Integrand size = 21, antiderivative size = 65 \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b^{3/2}} \] Output:
1/3*x^(1/2)*(b*x^5+a*x^2)^(1/2)/b-1/3*a*arctanh(b^(1/2)*x^(5/2)/(b*x^5+a*x ^2)^(1/2))/b^(3/2)
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26 \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {\sqrt {b} x^{5/2} \left (a+b x^3\right )-a x \sqrt {a+b x^3} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{3 b^{3/2} \sqrt {x^2 \left (a+b x^3\right )}} \] Input:
Integrate[x^(9/2)/Sqrt[a*x^2 + b*x^5],x]
Output:
(Sqrt[b]*x^(5/2)*(a + b*x^3) - a*x*Sqrt[a + b*x^3]*Log[Sqrt[b]*x^(3/2) + S qrt[a + b*x^3]])/(3*b^(3/2)*Sqrt[x^2*(a + b*x^3)])
Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1930, 1935, 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.
\(\displaystyle \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \int \frac {x^{3/2}}{\sqrt {b x^5+a x^2}}dx}{2 b}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \int \frac {1}{1-\frac {b x^5}{b x^5+a x^2}}d\frac {x^{5/2}}{\sqrt {b x^5+a x^2}}}{3 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^5}}{3 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a x^2+b x^5}}\right )}{3 b^{3/2}}\) |
Input:
Int[x^(9/2)/Sqrt[a*x^2 + b*x^5],x]
Output:
(Sqrt[x]*Sqrt[a*x^2 + b*x^5])/(3*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a* x^2 + b*x^5]])/(3*b^(3/2))
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[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 0.84 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {x^{\frac {3}{2}} \left (b \,x^{3}+a \right ) \left (x \sqrt {x \left (b \,x^{3}+a \right )}\, \sqrt {b}-\operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right ) a \right )}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, b^{\frac {3}{2}}}\) | \(79\) |
risch | \(\frac {x^{\frac {5}{2}} \left (b \,x^{3}+a \right )}{3 b \sqrt {x^{2} \left (b \,x^{3}+a \right )}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right ) \sqrt {x}\, \sqrt {x \left (b \,x^{3}+a \right )}}{3 b^{\frac {3}{2}} \sqrt {x^{2} \left (b \,x^{3}+a \right )}}\) | \(82\) |
Input:
int(x^(9/2)/(b*x^5+a*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*x^(3/2)*(b*x^3+a)*(x*(x*(b*x^3+a))^(1/2)*b^(1/2)-arctanh((x*(b*x^3+a)) ^(1/2)/x^2/b^(1/2))*a)/(b*x^5+a*x^2)^(1/2)/(x*(b*x^3+a))^(1/2)/b^(3/2)
Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\left [\frac {a \sqrt {b} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} + 4 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} + a\right )} \sqrt {b} \sqrt {x} - a^{2}\right ) + 4 \, \sqrt {b x^{5} + a x^{2}} b \sqrt {x}}{12 \, b^{2}}, \frac {a \sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {-b} \sqrt {x}}{2 \, b x^{3} + a}\right ) + 2 \, \sqrt {b x^{5} + a x^{2}} b \sqrt {x}}{6 \, b^{2}}\right ] \] Input:
integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")
Output:
[1/12*(a*sqrt(b)*log(-8*b^2*x^6 - 8*a*b*x^3 + 4*sqrt(b*x^5 + a*x^2)*(2*b*x ^3 + a)*sqrt(b)*sqrt(x) - a^2) + 4*sqrt(b*x^5 + a*x^2)*b*sqrt(x))/b^2, 1/6 *(a*sqrt(-b)*arctan(2*sqrt(b*x^5 + a*x^2)*sqrt(-b)*sqrt(x)/(2*b*x^3 + a)) + 2*sqrt(b*x^5 + a*x^2)*b*sqrt(x))/b^2]
\[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{\frac {9}{2}}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \] Input:
integrate(x**(9/2)/(b*x**5+a*x**2)**(1/2),x)
Output:
Integral(x**(9/2)/sqrt(x**2*(a + b*x**3)), x)
\[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\int { \frac {x^{\frac {9}{2}}}{\sqrt {b x^{5} + a x^{2}}} \,d x } \] Input:
integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(9/2)/sqrt(b*x^5 + a*x^2), x)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {\sqrt {b x^{3} + a} x^{\frac {3}{2}}}{3 \, b \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{3 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")
Output:
1/3*sqrt(b*x^3 + a)*x^(3/2)/(b*sgn(x)) + 1/3*a*log(abs(-sqrt(b)*x^(3/2) + sqrt(b*x^3 + a)))/(b^(3/2)*sgn(x))
Timed out. \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\int \frac {x^{9/2}}{\sqrt {b\,x^5+a\,x^2}} \,d x \] Input:
int(x^(9/2)/(a*x^2 + b*x^5)^(1/2),x)
Output:
int(x^(9/2)/(a*x^2 + b*x^5)^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \frac {x^{9/2}}{\sqrt {a x^2+b x^5}} \, dx=\frac {2 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b x +\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a -\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a}{6 b^{2}} \] Input:
int(x^(9/2)/(b*x^5+a*x^2)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(a + b*x**3)*b*x + sqrt(b)*log(sqrt(a + b*x**3) - sqrt(x)*s qrt(b)*x)*a - sqrt(b)*log(sqrt(a + b*x**3) + sqrt(x)*sqrt(b)*x)*a)/(6*b**2 )