Integrand size = 17, antiderivative size = 408 \[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=-\frac {88 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {88 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:
-88/1105*b^5*(b+a*x^(2/3))*x^(1/3)/a^(7/2)/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x^ (1/3)+a*x)^(1/2)+88/3315*b^4*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^3-88/4641*b^3 *x*(b*x^(1/3)+a*x)^(1/2)/a^2+24/1547*b^2*x^(5/3)*(b*x^(1/3)+a*x)^(1/2)/a+1 2/119*b*x^(7/3)*(b*x^(1/3)+a*x)^(1/2)+2/7*x^2*(b*x^(1/3)+a*x)^(3/2)+88/110 5*b^(21/4)*(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^(15/4)/(b*x^(1/3)+a*x)^(1/2)-44/1105*b^(21/4)*(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)*Inverse JacobiAM(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)),1/2*2^(1/2))/a^(15/4)/(b*x^(1/3 )+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.30 \[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {2 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x} \left (\left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}} \left (77 b^2-143 a b x^{2/3}+221 a^2 x^{4/3}\right )-77 b^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )\right )}{1547 a^3 \sqrt {1+\frac {a x^{2/3}}{b}}} \] Input:
Integrate[x*(b*x^(1/3) + a*x)^(3/2),x]
Output:
(2*x^(1/3)*Sqrt[b*x^(1/3) + a*x]*((b + a*x^(2/3))^2*Sqrt[1 + (a*x^(2/3))/b ]*(77*b^2 - 143*a*b*x^(2/3) + 221*a^2*x^(4/3)) - 77*b^4*Hypergeometric2F1[ -3/2, 3/4, 7/4, -((a*x^(2/3))/b)]))/(1547*a^3*Sqrt[1 + (a*x^(2/3))/b])
Time = 1.04 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1924, 1927, 1927, 1930, 1930, 1930, 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.
| \(\displaystyle \int x \left (a x+b \sqrt [3]{x}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int x^{5/3} \left (\sqrt [3]{x} b+a x\right )^{3/2}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle 3 \left (\frac {2}{7} b \int x^2 \sqrt {\sqrt [3]{x} b+a x}d\sqrt [3]{x}+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \int \frac {x^{7/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \int \frac {x^{5/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \int \frac {x}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {3 b \int \frac {\sqrt [3]{x}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{5 a}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {3 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {\sqrt [6]{x}}{\sqrt {x^{2/3} a+b}}d\sqrt [3]{x}}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {b} \sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 3 \left (\frac {2}{7} b \left (\frac {2}{17} b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x^{4/3}+b}}-\frac {\sqrt [6]{x} \sqrt {a x^{4/3}+b}}{\sqrt {a} x^{2/3}+\sqrt {b}}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )+\frac {2}{17} x^{7/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{21} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
Input:
Int[x*(b*x^(1/3) + a*x)^(3/2),x]
Output:
3*((2*x^2*(b*x^(1/3) + a*x)^(3/2))/21 + (2*b*((2*x^(7/3)*Sqrt[b*x^(1/3) + a*x])/17 + (2*b*((2*x^(5/3)*Sqrt[b*x^(1/3) + a*x])/(13*a) - (11*b*((2*x*Sq rt[b*x^(1/3) + a*x])/(9*a) - (7*b*((2*x^(1/3)*Sqrt[b*x^(1/3) + a*x])/(5*a) - (6*b*Sqrt[b + a*x^(2/3)]*x^(1/6)*(-((-((x^(1/6)*Sqrt[b + a*x^(4/3)])/(S qrt[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)*Sqrt [b + a*x^(4/3)])))/(5*a*Sqrt[b*x^(1/3) + a*x])))/(9*a)))/(13*a)))/17))/7)
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[(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^(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[((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 = 1.74 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(\frac {\frac {622 x^{\frac {8}{3}} a^{4} b^{2}}{1547}+\frac {80 x^{\frac {10}{3}} a^{5} b}{119}-\frac {16 a^{3} b^{3} x^{2}}{4641}-\frac {88 b^{6} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {44 b^{6} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {2 a^{6} x^{4}}{7}+\frac {88 x^{\frac {2}{3}} a \,b^{5}}{3315}+\frac {176 x^{\frac {4}{3}} a^{2} b^{4}}{23205}}{a^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(261\) |
| derivativedivides | \(\frac {2 a \,x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7}+\frac {46 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119}+\frac {24 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a}-\frac {88 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{2}}+\frac {88 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3315 a^{3}}-\frac {44 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{1105 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(271\) |
Input:
int(x*(b*x^(1/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/23205/a^4*(4665*x^(8/3)*a^4*b^2+7800*x^(10/3)*a^5*b-40*a^3*b^3*x^2-924*b ^6*((x^(1/3)*a+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)*a-(-a*b)^(1/ 2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticE(((x^(1/3 )*a+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+462*b^6*((x^(1/3)*a+(-a *b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)*a-(-a*b)^(1/2))/(-a*b)^(1/2))^ (1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticF(((x^(1/3)*a+(-a*b)^(1/2))/ (-a*b)^(1/2))^(1/2),1/2*2^(1/2))+3315*a^6*x^4+308*x^(2/3)*a*b^5+88*x^(4/3) *a^2*b^4)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)
\[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(b*x^(1/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
integral((a*x^2 + b*x^(4/3))*sqrt(a*x + b*x^(1/3)), x)
\[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int x \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x*(b*x**(1/3)+a*x)**(3/2),x)
Output:
Integral(x*(a*x + b*x**(1/3))**(3/2), x)
\[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(b*x^(1/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate((a*x + b*x^(1/3))^(3/2)*x, x)
\[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(b*x^(1/3)+a*x)^(3/2),x, algorithm="giac")
Output:
integrate((a*x + b*x^(1/3))^(3/2)*x, x)
Timed out. \[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int x\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2} \,d x \] Input:
int(x*(a*x + b*x^(1/3))^(3/2),x)
Output:
int(x*(a*x + b*x^(1/3))^(3/2), x)
\[ \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {\frac {24 x^{\frac {11}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{2} b^{2}}{1547}+\frac {46 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, a^{3} b \,x^{2}}{119}+\frac {88 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, b^{4}}{3315}+\frac {2 x^{\frac {19}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{4}}{7}-\frac {88 x^{\frac {7}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a \,b^{3}}{4641}-\frac {44 \left (\int \frac {x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {2}{3}} b +x^{\frac {4}{3}} a}d x \right ) b^{5}}{3315}}{a^{3}} \] Input:
int(x*(b*x^(1/3)+a*x)^(3/2),x)
Output:
(2*(180*x**(5/6)*sqrt(x**(2/3)*a + b)*a**2*b**2*x + 4485*sqrt(x)*sqrt(x**( 2/3)*a + b)*a**3*b*x**2 + 308*sqrt(x)*sqrt(x**(2/3)*a + b)*b**4 + 3315*x** (1/6)*sqrt(x**(2/3)*a + b)*a**4*x**3 - 220*x**(1/6)*sqrt(x**(2/3)*a + b)*a *b**3*x - 154*int((x**(1/6)*sqrt(x**(2/3)*a + b))/(x**(2/3)*b + x**(1/3)*a *x),x)*b**5))/(23205*a**3)