Integrand size = 17, antiderivative size = 134 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\frac {4}{7} a \sqrt {a x+b x^3}+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}+\frac {4 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{7 \sqrt [4]{b} \sqrt {a x+b x^3}} \]
2/7*(b*x^3+a*x)^(3/2)/x+4/7*a*(b*x^3+a*x)^(1/2)+4/7*a^(7/4)*(cos(2*arctan( b^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x^(1/2)/a^(1/4)))* EllipticF(sin(2*arctan(b^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*b ^(1/2))*x^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(1/4)/(b*x^3+a*x )^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\frac {2 a \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}} \]
(2*a*Sqrt[x*(a + b*x^2)]*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)])/ Sqrt[1 + (b*x^2)/a]
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1927, 1927, 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.
\(\displaystyle \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {6}{7} a \int \frac {\sqrt {b x^3+a x}}{x}dx+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {6}{7} a \left (\frac {2}{3} a \int \frac {1}{\sqrt {b x^3+a x}}dx+\frac {2}{3} \sqrt {a x+b x^3}\right )+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle \frac {6}{7} a \left (\frac {2 a \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {x} \sqrt {b x^2+a}}dx}{3 \sqrt {a x+b x^3}}+\frac {2}{3} \sqrt {a x+b x^3}\right )+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {6}{7} a \left (\frac {4 a \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}}{3 \sqrt {a x+b x^3}}+\frac {2}{3} \sqrt {a x+b x^3}\right )+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {6}{7} a \left (\frac {2 a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a x+b x^3}}+\frac {2}{3} \sqrt {a x+b x^3}\right )+\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x}\) |
(2*(a*x + b*x^3)^(3/2))/(7*x) + (6*a*((2*Sqrt[a*x + b*x^3])/3 + (2*a^(3/4) *Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E llipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*b^(1/4)*Sqrt[a*x + b*x^3])))/7
3.1.50.3.1 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[((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]
Time = 2.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {2 \left (b \,x^{2}+3 a \right ) x \left (b \,x^{2}+a \right )}{7 \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 b \sqrt {b \,x^{3}+a x}}\) | \(143\) |
default | \(\frac {2 b \,x^{2} \sqrt {b \,x^{3}+a x}}{7}+\frac {6 a \sqrt {b \,x^{3}+a x}}{7}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 b \sqrt {b \,x^{3}+a x}}\) | \(144\) |
elliptic | \(\frac {2 b \,x^{2} \sqrt {b \,x^{3}+a x}}{7}+\frac {6 a \sqrt {b \,x^{3}+a x}}{7}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 b \sqrt {b \,x^{3}+a x}}\) | \(144\) |
2/7*(b*x^2+3*a)*x*(b*x^2+a)/(x*(b*x^2+a))^(1/2)+4/7*a^2*(-a*b)^(1/2)/b*((x +(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2) *b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*x^3+a*x)^(1/2)*EllipticF(((x+(-a*b) ^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\frac {2 \, {\left (4 \, a^{2} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (b^{2} x^{2} + 3 \, a b\right )} \sqrt {b x^{3} + a x}\right )}}{7 \, b} \]
2/7*(4*a^2*sqrt(b)*weierstrassPInverse(-4*a/b, 0, x) + (b^2*x^2 + 3*a*b)*s qrt(b*x^3 + a*x))/b
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\int \frac {\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\int { \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x^2} \, dx=\int \frac {{\left (b\,x^3+a\,x\right )}^{3/2}}{x^2} \,d x \]