Integrand size = 15, antiderivative size = 208 \[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=-\frac {8 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^2}+\frac {24 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a}+\frac {12}{55} b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{5} x \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {4 b^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{77 a^{9/4} \sqrt {b \sqrt [3]{x}+a x}} \]
2/5*x*(b*x^(1/3)+a*x)^(3/2)-8/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^2+24/385*b^2* x^(2/3)*(b*x^(1/3)+a*x)^(1/2)/a+12/55*b*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)+4/77 *b^(15/4)*x^(1/6)*(cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*a rctan(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^(9/4)/(b*x^(1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.51 \[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (-\left (\left (5 b-11 a x^{2/3}\right ) \left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}}\right )+5 b^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{55 a^2 \sqrt {1+\frac {a x^{2/3}}{b}}} \]
(2*Sqrt[b*x^(1/3) + a*x]*(-((5*b - 11*a*x^(2/3))*(b + a*x^(2/3))^2*Sqrt[1 + (a*x^(2/3))/b]) + 5*b^3*Hypergeometric2F1[-3/2, 1/4, 5/4, -((a*x^(2/3))/ b)]))/(55*a^2*Sqrt[1 + (a*x^(2/3))/b])
Time = 0.41 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1910, 1924, 1927, 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.
\(\displaystyle \int \left (a x+b \sqrt [3]{x}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1910 |
\(\displaystyle \frac {2}{5} b \int \sqrt [3]{x} \sqrt {\sqrt [3]{x} b+a x}dx+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 1924 |
\(\displaystyle \frac {6}{5} b \int x \sqrt {\sqrt [3]{x} b+a x}d\sqrt [3]{x}+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \int \frac {x^{4/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \int \frac {x^{2/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{7 a}\right )+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{3 a}\right )}{7 a}\right )+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {6}{5} b \left (\frac {2}{11} b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}\right )}{7 a}\right )+\frac {2}{11} x^{4/3} \sqrt {a x+b \sqrt [3]{x}}\right )+\frac {2}{5} x \left (a x+b \sqrt [3]{x}\right )^{3/2}\) |
(2*x*(b*x^(1/3) + a*x)^(3/2))/5 + (6*b*((2*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/ 11 + (2*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))/5
3.2.42.3.1 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[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[((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 = 2.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {\frac {262 a^{3} b^{2} x^{\frac {5}{3}}}{385}+\frac {56 a^{4} b \,x^{\frac {7}{3}}}{55}+\frac {4 b^{4} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77}-\frac {16 a^{2} b^{3} x}{385}+\frac {2 a^{5} x^{3}}{5}-\frac {8 a \,b^{4} x^{\frac {1}{3}}}{77}}{a^{3} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(163\) |
derivativedivides | \(\frac {2 a \,x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{5}+\frac {34 b \,x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{55}+\frac {24 b^{2} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 a}-\frac {8 b^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 a^{2}}+\frac {4 b^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(196\) |
2/385*(131*a^3*b^2*x^(5/3)+196*a^4*b*x^(7/3)+10*b^4*(-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))-8*a^2*b^3*x+77*a^5*x^3-20*a*b^4*x^( 1/3))/a^3/(x^(1/3)*(b+a*x^(2/3)))^(1/2)
\[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} \,d x } \]
Time = 9.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.19 \[ \int \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx=\frac {2\,x\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {9}{4};\ \frac {13}{4};\ -\frac {a\,x^{2/3}}{b}\right )}{3\,{\left (\frac {a\,x^{2/3}}{b}+1\right )}^{3/2}} \]