\frac {15}{2}} x^{12} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} - \frac {12 i a b^{\frac {17}{2}} x^{16} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} + \frac {8 i b^{\frac {19}{2}} x^{20} \sqrt {- \frac {a}{b x^{4}} + 1}}{210 a^{5} b^{4} x^{12} - 420 a^{4} b^{5} x^{16} + 210 a^{3} b^{6} x^{20}} & \text {otherwise} \end {cases} \] Input:
integrate((-b*x**4+a)**(1/2)/x**15,x)
Output:
Piecewise((-15*a**5*b**(9/2)*sqrt(a/(b*x**4) - 1)/(210*a**5*b**4*x**12 - 4 20*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 33*a**4*b**(11/2)*x**4*sqrt(a/ (b*x**4) - 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x **20) - 17*a**3*b**(13/2)*x**8*sqrt(a/(b*x**4) - 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 3*a**2*b**(15/2)*x**12*sqrt( a/(b*x**4) - 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6 *x**20) - 12*a*b**(17/2)*x**16*sqrt(a/(b*x**4) - 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 8*b**(19/2)*x**20*sqrt(a/(b* x**4) - 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**2 0), Abs(a/(b*x**4)) > 1), (-15*I*a**5*b**(9/2)*sqrt(-a/(b*x**4) + 1)/(210* a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 33*I*a**4*b **(11/2)*x**4*sqrt(-a/(b*x**4) + 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x **16 + 210*a**3*b**6*x**20) - 17*I*a**3*b**(13/2)*x**8*sqrt(-a/(b*x**4) + 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 3*I *a**2*b**(15/2)*x**12*sqrt(-a/(b*x**4) + 1)/(210*a**5*b**4*x**12 - 420*a** 4*b**5*x**16 + 210*a**3*b**6*x**20) - 12*I*a*b**(17/2)*x**16*sqrt(-a/(b*x* *4) + 1)/(210*a**5*b**4*x**12 - 420*a**4*b**5*x**16 + 210*a**3*b**6*x**20) + 8*I*b**(19/2)*x**20*sqrt(-a/(b*x**4) + 1)/(210*a**5*b**4*x**12 - 420*a* *4*b**5*x**16 + 210*a**3*b**6*x**20), True))
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=-\frac {\frac {35 \, {\left (-b x^{4} + a\right )}^{\frac {3}{2}} b^{2}}{x^{6}} + \frac {42 \, {\left (-b x^{4} + a\right )}^{\frac {5}{2}} b}{x^{10}} + \frac {15 \, {\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{x^{14}}}{210 \, a^{3}} \] Input:
integrate((-b*x^4+a)^(1/2)/x^15,x, algorithm="maxima")
Output:
-1/210*(35*(-b*x^4 + a)^(3/2)*b^2/x^6 + 42*(-b*x^4 + a)^(5/2)*b/x^10 + 15* (-b*x^4 + a)^(7/2)/x^14)/a^3
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (59) = 118\).
Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.65 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=-\frac {8 \, {\left (70 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{8} \sqrt {-b} b^{3} + 35 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{6} a \sqrt {-b} b^{3} + 21 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{4} a^{2} \sqrt {-b} b^{3} - 7 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} a^{3} \sqrt {-b} b^{3} + a^{4} \sqrt {-b} b^{3}\right )}}{105 \, {\left ({\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} - a\right )}^{7}} \] Input:
integrate((-b*x^4+a)^(1/2)/x^15,x, algorithm="giac")
Output:
-8/105*(70*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^8*sqrt(-b)*b^3 + 35*(sqrt(-b) *x^2 - sqrt(-b*x^4 + a))^6*a*sqrt(-b)*b^3 + 21*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^4*a^2*sqrt(-b)*b^3 - 7*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2*a^3*sqrt (-b)*b^3 + a^4*sqrt(-b)*b^3)/((sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2 - a)^7
Time = 0.76 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=\frac {b\,\sqrt {a-b\,x^4}}{70\,a\,x^{10}}-\frac {\sqrt {a-b\,x^4}}{14\,x^{14}}+\frac {4\,b^3\,\sqrt {a-b\,x^4}}{105\,a^3\,x^2}+\frac {2\,b^2\,\sqrt {a-b\,x^4}}{105\,a^2\,x^6} \] Input:
int((a - b*x^4)^(1/2)/x^15,x)
Output:
(b*(a - b*x^4)^(1/2))/(70*a*x^10) - (a - b*x^4)^(1/2)/(14*x^14) + (4*b^3*( a - b*x^4)^(1/2))/(105*a^3*x^2) + (2*b^2*(a - b*x^4)^(1/2))/(105*a^2*x^6)
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {a-b x^4}}{x^{15}} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{3} x^{12}+4 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}-15 a^{3}\right )}{210 a^{3} x^{14}} \] Input:
int((-b*x^4+a)^(1/2)/x^15,x)
Output:
(sqrt(a - b*x**4)*( - 15*a**3 + 3*a**2*b*x**4 + 4*a*b**2*x**8 + 8*b**3*x** 12))/(210*a**3*x**14)