Integrand size = 19, antiderivative size = 319 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\frac {8 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 \sqrt {a} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {4}{5} b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{3} \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {8 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{3/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {4 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{5 a^{3/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:
8/5*b^2*(b+a*x^(2/3))*x^(1/3)/a^(1/2)/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x^(1/3) +a*x)^(1/2)+4/5*b*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)+2/3*(b*x^(1/3)+a*x)^(3/2)- 8/5*b^(9/4)*(b^(1/2)+a^(1/2)*x^(1/3))*((b+a*x^(2/3))/(b^(1/2)+a^(1/2)*x^(1 /3))^2)^(1/2)*x^(1/6)*EllipticE(sin(2*arctan(a^(1/4)*x^(1/6)/b^(1/4))),1/2 *2^(1/2))/a^(3/4)/(b*x^(1/3)+a*x)^(1/2)+4/5*b^(9/4)*(b^(1/2)+a^(1/2)*x^(1/ 3))*((b+a*x^(2/3))/(b^(1/2)+a^(1/2)*x^(1/3))^2)^(1/2)*x^(1/6)*InverseJacob iAM(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)),1/2*2^(1/2))/a^(3/4)/(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 = 0.19 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\frac {2 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )}{\sqrt {1+\frac {a x^{2/3}}{b}}} \] Input:
Integrate[(b*x^(1/3) + a*x)^(3/2)/x,x]
Output:
(2*b*x^(1/3)*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-3/2, 3/4, 7/4, -((a* x^(2/3))/b)])/Sqrt[1 + (a*x^(2/3))/b]
Time = 0.75 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1924, 1927, 1910, 1938, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {\left (a x+b \sqrt [3]{x}\right )^{3/2}}{x} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int \frac {\left (\sqrt [3]{x} b+a x\right )^{3/2}}{\sqrt [3]{x}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle 3 \left (\frac {2}{3} b \int \sqrt {\sqrt [3]{x} b+a x}d\sqrt [3]{x}+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1910 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {2}{5} b \int \frac {\sqrt [3]{x}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {\sqrt [6]{x}}{\sqrt {x^{2/3} a+b}}d\sqrt [3]{x}}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {b} \sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 3 \left (\frac {2}{3} b \left (\frac {4 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x^{4/3}+b}}-\frac {\sqrt [6]{x} \sqrt {a x^{4/3}+b}}{\sqrt {a} x^{2/3}+\sqrt {b}}}{\sqrt {a}}\right )}{5 \sqrt {a x+b \sqrt [3]{x}}}+\frac {2}{5} \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{9} \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
Input:
Int[(b*x^(1/3) + a*x)^(3/2)/x,x]
Output:
3*((2*(b*x^(1/3) + a*x)^(3/2))/9 + (2*b*((2*x^(1/3)*Sqrt[b*x^(1/3) + a*x]) /5 + (4*b*Sqrt[b + a*x^(2/3)]*x^(1/6)*(-((-((x^(1/6)*Sqrt[b + a*x^(4/3)])/ (Sqrt[b] + Sqrt[a]*x^(2/3))) + (b^(1/4)*(Sqrt[b] + Sqrt[a]*x^(2/3))*Sqrt[( b + a*x^(4/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2]*EllipticE[2*ArcTan[(a^(1/4)* x^(1/6))/b^(1/4)], 1/2])/(a^(1/4)*Sqrt[b + a*x^(4/3)]))/Sqrt[a]) + (b^(1/4 )*(Sqrt[b] + Sqrt[a]*x^(2/3))*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])/(2*a^(3/4)*Sq rt[b + a*x^(4/3)])))/(5*Sqrt[b*x^(1/3) + a*x])))/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Simp[a*(n - j)*(p/(n*p + 1)) Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
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 + 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[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Time = 0.72 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(\frac {2 a x \sqrt {b \,x^{\frac {1}{3}}+a x}}{3}+\frac {22 b \,x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{15}+\frac {4 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{5 a \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(205\) |
| default | \(\frac {\frac {8 b^{3} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {4 b^{3} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {22 x^{\frac {2}{3}} a \,b^{2}}{15}+\frac {32 x^{\frac {4}{3}} a^{2} b}{15}+\frac {2 a^{3} x^{2}}{3}}{a \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(228\) |
Input:
int((b*x^(1/3)+a*x)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
2/3*a*x*(b*x^(1/3)+a*x)^(1/2)+22/15*b*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)+4/5*b^ 2/a*(-a*b)^(1/2)*((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-2*(x^ (1/3)-1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1 /2)/(b*x^(1/3)+a*x)^(1/2)*(-2/a*(-a*b)^(1/2)*EllipticE(((x^(1/3)+1/a*(-a*b )^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/a*(-a*b)^(1/2)*EllipticF(((x ^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)))
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(3/2)/x,x, algorithm="fricas")
Output:
integral((a*x + b*x^(1/3))^(3/2)/x, x)
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\int \frac {\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}{x}\, dx \] Input:
integrate((b*x**(1/3)+a*x)**(3/2)/x,x)
Output:
Integral((a*x + b*x**(1/3))**(3/2)/x, x)
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(3/2)/x,x, algorithm="maxima")
Output:
integrate((a*x + b*x^(1/3))^(3/2)/x, x)
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(3/2)/x,x, algorithm="giac")
Output:
integrate((a*x + b*x^(1/3))^(3/2)/x, x)
Timed out. \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x} \,d x \] Input:
int((a*x + b*x^(1/3))^(3/2)/x,x)
Output:
int((a*x + b*x^(1/3))^(3/2)/x, x)
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x} \, dx=\frac {22 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, b}{15}+\frac {2 x^{\frac {7}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a}{3}+\frac {4 \left (\int \frac {x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {2}{3}} b +x^{\frac {4}{3}} a}d x \right ) b^{2}}{15} \] Input:
int((b*x^(1/3)+a*x)^(3/2)/x,x)
Output:
(2*(11*sqrt(x)*sqrt(x**(2/3)*a + b)*b + 5*x**(1/6)*sqrt(x**(2/3)*a + b)*a* x + 2*int((x**(1/6)*sqrt(x**(2/3)*a + b))/(x**(2/3)*b + x**(1/3)*a*x),x)*b **2))/15