Integrand size = 19, antiderivative size = 239 \[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{154 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]
-3*x^3/a/(b*x^(1/3)+a*x)^(1/2)-663/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^5+1989/3 85*b^2*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a^4-221/55*b*x^(4/3)*(b*x^(1/3)+a*x)^ (1/2)/a^3+17/5*x^2*(b*x^(1/3)+a*x)^(1/2)/a^2+663/154*b^(15/4)*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 = 10.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.50 \[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {-3315 b^4 \sqrt [3]{x}-1326 a b^3 x+442 a^2 b^2 x^{5/3}-238 a^3 b x^{7/3}+154 a^4 x^3+3315 b^4 \sqrt {1+\frac {a x^{2/3}}{b}} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )}{385 a^5 \sqrt {b \sqrt [3]{x}+a x}} \]
(-3315*b^4*x^(1/3) - 1326*a*b^3*x + 442*a^2*b^2*x^(5/3) - 238*a^3*b*x^(7/3 ) + 154*a^4*x^3 + 3315*b^4*Sqrt[1 + (a*x^(2/3))/b]*x^(1/3)*Hypergeometric2 F1[1/4, 1/2, 5/4, -((a*x^(2/3))/b)])/(385*a^5*Sqrt[b*x^(1/3) + a*x])
Time = 0.50 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1924, 1928, 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.
| \(\displaystyle \int \frac {x^3}{\left (a x+b \sqrt [3]{x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int \frac {x^{11/3}}{\left (\sqrt [3]{x} b+a x\right )^{3/2}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1928 |
| \(\displaystyle 3 \left (\frac {17 \int \frac {x^{8/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \int \frac {x^2}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \int \frac {x^{4/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \int \frac {x^{2/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{7 a}\right )}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{3 a}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {17 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{2 a}-\frac {x^3}{a \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
3*(-(x^3/(a*Sqrt[b*x^(1/3) + a*x])) + (17*((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]*EllipticF[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)))/(2*a))
3.2.59.3.1 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^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(n - j)*( p + 1))), x] - Simp[c^n*((m + j*p - n + j + 1)/(b*(n - j)*(p + 1))) Int[( c*x)^(m - n)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && !In tegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1] & & GtQ[m + j*p + 1, n - j]
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 = 3.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(-\frac {3 x^{\frac {1}{3}} b^{4}}{a^{5} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 a^{2}}-\frac {56 b \,x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 a^{3}}+\frac {834 b^{2} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 a^{4}}-\frac {432 b^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 a^{5}}+\frac {663 b^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{154 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(228\) |
| default | \(-\frac {-884 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{\frac {5}{3}} a^{3} b^{2}+476 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{\frac {7}{3}} a^{4} b -3315 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b^{4}+2652 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x \,a^{2} b^{3}-308 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} x^{3}+4320 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{\frac {1}{3}} a \,b^{4}+2310 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {1}{3}} a \,b^{4}}{770 x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right ) a^{6}}\) | \(260\) |
-3*x^(1/3)/a^5*b^4/((x^(2/3)+b/a)*x^(1/3)*a)^(1/2)+2/5*x^2*(b*x^(1/3)+a*x) ^(1/2)/a^2-56/55*b*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^3+834/385*b^2*x^(2/3)*( b*x^(1/3)+a*x)^(1/2)/a^4-432/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^5+663/154*b^4/ a^6*(-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/(-a*b)^(1/2))^(1 /2)/(b*x^(1/3)+a*x)^(1/2)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^( 1/2))^(1/2),1/2*2^(1/2))
\[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
integral((a^4*x^5 + 3*a^2*b^2*x^(11/3) - 2*a*b^3*x^3 - (2*a^3*b*x^4 - b^4* x^2)*x^(1/3))*sqrt(a*x + b*x^(1/3))/(a^6*x^4 + 2*a^3*b^3*x^2 + b^6), x)
\[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]