Integrand size = 17, antiderivative size = 116 \[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=-\frac {2 \sqrt {a x+b x^3}}{3 x^2}+\frac {2 b^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a x+b x^3}} \] Output:
-2/3*(b*x^3+a*x)^(1/2)/x^2+2/3*b^(3/4)*x^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2 +a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x^(1/2)/ a^(1/4)),1/2*2^(1/2))/a^(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 = 10.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=-\frac {2 \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\frac {b x^2}{a}\right )}{3 x^2 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[Sqrt[a*x + b*x^3]/x^3,x]
Output:
(-2*Sqrt[x*(a + b*x^2)]*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^2)/a)])/ (3*x^2*Sqrt[1 + (b*x^2)/a])
Time = 0.41 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1926, 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.
\(\displaystyle \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx\) |
\(\Big \downarrow \) 1926 |
\(\displaystyle \frac {2}{3} b \int \frac {1}{\sqrt {b x^3+a x}}dx-\frac {2 \sqrt {a x+b x^3}}{3 x^2}\) |
\(\Big \downarrow \) 1917 |
\(\displaystyle \frac {2 b \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {x} \sqrt {b x^2+a}}dx}{3 \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{3 x^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {4 b \sqrt {x} \sqrt {a+b x^2} \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}}{3 \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{3 x^2}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 b^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{3 x^2}\) |
Input:
Int[Sqrt[a*x + b*x^3]/x^3,x]
Output:
(-2*Sqrt[a*x + b*x^3])/(3*x^2) + (2*b^(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*a^(1/4)*Sqrt[a*x + b*x^3])
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 + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
Time = 0.50 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {2 \sqrt {b \,x^{3}+a x}}{3 x^{2}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {b \,x^{3}+a x}}\) | \(123\) |
elliptic | \(-\frac {2 \sqrt {b \,x^{3}+a x}}{3 x^{2}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {b \,x^{3}+a x}}\) | \(123\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )}{3 x \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {2 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {b \,x^{3}+a x}}\) | \(130\) |
Input:
int((b*x^3+a*x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-2/3*(b*x^3+a*x)^(1/2)/x^2+2/3*(-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)*(-1/(-a*b)^(1/2)* b*x)^(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))
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\frac {2 \, {\left (2 \, \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - \sqrt {b x^{3} + a x}\right )}}{3 \, x^{2}} \] Input:
integrate((b*x^3+a*x)^(1/2)/x^3,x, algorithm="fricas")
Output:
2/3*(2*sqrt(b)*x^2*weierstrassPInverse(-4*a/b, 0, x) - sqrt(b*x^3 + a*x))/ x^2
\[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\int \frac {\sqrt {x \left (a + b x^{2}\right )}}{x^{3}}\, dx \] Input:
integrate((b*x**3+a*x)**(1/2)/x**3,x)
Output:
Integral(sqrt(x*(a + b*x**2))/x**3, x)
\[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x^{3}} \,d x } \] Input:
integrate((b*x^3+a*x)^(1/2)/x^3,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a*x)/x^3, x)
\[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\int { \frac {\sqrt {b x^{3} + a x}}{x^{3}} \,d x } \] Input:
integrate((b*x^3+a*x)^(1/2)/x^3,x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a*x)/x^3, x)
Timed out. \[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\int \frac {\sqrt {b\,x^3+a\,x}}{x^3} \,d x \] Input:
int((a*x + b*x^3)^(1/2)/x^3,x)
Output:
int((a*x + b*x^3)^(1/2)/x^3, x)
\[ \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}-2 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{5}+a \,x^{3}}d x \right ) a x}{\sqrt {x}\, x} \] Input:
int((b*x^3+a*x)^(1/2)/x^3,x)
Output:
( - 2*(sqrt(a + b*x**2) + sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2))/(a*x**3 + b*x**5),x)*a*x))/(sqrt(x)*x)