Integrand size = 15, antiderivative size = 94 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac {4 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac {2 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac {4 a}{b^5 \sqrt {a+b x^2}}+\frac {\sqrt {a+b x^2}}{b^5} \] Output:
-1/7*a^4/b^5/(b*x^2+a)^(7/2)+4/5*a^3/b^5/(b*x^2+a)^(5/2)-2*a^2/b^5/(b*x^2+ a)^(3/2)+4*a/b^5/(b*x^2+a)^(1/2)+(b*x^2+a)^(1/2)/b^5
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {128 a^4+448 a^3 b x^2+560 a^2 b^2 x^4+280 a b^3 x^6+35 b^4 x^8}{35 b^5 \left (a+b x^2\right )^{7/2}} \] Input:
Integrate[x^9/(a + b*x^2)^(9/2),x]
Output:
(128*a^4 + 448*a^3*b*x^2 + 560*a^2*b^2*x^4 + 280*a*b^3*x^6 + 35*b^4*x^8)/( 35*b^5*(a + b*x^2)^(7/2))
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {243, 53, 2009}
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.
\(\displaystyle \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (b x^2+a\right )^{9/2}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^4}{b^4 \left (b x^2+a\right )^{9/2}}-\frac {4 a^3}{b^4 \left (b x^2+a\right )^{7/2}}+\frac {6 a^2}{b^4 \left (b x^2+a\right )^{5/2}}-\frac {4 a}{b^4 \left (b x^2+a\right )^{3/2}}+\frac {1}{b^4 \sqrt {b x^2+a}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 a^4}{7 b^5 \left (a+b x^2\right )^{7/2}}+\frac {8 a^3}{5 b^5 \left (a+b x^2\right )^{5/2}}-\frac {4 a^2}{b^5 \left (a+b x^2\right )^{3/2}}+\frac {8 a}{b^5 \sqrt {a+b x^2}}+\frac {2 \sqrt {a+b x^2}}{b^5}\right )\) |
Input:
Int[x^9/(a + b*x^2)^(9/2),x]
Output:
((-2*a^4)/(7*b^5*(a + b*x^2)^(7/2)) + (8*a^3)/(5*b^5*(a + b*x^2)^(5/2)) - (4*a^2)/(b^5*(a + b*x^2)^(3/2)) + (8*a)/(b^5*Sqrt[a + b*x^2]) + (2*Sqrt[a + b*x^2])/b^5)/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.43 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(\frac {35 b^{4} x^{8}+280 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+448 a^{3} b \,x^{2}+128 a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}\) | \(58\) |
trager | \(\frac {35 b^{4} x^{8}+280 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+448 a^{3} b \,x^{2}+128 a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}\) | \(58\) |
pseudoelliptic | \(\frac {35 b^{4} x^{8}+280 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+448 a^{3} b \,x^{2}+128 a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}\) | \(58\) |
orering | \(\frac {35 b^{4} x^{8}+280 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+448 a^{3} b \,x^{2}+128 a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}\) | \(58\) |
risch | \(\frac {\sqrt {b \,x^{2}+a}}{b^{5}}+\frac {\sqrt {b \,x^{2}+a}\, \left (140 b^{3} x^{6}+350 a \,b^{2} x^{4}+308 a^{2} b \,x^{2}+93 a^{3}\right ) a}{35 b^{5} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )}\) | \(104\) |
default | \(\frac {x^{8}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 a \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )}{b}\) | \(105\) |
Input:
int(x^9/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/35*(35*b^4*x^8+280*a*b^3*x^6+560*a^2*b^2*x^4+448*a^3*b*x^2+128*a^4)/(b*x ^2+a)^(7/2)/b^5
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.09 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (35 \, b^{4} x^{8} + 280 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 448 \, a^{3} b x^{2} + 128 \, a^{4}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \] Input:
integrate(x^9/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
1/35*(35*b^4*x^8 + 280*a*b^3*x^6 + 560*a^2*b^2*x^4 + 448*a^3*b*x^2 + 128*a ^4)*sqrt(b*x^2 + a)/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (87) = 174\).
Time = 0.78 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.83 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\begin {cases} \frac {128 a^{4}}{35 a^{3} b^{5} \sqrt {a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt {a + b x^{2}} + 105 a b^{7} x^{4} \sqrt {a + b x^{2}} + 35 b^{8} x^{6} \sqrt {a + b x^{2}}} + \frac {448 a^{3} b x^{2}}{35 a^{3} b^{5} \sqrt {a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt {a + b x^{2}} + 105 a b^{7} x^{4} \sqrt {a + b x^{2}} + 35 b^{8} x^{6} \sqrt {a + b x^{2}}} + \frac {560 a^{2} b^{2} x^{4}}{35 a^{3} b^{5} \sqrt {a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt {a + b x^{2}} + 105 a b^{7} x^{4} \sqrt {a + b x^{2}} + 35 b^{8} x^{6} \sqrt {a + b x^{2}}} + \frac {280 a b^{3} x^{6}}{35 a^{3} b^{5} \sqrt {a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt {a + b x^{2}} + 105 a b^{7} x^{4} \sqrt {a + b x^{2}} + 35 b^{8} x^{6} \sqrt {a + b x^{2}}} + \frac {35 b^{4} x^{8}}{35 a^{3} b^{5} \sqrt {a + b x^{2}} + 105 a^{2} b^{6} x^{2} \sqrt {a + b x^{2}} + 105 a b^{7} x^{4} \sqrt {a + b x^{2}} + 35 b^{8} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{10}}{10 a^{\frac {9}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**9/(b*x**2+a)**(9/2),x)
Output:
Piecewise((128*a**4/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sq rt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 448*a**3*b*x**2/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6* x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sq rt(a + b*x**2)) + 560*a**2*b**2*x**4/(35*a**3*b**5*sqrt(a + b*x**2) + 105* a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b* *8*x**6*sqrt(a + b*x**2)) + 280*a*b**3*x**6/(35*a**3*b**5*sqrt(a + b*x**2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x**2) + 35*b**8*x**6*sqrt(a + b*x**2)) + 35*b**4*x**8/(35*a**3*b**5*sqrt(a + b*x **2) + 105*a**2*b**6*x**2*sqrt(a + b*x**2) + 105*a*b**7*x**4*sqrt(a + b*x* *2) + 35*b**8*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**10/(10*a**(9/2)), Tru e))
Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x^{8}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, a x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {16 \, a^{2} x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {64 \, a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} + \frac {128 \, a^{4}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}} \] Input:
integrate(x^9/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
x^8/((b*x^2 + a)^(7/2)*b) + 8*a*x^6/((b*x^2 + a)^(7/2)*b^2) + 16*a^2*x^4/( (b*x^2 + a)^(7/2)*b^3) + 64/5*a^3*x^2/((b*x^2 + a)^(7/2)*b^4) + 128/35*a^4 /((b*x^2 + a)^(7/2)*b^5)
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.82 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {35 \, \sqrt {b x^{2} + a}}{b} + \frac {140 \, {\left (b x^{2} + a\right )}^{3} a - 70 \, {\left (b x^{2} + a\right )}^{2} a^{2} + 28 \, {\left (b x^{2} + a\right )} a^{3} - 5 \, a^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b}}{35 \, b^{4}} \] Input:
integrate(x^9/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
1/35*(35*sqrt(b*x^2 + a)/b + (140*(b*x^2 + a)^3*a - 70*(b*x^2 + a)^2*a^2 + 28*(b*x^2 + a)*a^3 - 5*a^4)/((b*x^2 + a)^(7/2)*b))/b^4
Time = 0.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\sqrt {b\,x^2+a}}{b^5}+\frac {4\,a}{b^5\,\sqrt {b\,x^2+a}}-\frac {2\,a^2}{b^5\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {4\,a^3}{5\,b^5\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {a^4}{7\,b^5\,{\left (b\,x^2+a\right )}^{7/2}} \] Input:
int(x^9/(a + b*x^2)^(9/2),x)
Output:
(a + b*x^2)^(1/2)/b^5 + (4*a)/(b^5*(a + b*x^2)^(1/2)) - (2*a^2)/(b^5*(a + b*x^2)^(3/2)) + (4*a^3)/(5*b^5*(a + b*x^2)^(5/2)) - a^4/(7*b^5*(a + b*x^2) ^(7/2))
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.04 \[ \int \frac {x^9}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {\sqrt {b \,x^{2}+a}\, \left (35 b^{4} x^{8}+280 a \,b^{3} x^{6}+560 a^{2} b^{2} x^{4}+448 a^{3} b \,x^{2}+128 a^{4}\right )}{35 b^{5} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(x^9/(b*x^2+a)^(9/2),x)
Output:
(sqrt(a + b*x**2)*(128*a**4 + 448*a**3*b*x**2 + 560*a**2*b**2*x**4 + 280*a *b**3*x**6 + 35*b**4*x**8))/(35*b**5*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x **4 + 4*a*b**3*x**6 + b**4*x**8))