Integrand size = 15, antiderivative size = 68 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}+\frac {4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7} \] Output:
-1/11*(b*x^2+a)^(7/2)/a/x^11+4/99*b*(b*x^2+a)^(7/2)/a^2/x^9-8/693*b^2*(b*x ^2+a)^(7/2)/a^3/x^7
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (-63 a^2+28 a b x^2-8 b^2 x^4\right )}{693 a^3 x^{11}} \] Input:
Integrate[(a + b*x^2)^(5/2)/x^12,x]
Output:
((a + b*x^2)^(7/2)*(-63*a^2 + 28*a*b*x^2 - 8*b^2*x^4))/(693*a^3*x^11)
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {245, 245, 242}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {4 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^{10}}dx}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{5/2}}{x^8}dx}{9 a}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac {\left (a+b x^2\right )^{7/2}}{9 a x^9}\right )}{11 a}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}\) |
Input:
Int[(a + b*x^2)^(5/2)/x^12,x]
Output:
-1/11*(a + b*x^2)^(7/2)/(a*x^11) - (4*b*(-1/9*(a + b*x^2)^(7/2)/(a*x^9) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)))/(11*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (8 b^{2} x^{4}-28 a b \,x^{2}+63 a^{2}\right )}{693 x^{11} a^{3}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (8 b^{2} x^{4}-28 a b \,x^{2}+63 a^{2}\right )}{693 x^{11} a^{3}}\) | \(39\) |
orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (8 b^{2} x^{4}-28 a b \,x^{2}+63 a^{2}\right )}{693 x^{11} a^{3}}\) | \(39\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 a \,x^{11}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\) | \(61\) |
trager | \(-\frac {\left (8 b^{5} x^{10}-4 a \,b^{4} x^{8}+3 a^{2} b^{3} x^{6}+113 a^{3} b^{2} x^{4}+161 a^{4} b \,x^{2}+63 a^{5}\right ) \sqrt {b \,x^{2}+a}}{693 x^{11} a^{3}}\) | \(72\) |
risch | \(-\frac {\left (8 b^{5} x^{10}-4 a \,b^{4} x^{8}+3 a^{2} b^{3} x^{6}+113 a^{3} b^{2} x^{4}+161 a^{4} b \,x^{2}+63 a^{5}\right ) \sqrt {b \,x^{2}+a}}{693 x^{11} a^{3}}\) | \(72\) |
Input:
int((b*x^2+a)^(5/2)/x^12,x,method=_RETURNVERBOSE)
Output:
-1/693*(b*x^2+a)^(7/2)*(8*b^2*x^4-28*a*b*x^2+63*a^2)/x^11/a^3
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {{\left (8 \, b^{5} x^{10} - 4 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + 113 \, a^{3} b^{2} x^{4} + 161 \, a^{4} b x^{2} + 63 \, a^{5}\right )} \sqrt {b x^{2} + a}}{693 \, a^{3} x^{11}} \] Input:
integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="fricas")
Output:
-1/693*(8*b^5*x^10 - 4*a*b^4*x^8 + 3*a^2*b^3*x^6 + 113*a^3*b^2*x^4 + 161*a ^4*b*x^2 + 63*a^5)*sqrt(b*x^2 + a)/(a^3*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (61) = 122\).
Time = 1.25 (sec) , antiderivative size = 481, normalized size of antiderivative = 7.07 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=- \frac {63 a^{7} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x^{2} \cdot \left (693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}\right )} - \frac {287 a^{6} b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {498 a^{5} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {390 a^{4} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {115 a^{3} b^{\frac {17}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {3 a^{2} b^{\frac {19}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {12 a b^{\frac {21}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {8 b^{\frac {23}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} \] Input:
integrate((b*x**2+a)**(5/2)/x**12,x)
Output:
-63*a**7*b**(9/2)*sqrt(a/(b*x**2) + 1)/(x**2*(693*a**5*b**4*x**8 + 1386*a* *4*b**5*x**10 + 693*a**3*b**6*x**12)) - 287*a**6*b**(11/2)*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12) - 498*a**5*b**(13/2)*x**2*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a* *4*b**5*x**10 + 693*a**3*b**6*x**12) - 390*a**4*b**(15/2)*x**4*sqrt(a/(b*x **2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12 ) - 115*a**3*b**(17/2)*x**6*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 138 6*a**4*b**5*x**10 + 693*a**3*b**6*x**12) - 3*a**2*b**(19/2)*x**8*sqrt(a/(b *x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x** 12) - 12*a*b**(21/2)*x**10*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386 *a**4*b**5*x**10 + 693*a**3*b**6*x**12) - 8*b**(23/2)*x**12*sqrt(a/(b*x**2 ) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12)
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=-\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{693 \, a^{3} x^{7}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{99 \, a^{2} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{11 \, a x^{11}} \] Input:
integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="maxima")
Output:
-8/693*(b*x^2 + a)^(7/2)*b^2/(a^3*x^7) + 4/99*(b*x^2 + a)^(7/2)*b/(a^2*x^9 ) - 1/11*(b*x^2 + a)^(7/2)/(a*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.62 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=\frac {16 \, {\left (462 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} b^{\frac {11}{2}} + 1155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a b^{\frac {11}{2}} + 2541 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{2} b^{\frac {11}{2}} + 2079 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} b^{\frac {11}{2}} + 1485 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {11}{2}} + 297 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {11}{2}} + 55 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {11}{2}} - 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {11}{2}} + a^{8} b^{\frac {11}{2}}\right )}}{693 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{11}} \] Input:
integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="giac")
Output:
16/693*(462*(sqrt(b)*x - sqrt(b*x^2 + a))^16*b^(11/2) + 1155*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^(11/2) + 2541*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2 *b^(11/2) + 2079*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(11/2) + 1485*(sqr t(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(11/2) + 297*(sqrt(b)*x - sqrt(b*x^2 + a ))^6*a^5*b^(11/2) + 55*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*b^(11/2) - 11*( sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(11/2) + a^8*b^(11/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^11
Time = 1.57 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=\frac {4\,b^4\,\sqrt {b\,x^2+a}}{693\,a^2\,x^3}-\frac {113\,b^2\,\sqrt {b\,x^2+a}}{693\,x^7}-\frac {b^3\,\sqrt {b\,x^2+a}}{231\,a\,x^5}-\frac {a^2\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {8\,b^5\,\sqrt {b\,x^2+a}}{693\,a^3\,x}-\frac {23\,a\,b\,\sqrt {b\,x^2+a}}{99\,x^9} \] Input:
int((a + b*x^2)^(5/2)/x^12,x)
Output:
(4*b^4*(a + b*x^2)^(1/2))/(693*a^2*x^3) - (113*b^2*(a + b*x^2)^(1/2))/(693 *x^7) - (b^3*(a + b*x^2)^(1/2))/(231*a*x^5) - (a^2*(a + b*x^2)^(1/2))/(11* x^11) - (8*b^5*(a + b*x^2)^(1/2))/(693*a^3*x) - (23*a*b*(a + b*x^2)^(1/2)) /(99*x^9)
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx=\frac {-63 \sqrt {b \,x^{2}+a}\, a^{5}-161 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-113 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-3 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}+4 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-8 \sqrt {b \,x^{2}+a}\, b^{5} x^{10}+8 \sqrt {b}\, b^{5} x^{11}}{693 a^{3} x^{11}} \] Input:
int((b*x^2+a)^(5/2)/x^12,x)
Output:
( - 63*sqrt(a + b*x**2)*a**5 - 161*sqrt(a + b*x**2)*a**4*b*x**2 - 113*sqrt (a + b*x**2)*a**3*b**2*x**4 - 3*sqrt(a + b*x**2)*a**2*b**3*x**6 + 4*sqrt(a + b*x**2)*a*b**4*x**8 - 8*sqrt(a + b*x**2)*b**5*x**10 + 8*sqrt(b)*b**5*x* *11)/(693*a**3*x**11)