Integrand size = 15, antiderivative size = 178 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=-\frac {9 a^6 x \sqrt {a+b x^2}}{2048 b^2}+\frac {3 a^5 x^3 \sqrt {a+b x^2}}{1024 b}+\frac {3}{256} a^4 x^5 \sqrt {a+b x^2}+\frac {3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac {3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac {3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac {9 a^7 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2048 b^{5/2}} \] Output:
-9/2048*a^6*x*(b*x^2+a)^(1/2)/b^2+3/1024*a^5*x^3*(b*x^2+a)^(1/2)/b+3/256*a ^4*x^5*(b*x^2+a)^(1/2)+3/128*a^3*x^5*(b*x^2+a)^(3/2)+3/80*a^2*x^5*(b*x^2+a )^(5/2)+3/56*a*x^5*(b*x^2+a)^(7/2)+1/14*x^5*(b*x^2+a)^(9/2)+9/2048*a^7*arc tanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
Time = 0.58 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.71 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\frac {\sqrt {a+b x^2} \left (-315 a^6 x+210 a^5 b x^3+14168 a^4 b^2 x^5+39056 a^3 b^3 x^7+44928 a^2 b^4 x^9+24320 a b^5 x^{11}+5120 b^6 x^{13}\right )}{71680 b^2}+\frac {9 a^7 \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{1024 b^{5/2}} \] Input:
Integrate[x^4*(a + b*x^2)^(9/2),x]
Output:
(Sqrt[a + b*x^2]*(-315*a^6*x + 210*a^5*b*x^3 + 14168*a^4*b^2*x^5 + 39056*a ^3*b^3*x^7 + 44928*a^2*b^4*x^9 + 24320*a*b^5*x^11 + 5120*b^6*x^13))/(71680 *b^2) + (9*a^7*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/(1024*b^ (5/2))
Time = 0.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {248, 248, 248, 248, 248, 262, 262, 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.
| \(\displaystyle \int x^4 \left (a+b x^2\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {9}{14} a \int x^4 \left (b x^2+a\right )^{7/2}dx+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \int x^4 \left (b x^2+a\right )^{5/2}dx+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \int x^4 \left (b x^2+a\right )^{3/2}dx+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \int x^4 \sqrt {b x^2+a}dx+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \int \frac {x^4}{\sqrt {b x^2+a}}dx+\frac {1}{6} x^5 \sqrt {a+b x^2}\right )+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \int \frac {x^2}{\sqrt {b x^2+a}}dx}{4 b}\right )+\frac {1}{6} x^5 \sqrt {a+b x^2}\right )+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{4 b}\right )+\frac {1}{6} x^5 \sqrt {a+b x^2}\right )+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{4 b}\right )+\frac {1}{6} x^5 \sqrt {a+b x^2}\right )+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {9}{14} a \left (\frac {7}{12} a \left (\frac {1}{2} a \left (\frac {3}{8} a \left (\frac {1}{6} a \left (\frac {x^3 \sqrt {a+b x^2}}{4 b}-\frac {3 a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )}{4 b}\right )+\frac {1}{6} x^5 \sqrt {a+b x^2}\right )+\frac {1}{8} x^5 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{12} x^5 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{14} x^5 \left (a+b x^2\right )^{9/2}\) |
Input:
Int[x^4*(a + b*x^2)^(9/2),x]
Output:
(x^5*(a + b*x^2)^(9/2))/14 + (9*a*((x^5*(a + b*x^2)^(7/2))/12 + (7*a*((x^5 *(a + b*x^2)^(5/2))/10 + (a*((x^5*(a + b*x^2)^(3/2))/8 + (3*a*((x^5*Sqrt[a + b*x^2])/6 + (a*((x^3*Sqrt[a + b*x^2])/(4*b) - (3*a*((x*Sqrt[a + b*x^2]) /(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(4*b)))/6) )/8))/2))/12))/14
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(-\frac {x \left (-5120 b^{6} x^{12}-24320 a \,b^{5} x^{10}-44928 a^{2} b^{4} x^{8}-39056 a^{3} b^{3} x^{6}-14168 a^{4} b^{2} x^{4}-210 a^{5} b \,x^{2}+315 a^{6}\right ) \sqrt {b \,x^{2}+a}}{71680 b^{2}}+\frac {9 a^{7} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2048 b^{\frac {5}{2}}}\) | \(106\) |
| pseudoelliptic | \(\frac {\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a^{7}}{2048}-\frac {9 \left (-\frac {4864 a \,b^{\frac {11}{2}} x^{10}}{63}-\frac {1024 b^{\frac {13}{2}} x^{12}}{63}+a^{2} \left (-\frac {4992 b^{\frac {9}{2}} x^{8}}{35}-\frac {39056 a \,b^{\frac {7}{2}} x^{6}}{315}-\frac {2024 a^{2} b^{\frac {5}{2}} x^{4}}{45}-\frac {2 a^{3} b^{\frac {3}{2}} x^{2}}{3}+a^{4} \sqrt {b}\right )\right ) x \sqrt {b \,x^{2}+a}}{2048}}{b^{\frac {5}{2}}}\) | \(109\) |
| default | \(\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{14 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{12 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\right )}{12 b}\right )}{14 b}\) | \(146\) |
Input:
int(x^4*(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
-1/71680*x*(-5120*b^6*x^12-24320*a*b^5*x^10-44928*a^2*b^4*x^8-39056*a^3*b^ 3*x^6-14168*a^4*b^2*x^4-210*a^5*b*x^2+315*a^6)*(b*x^2+a)^(1/2)/b^2+9/2048* a^7/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))
Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.31 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\left [\frac {315 \, a^{7} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt {b x^{2} + a}}{143360 \, b^{3}}, -\frac {315 \, a^{7} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt {b x^{2} + a}}{71680 \, b^{3}}\right ] \] Input:
integrate(x^4*(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/143360*(315*a^7*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(5120*b^7*x^13 + 24320*a*b^6*x^11 + 44928*a^2*b^5*x^9 + 39056*a^3*b^4 *x^7 + 14168*a^4*b^3*x^5 + 210*a^5*b^2*x^3 - 315*a^6*b*x)*sqrt(b*x^2 + a)) /b^3, -1/71680*(315*a^7*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (512 0*b^7*x^13 + 24320*a*b^6*x^11 + 44928*a^2*b^5*x^9 + 39056*a^3*b^4*x^7 + 14 168*a^4*b^3*x^5 + 210*a^5*b^2*x^3 - 315*a^6*b*x)*sqrt(b*x^2 + a))/b^3]
Timed out. \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\text {Timed out} \] Input:
integrate(x**4*(b*x**2+a)**(9/2),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.79 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{3}}{14 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} a x}{56 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {9}{2}} a^{2} x}{560 \, b^{2}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x}{4480 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4} x}{1280 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5} x}{1024 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a} a^{6} x}{2048 \, b^{2}} + \frac {9 \, a^{7} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2048 \, b^{\frac {5}{2}}} \] Input:
integrate(x^4*(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
1/14*(b*x^2 + a)^(11/2)*x^3/b - 1/56*(b*x^2 + a)^(11/2)*a*x/b^2 + 1/560*(b *x^2 + a)^(9/2)*a^2*x/b^2 + 9/4480*(b*x^2 + a)^(7/2)*a^3*x/b^2 + 3/1280*(b *x^2 + a)^(5/2)*a^4*x/b^2 + 3/1024*(b*x^2 + a)^(3/2)*a^5*x/b^2 + 9/2048*sq rt(b*x^2 + a)*a^6*x/b^2 + 9/2048*a^7*arcsinh(b*x/sqrt(a*b))/b^(5/2)
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.67 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=-\frac {9 \, a^{7} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2048 \, b^{\frac {5}{2}}} - \frac {1}{71680} \, {\left (\frac {315 \, a^{6}}{b^{2}} - 2 \, {\left (\frac {105 \, a^{5}}{b} + 4 \, {\left (1771 \, a^{4} + 2 \, {\left (2441 \, a^{3} b + 8 \, {\left (351 \, a^{2} b^{2} + 10 \, {\left (4 \, b^{4} x^{2} + 19 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \] Input:
integrate(x^4*(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-9/2048*a^7*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2) - 1/71680*(315* a^6/b^2 - 2*(105*a^5/b + 4*(1771*a^4 + 2*(2441*a^3*b + 8*(351*a^2*b^2 + 10 *(4*b^4*x^2 + 19*a*b^3)*x^2)*x^2)*x^2)*x^2)*x^2)*sqrt(b*x^2 + a)*x
Timed out. \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\int x^4\,{\left (b\,x^2+a\right )}^{9/2} \,d x \] Input:
int(x^4*(a + b*x^2)^(9/2),x)
Output:
int(x^4*(a + b*x^2)^(9/2), x)
Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx=\frac {-315 \sqrt {b \,x^{2}+a}\, a^{6} b x +210 \sqrt {b \,x^{2}+a}\, a^{5} b^{2} x^{3}+14168 \sqrt {b \,x^{2}+a}\, a^{4} b^{3} x^{5}+39056 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x^{7}+44928 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{9}+24320 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{11}+5120 \sqrt {b \,x^{2}+a}\, b^{7} x^{13}+315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{7}}{71680 b^{3}} \] Input:
int(x^4*(b*x^2+a)^(9/2),x)
Output:
( - 315*sqrt(a + b*x**2)*a**6*b*x + 210*sqrt(a + b*x**2)*a**5*b**2*x**3 + 14168*sqrt(a + b*x**2)*a**4*b**3*x**5 + 39056*sqrt(a + b*x**2)*a**3*b**4*x **7 + 44928*sqrt(a + b*x**2)*a**2*b**5*x**9 + 24320*sqrt(a + b*x**2)*a*b** 6*x**11 + 5120*sqrt(a + b*x**2)*b**7*x**13 + 315*sqrt(b)*log((sqrt(a + b*x **2) + sqrt(b)*x)/sqrt(a))*a**7)/(71680*b**3)