Integrand size = 15, antiderivative size = 72 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=-\frac {81}{256} x \sqrt {-9+4 x^2}-\frac {3}{32} x^3 \sqrt {-9+4 x^2}+\frac {1}{6} x^5 \sqrt {-9+4 x^2}-\frac {729}{512} \text {arctanh}\left (\frac {2 x}{\sqrt {-9+4 x^2}}\right ) \] Output:
-81/256*x*(4*x^2-9)^(1/2)-3/32*x^3*(4*x^2-9)^(1/2)+1/6*x^5*(4*x^2-9)^(1/2) -729/512*arctanh(2*x/(4*x^2-9)^(1/2))
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.68 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\frac {1}{768} x \sqrt {-9+4 x^2} \left (-243-72 x^2+128 x^4\right )+\frac {729}{512} \log \left (-2 x+\sqrt {-9+4 x^2}\right ) \] Input:
Integrate[x^4*Sqrt[-9 + 4*x^2],x]
Output:
(x*Sqrt[-9 + 4*x^2]*(-243 - 72*x^2 + 128*x^4))/768 + (729*Log[-2*x + Sqrt[ -9 + 4*x^2]])/512
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
\(\displaystyle \int x^4 \sqrt {4 x^2-9} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {1}{6} x^5 \sqrt {4 x^2-9}-\frac {3}{2} \int \frac {x^4}{\sqrt {4 x^2-9}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^5 \sqrt {4 x^2-9}-\frac {3}{2} \left (\frac {27}{16} \int \frac {x^2}{\sqrt {4 x^2-9}}dx+\frac {1}{16} \sqrt {4 x^2-9} x^3\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^5 \sqrt {4 x^2-9}-\frac {3}{2} \left (\frac {27}{16} \left (\frac {9}{8} \int \frac {1}{\sqrt {4 x^2-9}}dx+\frac {1}{8} \sqrt {4 x^2-9} x\right )+\frac {1}{16} \sqrt {4 x^2-9} x^3\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{6} x^5 \sqrt {4 x^2-9}-\frac {3}{2} \left (\frac {27}{16} \left (\frac {9}{8} \int \frac {1}{1-\frac {4 x^2}{4 x^2-9}}d\frac {x}{\sqrt {4 x^2-9}}+\frac {1}{8} \sqrt {4 x^2-9} x\right )+\frac {1}{16} \sqrt {4 x^2-9} x^3\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{6} x^5 \sqrt {4 x^2-9}-\frac {3}{2} \left (\frac {27}{16} \left (\frac {9}{16} \text {arctanh}\left (\frac {2 x}{\sqrt {4 x^2-9}}\right )+\frac {1}{8} \sqrt {4 x^2-9} x\right )+\frac {1}{16} \sqrt {4 x^2-9} x^3\right )\) |
Input:
Int[x^4*Sqrt[-9 + 4*x^2],x]
Output:
(x^5*Sqrt[-9 + 4*x^2])/6 - (3*((x^3*Sqrt[-9 + 4*x^2])/16 + (27*((x*Sqrt[-9 + 4*x^2])/8 + (9*ArcTanh[(2*x)/Sqrt[-9 + 4*x^2]])/16))/16))/2
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 = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.58
method | result | size |
trager | \(\frac {x \left (128 x^{4}-72 x^{2}-243\right ) \sqrt {4 x^{2}-9}}{768}-\frac {729 \ln \left (\sqrt {4 x^{2}-9}+2 x \right )}{512}\) | \(42\) |
risch | \(\frac {x \left (128 x^{4}-72 x^{2}-243\right ) \sqrt {4 x^{2}-9}}{768}-\frac {729 \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right ) \sqrt {4}}{1024}\) | \(47\) |
default | \(\frac {x^{3} \left (4 x^{2}-9\right )^{\frac {3}{2}}}{24}+\frac {9 x \left (4 x^{2}-9\right )^{\frac {3}{2}}}{128}+\frac {81 x \sqrt {4 x^{2}-9}}{256}-\frac {729 \ln \left (\sqrt {4}\, x +\sqrt {4 x^{2}-9}\right ) \sqrt {4}}{1024}\) | \(61\) |
meijerg | \(\frac {729 i \sqrt {\operatorname {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (\frac {i \sqrt {\pi }\, x \left (-\frac {640}{81} x^{4}+\frac {40}{9} x^{2}+15\right ) \sqrt {-\frac {4 x^{2}}{9}+1}}{90}-\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 x}{3}\right )}{4}\right )}{128 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\) | \(68\) |
pseudoelliptic | \(\frac {531441 \ln \left (\frac {\sqrt {4 x^{2}-9}-2 x}{x}\right )-531441 \ln \left (\frac {\sqrt {4 x^{2}-9}+2 x}{x}\right )+\left (124416 x^{5}-69984 x^{3}-236196 x \right ) \sqrt {4 x^{2}-9}}{1024 \left (\sqrt {4 x^{2}-9}+2 x \right )^{3} \left (-\sqrt {4 x^{2}-9}+2 x \right )^{3}}\) | \(100\) |
Input:
int(x^4*(4*x^2-9)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/768*x*(128*x^4-72*x^2-243)*(4*x^2-9)^(1/2)-729/512*ln((4*x^2-9)^(1/2)+2* x)
Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.58 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\frac {1}{768} \, {\left (128 \, x^{5} - 72 \, x^{3} - 243 \, x\right )} \sqrt {4 \, x^{2} - 9} + \frac {729}{512} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 9}\right ) \] Input:
integrate(x^4*(4*x^2-9)^(1/2),x, algorithm="fricas")
Output:
1/768*(128*x^5 - 72*x^3 - 243*x)*sqrt(4*x^2 - 9) + 729/512*log(-2*x + sqrt (4*x^2 - 9))
Result contains complex when optimal does not.
Time = 5.01 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.29 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\begin {cases} \frac {2 x^{7}}{3 \sqrt {4 x^{2} - 9}} - \frac {15 x^{5}}{8 \sqrt {4 x^{2} - 9}} - \frac {27 x^{3}}{64 \sqrt {4 x^{2} - 9}} + \frac {729 x}{256 \sqrt {4 x^{2} - 9}} - \frac {729 \operatorname {acosh}{\left (\frac {2 x}{3} \right )}}{512} & \text {for}\: \left |{x^{2}}\right | > \frac {9}{4} \\- \frac {2 i x^{7}}{3 \sqrt {9 - 4 x^{2}}} + \frac {15 i x^{5}}{8 \sqrt {9 - 4 x^{2}}} + \frac {27 i x^{3}}{64 \sqrt {9 - 4 x^{2}}} - \frac {729 i x}{256 \sqrt {9 - 4 x^{2}}} + \frac {729 i \operatorname {asin}{\left (\frac {2 x}{3} \right )}}{512} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(4*x**2-9)**(1/2),x)
Output:
Piecewise((2*x**7/(3*sqrt(4*x**2 - 9)) - 15*x**5/(8*sqrt(4*x**2 - 9)) - 27 *x**3/(64*sqrt(4*x**2 - 9)) + 729*x/(256*sqrt(4*x**2 - 9)) - 729*acosh(2*x /3)/512, Abs(x**2) > 9/4), (-2*I*x**7/(3*sqrt(9 - 4*x**2)) + 15*I*x**5/(8* sqrt(9 - 4*x**2)) + 27*I*x**3/(64*sqrt(9 - 4*x**2)) - 729*I*x/(256*sqrt(9 - 4*x**2)) + 729*I*asin(2*x/3)/512, True))
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\frac {1}{24} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x^{3} + \frac {9}{128} \, {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} x + \frac {81}{256} \, \sqrt {4 \, x^{2} - 9} x - \frac {729}{512} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} - 9}\right ) \] Input:
integrate(x^4*(4*x^2-9)^(1/2),x, algorithm="maxima")
Output:
1/24*(4*x^2 - 9)^(3/2)*x^3 + 9/128*(4*x^2 - 9)^(3/2)*x + 81/256*sqrt(4*x^2 - 9)*x - 729/512*log(8*x + 4*sqrt(4*x^2 - 9))
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.61 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\frac {1}{768} \, {\left (8 \, {\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt {4 \, x^{2} - 9} x + \frac {729}{512} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 9} \right |}\right ) \] Input:
integrate(x^4*(4*x^2-9)^(1/2),x, algorithm="giac")
Output:
1/768*(8*(16*x^2 - 9)*x^2 - 243)*sqrt(4*x^2 - 9)*x + 729/512*log(abs(-2*x + sqrt(4*x^2 - 9)))
Timed out. \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\int x^4\,\sqrt {4\,x^2-9} \,d x \] Input:
int(x^4*(4*x^2 - 9)^(1/2),x)
Output:
int(x^4*(4*x^2 - 9)^(1/2), x)
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int x^4 \sqrt {-9+4 x^2} \, dx=\frac {\sqrt {4 x^{2}-9}\, x^{5}}{6}-\frac {3 \sqrt {4 x^{2}-9}\, x^{3}}{32}-\frac {81 \sqrt {4 x^{2}-9}\, x}{256}-\frac {729 \,\mathrm {log}\left (\frac {\sqrt {4 x^{2}-9}}{3}+\frac {2 x}{3}\right )}{512} \] Input:
int(x^4*(4*x^2-9)^(1/2),x)
Output:
(256*sqrt(4*x**2 - 9)*x**5 - 144*sqrt(4*x**2 - 9)*x**3 - 486*sqrt(4*x**2 - 9)*x - 2187*log((sqrt(4*x**2 - 9) + 2*x)/3))/1536