Integrand size = 17, antiderivative size = 140 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=-\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {5 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a x+b x^3}} \]
-10/21*a*(b*x^3+a*x)^(1/2)/b^2+2/7*x^2*(b*x^3+a*x)^(1/2)/b+5/21*a^(7/4)*(c os(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^(9/4 )/(b*x^3+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\frac {2 x \left (-5 a^2-2 a b x^2+3 b^2 x^4+5 a^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{21 b^2 \sqrt {x \left (a+b x^2\right )}} \]
(2*x*(-5*a^2 - 2*a*b*x^2 + 3*b^2*x^4 + 5*a^2*Sqrt[1 + (b*x^2)/a]*Hypergeom etric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^2*Sqrt[x*(a + b*x^2)])
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {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.
\(\displaystyle \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {5 a \int \frac {x^2}{\sqrt {b x^3+a x}}dx}{7 b}\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {5 a \left (\frac {2 \sqrt {a x+b x^3}}{3 b}-\frac {a \int \frac {1}{\sqrt {b x^3+a x}}dx}{3 b}\right )}{7 b}\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {5 a \left (\frac {2 \sqrt {a x+b x^3}}{3 b}-\frac {a \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {x} \sqrt {b x^2+a}}dx}{3 b \sqrt {a x+b x^3}}\right )}{7 b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {5 a \left (\frac {2 \sqrt {a x+b x^3}}{3 b}-\frac {2 a \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}}{3 b \sqrt {a x+b x^3}}\right )}{7 b}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {5 a \left (\frac {2 \sqrt {a x+b x^3}}{3 b}-\frac {a^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 b^{5/4} \sqrt {a x+b x^3}}\right )}{7 b}\) |
(2*x^2*Sqrt[a*x + b*x^3])/(7*b) - (5*a*((2*Sqrt[a*x + b*x^3])/(3*b) - (a^( 3/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^ 2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(3*b^(5/4)*Sqrt[a* x + b*x^3])))/(7*b)
3.1.57.3.1 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^(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 = 2.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {2 \left (-3 b \,x^{2}+5 a \right ) x \left (b \,x^{2}+a \right )}{21 b^{2} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(147\) |
default | \(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b}-\frac {10 a \sqrt {b \,x^{3}+a x}}{21 b^{2}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(149\) |
elliptic | \(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b}-\frac {10 a \sqrt {b \,x^{3}+a x}}{21 b^{2}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(149\) |
-2/21*(-3*b*x^2+5*a)/b^2*x*(b*x^2+a)/(x*(b*x^2+a))^(1/2)+5/21*a^2/b^3*(-a* b)^(1/2)*((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)*Elliptic F(((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 = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\frac {2 \, {\left (5 \, a^{2} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, b^{2} x^{2} - 5 \, a b\right )} \sqrt {b x^{3} + a x}\right )}}{21 \, b^{3}} \]
2/21*(5*a^2*sqrt(b)*weierstrassPInverse(-4*a/b, 0, x) + (3*b^2*x^2 - 5*a*b )*sqrt(b*x^3 + a*x))/b^3
\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int \frac {x^{4}}{\sqrt {x \left (a + b x^{2}\right )}}\, dx \]
\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{3} + a x}} \,d x } \]
\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{3} + a x}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int \frac {x^4}{\sqrt {b\,x^3+a\,x}} \,d x \]