Integrand size = 19, antiderivative size = 78 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-6 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right ) \] Output:
6*b*(b*x^(2/3)+a*x)^(1/2)/x^(1/3)+2*(b*x^(2/3)+a*x)^(3/2)/x-6*b^(3/2)*arct anh(b^(1/2)*x^(1/3)/(b*x^(2/3)+a*x)^(1/2))
Time = 10.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.13 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (\sqrt {b+a \sqrt [3]{x}} \left (4 b+a \sqrt [3]{x}\right )-3 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a \sqrt [3]{x}}}{\sqrt {b}}\right )\right )}{\sqrt {b+a \sqrt [3]{x}} \sqrt [3]{x}} \] Input:
Integrate[(b*x^(2/3) + a*x)^(3/2)/x^2,x]
Output:
(2*Sqrt[b*x^(2/3) + a*x]*(Sqrt[b + a*x^(1/3)]*(4*b + a*x^(1/3)) - 3*b^(3/2 )*ArcTanh[Sqrt[b + a*x^(1/3)]/Sqrt[b]]))/(Sqrt[b + a*x^(1/3)]*x^(1/3))
Time = 0.38 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1927, 1927, 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.
\(\displaystyle \int \frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle b \int \frac {\sqrt {x^{2/3} b+a x}}{x^{4/3}}dx+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x}\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle b \left (b \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx+\frac {6 \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}\right )+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle b \left (\frac {6 \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}-6 b \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}\right )+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle b \left (\frac {6 \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}-6 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )\right )+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x}\) |
Input:
Int[(b*x^(2/3) + a*x)^(3/2)/x^2,x]
Output:
(2*(b*x^(2/3) + a*x)^(3/2))/x + b*((6*Sqrt[b*x^(2/3) + a*x])/x^(1/3) - 6*S qrt[b]*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])
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*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 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 = 0.39 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (-3 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right )+\left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}+3 b \sqrt {x^{\frac {1}{3}} a +b}\right )}{x \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}}\) | \(67\) |
default | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (-3 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right )+\left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}+3 b \sqrt {x^{\frac {1}{3}} a +b}\right )}{x \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}}\) | \(67\) |
Input:
int((b*x^(2/3)+a*x)^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
2*(b*x^(2/3)+a*x)^(3/2)*(-3*b^(3/2)*arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))+( x^(1/3)*a+b)^(3/2)+3*b*(x^(1/3)*a+b)^(1/2))/x/(x^(1/3)*a+b)^(3/2)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate((b*x**(2/3)+a*x)**(3/2)/x**2,x)
Output:
Integral((a*x + b*x**(2/3))**(3/2)/x**2, x)
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^2,x, algorithm="maxima")
Output:
integrate((a*x + b*x^(2/3))^(3/2)/x^2, x)
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\frac {6 \, b^{2} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} + 6 \, \sqrt {a x^{\frac {1}{3}} + b} b - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )}}{\sqrt {-b}} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^2,x, algorithm="giac")
Output:
6*b^2*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/sqrt(-b) + 2*(a*x^(1/3) + b)^(3 /2) + 6*sqrt(a*x^(1/3) + b)*b - 2*(3*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt (-b)*b^(3/2))/sqrt(-b)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^2} \,d x \] Input:
int((a*x + b*x^(2/3))^(3/2)/x^2,x)
Output:
int((a*x + b*x^(2/3))^(3/2)/x^2, x)
Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx=2 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a +8 \sqrt {x^{\frac {1}{3}} a +b}\, b +3 \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) b -3 \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) b \] Input:
int((b*x^(2/3)+a*x)^(3/2)/x^2,x)
Output:
2*x**(1/3)*sqrt(x**(1/3)*a + b)*a + 8*sqrt(x**(1/3)*a + b)*b + 3*sqrt(b)*l og(sqrt(x**(1/3)*a + b) - sqrt(b))*b - 3*sqrt(b)*log(sqrt(x**(1/3)*a + b) + sqrt(b))*b