Integrand size = 21, antiderivative size = 95 \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=-\frac {3 a \sqrt {a x^2+b x^3}}{4 b^2 \sqrt {x}}+\frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{4 b^{5/2}} \] Output:
-3/4*a*(b*x^3+a*x^2)^(1/2)/b^2/x^(1/2)+1/2*x^(1/2)*(b*x^3+a*x^2)^(1/2)/b+3 /4*a^2*arctanh(b^(1/2)*x^(3/2)/(b*x^3+a*x^2)^(1/2))/b^(5/2)
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04 \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\frac {\sqrt {b} x^{3/2} \left (-3 a^2-a b x+2 b^2 x^2\right )+6 a^2 x \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{4 b^{5/2} \sqrt {x^2 (a+b x)}} \] Input:
Integrate[x^(5/2)/Sqrt[a*x^2 + b*x^3],x]
Output:
(Sqrt[b]*x^(3/2)*(-3*a^2 - a*b*x + 2*b^2*x^2) + 6*a^2*x*Sqrt[a + b*x]*ArcT anh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/(4*b^(5/2)*Sqrt[x^2*(a + b*x)])
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1930, 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.
\(\displaystyle \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}-\frac {3 a \int \frac {x^{3/2}}{\sqrt {b x^3+a x^2}}dx}{4 b}\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \int \frac {\sqrt {x}}{\sqrt {b x^3+a x^2}}dx}{2 b}\right )}{4 b}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \int \frac {1}{1-\frac {b x^3}{b x^3+a x^2}}d\frac {x^{3/2}}{\sqrt {b x^3+a x^2}}}{b}\right )}{4 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b x^3}}{2 b}-\frac {3 a \left (\frac {\sqrt {a x^2+b x^3}}{b \sqrt {x}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x^2+b x^3}}\right )}{b^{3/2}}\right )}{4 b}\) |
Input:
Int[x^(5/2)/Sqrt[a*x^2 + b*x^3],x]
Output:
(Sqrt[x]*Sqrt[a*x^2 + b*x^3])/(2*b) - (3*a*(Sqrt[a*x^2 + b*x^3]/(b*Sqrt[x] ) - (a*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a*x^2 + b*x^3]])/b^(3/2)))/(4*b)
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 = 0.40 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94
method | result | size |
risch | \(-\frac {\left (-2 b x +3 a \right ) x^{\frac {3}{2}} \left (b x +a \right )}{4 b^{2} \sqrt {x^{2} \left (b x +a \right )}}+\frac {3 a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x}\, \sqrt {x \left (b x +a \right )}}{8 b^{\frac {5}{2}} \sqrt {x^{2} \left (b x +a \right )}}\) | \(89\) |
default | \(\frac {\sqrt {x}\, \left (4 b^{\frac {7}{2}} x^{3}-2 a \,x^{2} b^{\frac {5}{2}}-6 b^{\frac {3}{2}} a^{2} x +3 \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {2 \sqrt {b \,x^{2}+a x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b \right )}{8 \sqrt {b \,x^{3}+a \,x^{2}}\, b^{\frac {7}{2}}}\) | \(92\) |
Input:
int(x^(5/2)/(b*x^3+a*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(-2*b*x+3*a)*x^(3/2)*(b*x+a)/b^2/(x^2*(b*x+a))^(1/2)+3/8*a^2/b^(5/2)* ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))/(x^2*(b*x+a))^(1/2)*x^(1/2)*(x*( b*x+a))^(1/2)
Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.76 \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} x \log \left (\frac {2 \, b x^{2} + a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {b} \sqrt {x}}{x}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {x}}{8 \, b^{3} x}, -\frac {3 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-b} \sqrt {x}}{b x^{2} + a x}\right ) - \sqrt {b x^{3} + a x^{2}} {\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt {x}}{4 \, b^{3} x}\right ] \] Input:
integrate(x^(5/2)/(b*x^3+a*x^2)^(1/2),x, algorithm="fricas")
Output:
[1/8*(3*a^2*sqrt(b)*x*log((2*b*x^2 + a*x + 2*sqrt(b*x^3 + a*x^2)*sqrt(b)*s qrt(x))/x) + 2*sqrt(b*x^3 + a*x^2)*(2*b^2*x - 3*a*b)*sqrt(x))/(b^3*x), -1/ 4*(3*a^2*sqrt(-b)*x*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-b)*sqrt(x)/(b*x^2 + a *x)) - sqrt(b*x^3 + a*x^2)*(2*b^2*x - 3*a*b)*sqrt(x))/(b^3*x)]
\[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\int \frac {x^{\frac {5}{2}}}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \] Input:
integrate(x**(5/2)/(b*x**3+a*x**2)**(1/2),x)
Output:
Integral(x**(5/2)/sqrt(x**2*(a + b*x)), x)
\[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\int { \frac {x^{\frac {5}{2}}}{\sqrt {b x^{3} + a x^{2}}} \,d x } \] Input:
integrate(x^(5/2)/(b*x^3+a*x^2)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(5/2)/sqrt(b*x^3 + a*x^2), x)
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\frac {3 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, b^{\frac {5}{2}}} + \frac {\sqrt {b x + a} \sqrt {x} {\left (\frac {2 \, x}{b} - \frac {3 \, a}{b^{2}}\right )} - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{b^{\frac {5}{2}}}}{4 \, \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^(5/2)/(b*x^3+a*x^2)^(1/2),x, algorithm="giac")
Output:
3/8*a^2*log(abs(a))*sgn(x)/b^(5/2) + 1/4*(sqrt(b*x + a)*sqrt(x)*(2*x/b - 3 *a/b^2) - 3*a^2*log(abs(-sqrt(b)*sqrt(x) + sqrt(b*x + a)))/b^(5/2))/sgn(x)
Timed out. \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\int \frac {x^{5/2}}{\sqrt {b\,x^3+a\,x^2}} \,d x \] Input:
int(x^(5/2)/(a*x^2 + b*x^3)^(1/2),x)
Output:
int(x^(5/2)/(a*x^2 + b*x^3)^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.60 \[ \int \frac {x^{5/2}}{\sqrt {a x^2+b x^3}} \, dx=\frac {-3 \sqrt {x}\, \sqrt {b x +a}\, a b +2 \sqrt {x}\, \sqrt {b x +a}\, b^{2} x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2}}{4 b^{3}} \] Input:
int(x^(5/2)/(b*x^3+a*x^2)^(1/2),x)
Output:
( - 3*sqrt(x)*sqrt(a + b*x)*a*b + 2*sqrt(x)*sqrt(a + b*x)*b**2*x + 3*sqrt( b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2)/(4*b**3)