Integrand size = 16, antiderivative size = 71 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a-b x^4}}{10 a x^{10}}-\frac {2 b \sqrt {a-b x^4}}{15 a^2 x^6}-\frac {4 b^2 \sqrt {a-b x^4}}{15 a^3 x^2} \] Output:
-1/10*(-b*x^4+a)^(1/2)/a/x^10-2/15*b*(-b*x^4+a)^(1/2)/a^2/x^6-4/15*b^2*(-b *x^4+a)^(1/2)/a^3/x^2
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=\frac {\sqrt {a-b x^4} \left (-3 a^2-4 a b x^4-8 b^2 x^8\right )}{30 a^3 x^{10}} \] Input:
Integrate[1/(x^11*Sqrt[a - b*x^4]),x]
Output:
(Sqrt[a - b*x^4]*(-3*a^2 - 4*a*b*x^4 - 8*b^2*x^8))/(30*a^3*x^10)
Time = 0.31 (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.
\(\displaystyle \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \int \frac {1}{x^7 \sqrt {a-b x^4}}dx}{5 a}-\frac {\sqrt {a-b x^4}}{10 a x^{10}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {4 b \left (\frac {2 b \int \frac {1}{x^3 \sqrt {a-b x^4}}dx}{3 a}-\frac {\sqrt {a-b x^4}}{6 a x^6}\right )}{5 a}-\frac {\sqrt {a-b x^4}}{10 a x^{10}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {4 b \left (-\frac {b \sqrt {a-b x^4}}{3 a^2 x^2}-\frac {\sqrt {a-b x^4}}{6 a x^6}\right )}{5 a}-\frac {\sqrt {a-b x^4}}{10 a x^{10}}\) |
Input:
Int[1/(x^11*Sqrt[a - b*x^4]),x]
Output:
-1/10*Sqrt[a - b*x^4]/(a*x^10) + (4*b*(-1/6*Sqrt[a - b*x^4]/(a*x^6) - (b*S qrt[a - b*x^4])/(3*a^2*x^2)))/(5*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 = 0.97 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
default | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
trager | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
elliptic | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
pseudoelliptic | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
orering | \(-\frac {\sqrt {-b \,x^{4}+a}\, \left (8 b^{2} x^{8}+4 a b \,x^{4}+3 a^{2}\right )}{30 a^{3} x^{10}}\) | \(40\) |
Input:
int(1/x^11/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/30*(-b*x^4+a)^(1/2)*(8*b^2*x^8+4*a*b*x^4+3*a^2)/a^3/x^10
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=-\frac {{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} + 3 \, a^{2}\right )} \sqrt {-b x^{4} + a}}{30 \, a^{3} x^{10}} \] Input:
integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="fricas")
Output:
-1/30*(8*b^2*x^8 + 4*a*b*x^4 + 3*a^2)*sqrt(-b*x^4 + a)/(a^3*x^10)
Result contains complex when optimal does not.
Time = 1.57 (sec) , antiderivative size = 609, normalized size of antiderivative = 8.58 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=\begin {cases} - \frac {3 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {2 a^{3} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {3 a^{2} b^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {12 a b^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {8 b^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{b x^{4}} - 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {3 i a^{4} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {2 i a^{3} b^{\frac {11}{2}} x^{4} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {3 i a^{2} b^{\frac {13}{2}} x^{8} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} + \frac {12 i a b^{\frac {15}{2}} x^{12} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} - \frac {8 i b^{\frac {17}{2}} x^{16} \sqrt {- \frac {a}{b x^{4}} + 1}}{30 a^{5} b^{4} x^{8} - 60 a^{4} b^{5} x^{12} + 30 a^{3} b^{6} x^{16}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**11/(-b*x**4+a)**(1/2),x)
Output:
Piecewise((-3*a**4*b**(9/2)*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a **4*b**5*x**12 + 30*a**3*b**6*x**16) + 2*a**3*b**(11/2)*x**4*sqrt(a/(b*x** 4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) - 3* a**2*b**(13/2)*x**8*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5 *x**12 + 30*a**3*b**6*x**16) + 12*a*b**(15/2)*x**12*sqrt(a/(b*x**4) - 1)/( 30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) - 8*b**(17/2) *x**16*sqrt(a/(b*x**4) - 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a **3*b**6*x**16), Abs(a/(b*x**4)) > 1), (-3*I*a**4*b**(9/2)*sqrt(-a/(b*x**4 ) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) + 2*I *a**3*b**(11/2)*x**4*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b* *5*x**12 + 30*a**3*b**6*x**16) - 3*I*a**2*b**(13/2)*x**8*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16) + 12*I* a*b**(15/2)*x**12*sqrt(-a/(b*x**4) + 1)/(30*a**5*b**4*x**8 - 60*a**4*b**5* x**12 + 30*a**3*b**6*x**16) - 8*I*b**(17/2)*x**16*sqrt(-a/(b*x**4) + 1)/(3 0*a**5*b**4*x**8 - 60*a**4*b**5*x**12 + 30*a**3*b**6*x**16), True))
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=-\frac {\frac {15 \, \sqrt {-b x^{4} + a} b^{2}}{x^{2}} + \frac {10 \, {\left (-b x^{4} + a\right )}^{\frac {3}{2}} b}{x^{6}} + \frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{x^{10}}}{30 \, a^{3}} \] Input:
integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="maxima")
Output:
-1/30*(15*sqrt(-b*x^4 + a)*b^2/x^2 + 10*(-b*x^4 + a)^(3/2)*b/x^6 + 3*(-b*x ^4 + a)^(5/2)/x^10)/a^3
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=\frac {8 \, {\left (10 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{4} - 5 \, {\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} a + a^{2}\right )} \sqrt {-b} b^{2}}{15 \, {\left ({\left (\sqrt {-b} x^{2} - \sqrt {-b x^{4} + a}\right )}^{2} - a\right )}^{5}} \] Input:
integrate(1/x^11/(-b*x^4+a)^(1/2),x, algorithm="giac")
Output:
8/15*(10*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^4 - 5*(sqrt(-b)*x^2 - sqrt(-b*x ^4 + a))^2*a + a^2)*sqrt(-b)*b^2/((sqrt(-b)*x^2 - sqrt(-b*x^4 + a))^2 - a) ^5
Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a-b\,x^4}\,\left (3\,a^2+4\,a\,b\,x^4+8\,b^2\,x^8\right )}{30\,a^3\,x^{10}} \] Input:
int(1/(x^11*(a - b*x^4)^(1/2)),x)
Output:
-((a - b*x^4)^(1/2)*(3*a^2 + 8*b^2*x^8 + 4*a*b*x^4))/(30*a^3*x^10)
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^{11} \sqrt {a-b x^4}} \, dx=\frac {\sqrt {-b \,x^{4}+a}\, \left (-8 b^{2} x^{8}-4 a b \,x^{4}-3 a^{2}\right )}{30 a^{3} x^{10}} \] Input:
int(1/x^11/(-b*x^4+a)^(1/2),x)
Output:
(sqrt(a - b*x**4)*( - 3*a**2 - 4*a*b*x**4 - 8*b**2*x**8))/(30*a**3*x**10)