Integrand size = 19, antiderivative size = 298 \[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {1768 b^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 a^5}-\frac {1768 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{168245 a^4}+\frac {1768 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{216315 a^3}-\frac {136 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{19665 a^2}+\frac {8 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a}+\frac {4}{69} b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{9} x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {884 b^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{100947 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]
2/9*x^3*(b*x^(1/3)+a*x)^(3/2)+1768/100947*b^6*(b*x^(1/3)+a*x)^(1/2)/a^5-17 68/168245*b^5*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^4+1768/216315*b^4*x^(4/3)*(b *x^(1/3)+a*x)^(1/2)/a^3-136/19665*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^2+8/1311 *b^2*x^(8/3)*(b*x^(1/3)+a*x)^(1/2)/a+4/69*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2) -884/100947*b^(27/4)*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)/a^(21/4)/(b*x^(1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.18 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.48 \[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (\left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}} \left (3315 b^4-7293 a b^3 x^{2/3}+12155 a^2 b^2 x^{4/3}-17765 a^3 b x^2+24035 a^4 x^{8/3}\right )-3315 b^6 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{216315 a^5 \sqrt {1+\frac {a x^{2/3}}{b}}} \]
(2*Sqrt[b*x^(1/3) + a*x]*((b + a*x^(2/3))^2*Sqrt[1 + (a*x^(2/3))/b]*(3315* b^4 - 7293*a*b^3*x^(2/3) + 12155*a^2*b^2*x^(4/3) - 17765*a^3*b*x^2 + 24035 *a^4*x^(8/3)) - 3315*b^6*Hypergeometric2F1[-3/2, 1/4, 5/4, -((a*x^(2/3))/b )]))/(216315*a^5*Sqrt[1 + (a*x^(2/3))/b])
Time = 0.60 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1924, 1927, 1927, 1930, 1930, 1930, 1930, 1930, 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.
| \(\displaystyle \int x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int x^{8/3} \left (\sqrt [3]{x} b+a x\right )^{3/2}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle 3 \left (\frac {2}{9} b \int x^3 \sqrt {\sqrt [3]{x} b+a x}d\sqrt [3]{x}+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \int \frac {x^{10/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \int \frac {x^{8/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \int \frac {x^2}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \int \frac {x^{4/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \int \frac {x^{2/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{3 a}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {2}{9} b \left (\frac {2}{23} b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )+\frac {2}{23} x^{10/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{27} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2}\right )\) |
3*((2*x^3*(b*x^(1/3) + a*x)^(3/2))/27 + (2*b*((2*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/23 + (2*b*((2*x^(8/3)*Sqrt[b*x^(1/3) + a*x])/(19*a) - (17*b*((2*x^2 *Sqrt[b*x^(1/3) + a*x])/(15*a) - (13*b*((2*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/ (11*a) - (9*b*((2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(7*a) - (5*b*((2*Sqrt[b*x ^(1/3) + a*x])/(3*a) - (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]*Elliptic F[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(3*a^(5/4)*Sqrt[b*x^(1/3) + a *x]*Sqrt[b + a*x^(4/3)])))/(7*a)))/(11*a)))/(15*a)))/(19*a)))/23))/9)
3.2.40.3.1 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 + 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]
Time = 3.36 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(\frac {\frac {1126 x^{\frac {11}{3}} a^{6} b^{2}}{3933}+\frac {104 x^{\frac {13}{3}} a^{7} b}{207}-\frac {16 a^{5} b^{3} x^{3}}{19665}-\frac {3536 x^{\frac {5}{3}} a^{3} b^{5}}{1514205}+\frac {272 x^{\frac {7}{3}} a^{4} b^{4}}{216315}+\frac {2 x^{5} a^{8}}{9}-\frac {884 b^{7} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{100947}+\frac {3536 a^{2} b^{6} x}{504735}+\frac {1768 x^{\frac {1}{3}} a \,b^{7}}{100947}}{a^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(196\) |
| derivativedivides | \(\frac {2 a \,x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9}+\frac {58 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207}+\frac {8 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a}-\frac {136 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{19665 a^{2}}+\frac {1768 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{216315 a^{3}}-\frac {1768 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{168245 a^{4}}+\frac {1768 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{100947 a^{5}}-\frac {884 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{100947 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(262\) |
2/1514205*(216755*x^(11/3)*a^6*b^2+380380*x^(13/3)*a^7*b-616*a^5*b^3*x^3-1 768*x^(5/3)*a^3*b^5+952*x^(7/3)*a^4*b^4+168245*x^5*a^8-6630*b^7*(-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))+5304*a^2*b^6*x+13260*x^ (1/3)*a*b^7)/a^6/(x^(1/3)*(b+a*x^(2/3)))^(1/2)
\[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{2} \,d x } \]
\[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int x^{2} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \]
\[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{2} \,d x } \]
\[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int x^2\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2} \,d x \]