Integrand size = 11, antiderivative size = 122 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {63 a^5 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b}} \] Output:
63/256*a^4*x*(b*x^2+a)^(1/2)+21/128*a^3*x*(b*x^2+a)^(3/2)+21/160*a^2*x*(b* x^2+a)^(5/2)+9/80*a*x*(b*x^2+a)^(7/2)+1/10*x*(b*x^2+a)^(9/2)+63/256*a^5*ar ctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.76 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {\sqrt {a+b x^2} \left (965 a^4 x+1490 a^3 b x^3+1368 a^2 b^2 x^5+656 a b^3 x^7+128 b^4 x^9\right )}{1280}-\frac {63 a^5 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{256 \sqrt {b}} \] Input:
Integrate[(a + b*x^2)^(9/2),x]
Output:
(Sqrt[a + b*x^2]*(965*a^4*x + 1490*a^3*b*x^3 + 1368*a^2*b^2*x^5 + 656*a*b^ 3*x^7 + 128*b^4*x^9))/1280 - (63*a^5*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/ (256*Sqrt[b])
Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {211, 211, 211, 211, 211, 224, 219}
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.
\(\displaystyle \int \left (a+b x^2\right )^{9/2} \, dx\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{10} a \int \left (b x^2+a\right )^{7/2}dx+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \int \left (b x^2+a\right )^{5/2}dx+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {9}{10} a \left (\frac {7}{8} a \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {1}{8} x \left (a+b x^2\right )^{7/2}\right )+\frac {1}{10} x \left (a+b x^2\right )^{9/2}\) |
Input:
Int[(a + b*x^2)^(9/2),x]
Output:
(x*(a + b*x^2)^(9/2))/10 + (9*a*((x*(a + b*x^2)^(7/2))/8 + (7*a*((x*(a + b *x^2)^(5/2))/6 + (5*a*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2]) /2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/6))/8))/10
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {x \left (128 b^{4} x^{8}+656 a \,b^{3} x^{6}+1368 a^{2} b^{2} x^{4}+1490 a^{3} b \,x^{2}+965 a^{4}\right ) \sqrt {b \,x^{2}+a}}{1280}+\frac {63 a^{5} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 \sqrt {b}}\) | \(81\) |
default | \(\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\) | \(100\) |
Input:
int((b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/1280*x*(128*b^4*x^8+656*a*b^3*x^6+1368*a^2*b^2*x^4+1490*a^3*b*x^2+965*a^ 4)*(b*x^2+a)^(1/2)+63/256*a^5*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)
Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.56 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\left [\frac {315 \, a^{5} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{2560 \, b}, -\frac {315 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{1280 \, b}\right ] \] Input:
integrate((b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/2560*(315*a^5*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(128*b^5*x^9 + 656*a*b^4*x^7 + 1368*a^2*b^3*x^5 + 1490*a^3*b^2*x^3 + 96 5*a^4*b*x)*sqrt(b*x^2 + a))/b, -1/1280*(315*a^5*sqrt(-b)*arctan(sqrt(-b)*x /sqrt(b*x^2 + a)) - (128*b^5*x^9 + 656*a*b^4*x^7 + 1368*a^2*b^3*x^5 + 1490 *a^3*b^2*x^3 + 965*a^4*b*x)*sqrt(b*x^2 + a))/b]
Time = 10.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.24 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {193 a^{\frac {9}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{256} + \frac {149 a^{\frac {7}{2}} b x^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{128} + \frac {171 a^{\frac {5}{2}} b^{2} x^{5} \sqrt {1 + \frac {b x^{2}}{a}}}{160} + \frac {41 a^{\frac {3}{2}} b^{3} x^{7} \sqrt {1 + \frac {b x^{2}}{a}}}{80} + \frac {\sqrt {a} b^{4} x^{9} \sqrt {1 + \frac {b x^{2}}{a}}}{10} + \frac {63 a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 \sqrt {b}} \] Input:
integrate((b*x**2+a)**(9/2),x)
Output:
193*a**(9/2)*x*sqrt(1 + b*x**2/a)/256 + 149*a**(7/2)*b*x**3*sqrt(1 + b*x** 2/a)/128 + 171*a**(5/2)*b**2*x**5*sqrt(1 + b*x**2/a)/160 + 41*a**(3/2)*b** 3*x**7*sqrt(1 + b*x**2/a)/80 + sqrt(a)*b**4*x**9*sqrt(1 + b*x**2/a)/10 + 6 3*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*sqrt(b))
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.72 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {1}{10} \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} x + \frac {9}{80} \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x + \frac {21}{160} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} x + \frac {21}{128} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x + \frac {63}{256} \, \sqrt {b x^{2} + a} a^{4} x + \frac {63 \, a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {b}} \] Input:
integrate((b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
1/10*(b*x^2 + a)^(9/2)*x + 9/80*(b*x^2 + a)^(7/2)*a*x + 21/160*(b*x^2 + a) ^(5/2)*a^2*x + 21/128*(b*x^2 + a)^(3/2)*a^3*x + 63/256*sqrt(b*x^2 + a)*a^4 *x + 63/256*a^5*arcsinh(b*x/sqrt(a*b))/sqrt(b)
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \left (a+b x^2\right )^{9/2} \, dx=-\frac {63 \, a^{5} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, \sqrt {b}} + \frac {1}{1280} \, {\left (965 \, a^{4} + 2 \, {\left (745 \, a^{3} b + 4 \, {\left (171 \, a^{2} b^{2} + 2 \, {\left (8 \, b^{4} x^{2} + 41 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \] Input:
integrate((b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-63/256*a^5*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/1280*(965*a ^4 + 2*(745*a^3*b + 4*(171*a^2*b^2 + 2*(8*b^4*x^2 + 41*a*b^3)*x^2)*x^2)*x^ 2)*sqrt(b*x^2 + a)*x
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.30 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {x\,{\left (b\,x^2+a\right )}^{9/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{9/2}} \] Input:
int((a + b*x^2)^(9/2),x)
Output:
(x*(a + b*x^2)^(9/2)*hypergeom([-9/2, 1/2], 3/2, -(b*x^2)/a))/((b*x^2)/a + 1)^(9/2)
Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^2\right )^{9/2} \, dx=\frac {965 \sqrt {b \,x^{2}+a}\, a^{4} b x +1490 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{3}+1368 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{5}+656 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{7}+128 \sqrt {b \,x^{2}+a}\, b^{5} x^{9}+315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5}}{1280 b} \] Input:
int((b*x^2+a)^(9/2),x)
Output:
(965*sqrt(a + b*x**2)*a**4*b*x + 1490*sqrt(a + b*x**2)*a**3*b**2*x**3 + 13 68*sqrt(a + b*x**2)*a**2*b**3*x**5 + 656*sqrt(a + b*x**2)*a*b**4*x**7 + 12 8*sqrt(a + b*x**2)*b**5*x**9 + 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b) *x)/sqrt(a))*a**5)/(1280*b)