Integrand size = 19, antiderivative size = 144 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac {4 a^{3/4} b^{3/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt {b \sqrt [3]{x}+a x}} \]
-2*(b*x^(1/3)+a*x)^(3/2)/x+4*a*(b*x^(1/3)+a*x)^(1/2)+4*a^(3/4)*b^(3/4)*x^( 1/6)*(cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*arctan(a^(1/4) *x^(1/6)/b^(1/4)))*EllipticF(sin(2*arctan(a^(1/4)*x^(1/6)/b^(1/4))),1/2*2^ (1/2))*(x^(1/3)*a^(1/2)+b^(1/2))*((b+a*x^(2/3))/(x^(1/3)*a^(1/2)+b^(1/2))^ 2)^(1/2)/(b*x^(1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=-\frac {2 b \sqrt {b \sqrt [3]{x}+a x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},-\frac {a x^{2/3}}{b}\right )}{\sqrt {1+\frac {a x^{2/3}}{b}} x^{2/3}} \]
(-2*b*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-3/2, -3/4, 1/4, -((a*x^(2/3 ))/b)])/(Sqrt[1 + (a*x^(2/3))/b]*x^(2/3))
Time = 0.33 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1924, 1926, 1927, 1917, 266, 761}
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.
\(\displaystyle \int \frac {\left (a x+b \sqrt [3]{x}\right )^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 1924 |
\(\displaystyle 3 \int \frac {\left (\sqrt [3]{x} b+a x\right )^{3/2}}{x^{4/3}}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 1926 |
\(\displaystyle 3 \left (2 a \int \frac {\sqrt {\sqrt [3]{x} b+a x}}{\sqrt [3]{x}}d\sqrt [3]{x}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle 3 \left (2 a \left (\frac {2}{3} b \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}+\frac {2}{3} \sqrt {a x+b \sqrt [3]{x}}\right )-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle 3 \left (2 a \left (\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{3} \sqrt {a x+b \sqrt [3]{x}}\right )-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 266 |
\(\displaystyle 3 \left (2 a \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{3} \sqrt {a x+b \sqrt [3]{x}}\right )-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 761 |
\(\displaystyle 3 \left (2 a \left (\frac {2 b^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}+\frac {2}{3} \sqrt {a x+b \sqrt [3]{x}}\right )-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x}\right )\) |
3*((-2*(b*x^(1/3) + a*x)^(3/2))/(3*x) + 2*a*((2*Sqrt[b*x^(1/3) + a*x])/3 + (2*b^(3/4)*(Sqrt[b] + Sqrt[a]*x^(2/3))*Sqrt[b + a*x^(2/3)]*x^(1/6)*Sqrt[( b + a*x^(4/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2]*EllipticF[2*ArcTan[(a^(1/4)* x^(1/6))/b^(1/4)], 1/2])/(3*a^(1/4)*Sqrt[b*x^(1/3) + a*x]*Sqrt[b + a*x^(4/ 3)])))
3.2.44.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]) Int[ x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp [1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j ] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1 ]
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*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]
Time = 2.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {4 x^{\frac {1}{3}} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b +2 x^{\frac {4}{3}} a^{2}-2 b^{2}}{x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(130\) |
derivativedivides | \(-\frac {2 b \sqrt {b \,x^{\frac {1}{3}}+a x}}{x^{\frac {2}{3}}}+2 a \sqrt {b \,x^{\frac {1}{3}}+a x}+\frac {4 b \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(147\) |
2/x^(1/3)*(2*x^(1/3)*(-a*b)^(1/2)*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^ (1/2)*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)*a/(-a*b)^ (1/2))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2 ^(1/2))*b+x^(4/3)*a^2-b^2)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=\int \frac {\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^2} \,d x \]