Integrand size = 19, antiderivative size = 113 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=-\frac {3 a \sqrt {b x^{2/3}+a x}}{4 x}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{8 b x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^2}+\frac {3 a^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{8 b^{3/2}} \]
-(b*x^(2/3)+a*x)^(3/2)/x^2+3/8*a^3*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x) ^(1/2))/b^(3/2)-3/4*a*(b*x^(2/3)+a*x)^(1/2)/x-3/8*a^2*(b*x^(2/3)+a*x)^(1/2 )/b/x^(2/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.54 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=\frac {6 a^3 \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^4 \sqrt [3]{x}} \]
(6*a^3*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 4, 7 /2, 1 + (a*x^(1/3))/b])/(5*b^4*x^(1/3))
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1926, 1926, 1931, 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.
\(\displaystyle \int \frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 1926 |
\(\displaystyle \frac {1}{2} a \int \frac {\sqrt {x^{2/3} b+a x}}{x^2}dx-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2}\) |
\(\Big \downarrow \) 1926 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2}\) |
\(\Big \downarrow \) 1931 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} a \left (\frac {1}{4} a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{2 x}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^2}\) |
-((b*x^(2/3) + a*x)^(3/2)/x^2) + (a*((-3*Sqrt[b*x^(2/3) + a*x])/(2*x) + (a *((-3*Sqrt[b*x^(2/3) + a*x])/(b*x^(2/3)) + (3*a*ArcTanh[(Sqrt[b]*x^(1/3))/ Sqrt[b*x^(2/3) + a*x]])/b^(3/2)))/4))/2
3.2.81.3.1 Defintions of rubi rules used
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 + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*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]) && LtQ[ m + j*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 = 1.75 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b \,a^{3} x -3 b^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}-8 b^{\frac {5}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+3 b^{\frac {7}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{8 x^{2} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {5}{2}}}\) | \(93\) |
default | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b \,a^{3} x -3 b^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}-8 b^{\frac {5}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+3 b^{\frac {7}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{8 x^{2} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {5}{2}}}\) | \(93\) |
1/8*(b*x^(2/3)+a*x)^(3/2)*(3*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b*a^3*x- 3*b^(3/2)*(b+a*x^(1/3))^(5/2)-8*b^(5/2)*(b+a*x^(1/3))^(3/2)+3*b^(7/2)*(b+a *x^(1/3))^(1/2))/x^2/(b+a*x^(1/3))^(3/2)/b^(5/2)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=\int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=-\frac {\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {3 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{4} + 8 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{4} b - 3 \, \sqrt {a x^{\frac {1}{3}} + b} a^{4} b^{2}}{a^{3} b x}}{8 \, a} \]
-1/8*(3*a^4*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b) + (3*(a*x^(1 /3) + b)^(5/2)*a^4 + 8*(a*x^(1/3) + b)^(3/2)*a^4*b - 3*sqrt(a*x^(1/3) + b) *a^4*b^2)/(a^3*b*x))/a
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^3} \,d x \]