Integrand size = 44, antiderivative size = 123 \[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=-\frac {4 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \]
-4/3*arctan(a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/(a^2*x^2+b^2))/a^(1/2)/b ^(1/2)-1/3*2^(1/2)*arctanh(2^(1/2)*a^(1/2)*b^(1/2)*(a^2*x^3+b^2*x)^(1/2)/( a^2*x^2+b^2))/a^(1/2)/b^(1/2)
Time = 0.89 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05 \[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (4 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{3 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]
-1/3*(Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(4*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt [b^2 + a^2*x^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[ b^2 + a^2*x^2]]))/(Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^2*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.92 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2467, 25, 2035, 7276, 2009}
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.
\(\displaystyle \int \frac {a^3 x^3+b^3}{\sqrt {a^2 x^3+b^2 x} \left (a^3 x^3-b^3\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int -\frac {b^3+a^3 x^3}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^3-a^3 x^3\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^3+a^3 x^3}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^3-a^3 x^3\right )}dx}{\sqrt {a^2 x^3+b^2 x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \frac {b^3+a^3 x^3}{\sqrt {b^2+a^2 x^2} \left (b^3-a^3 x^3\right )}d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \int \left (\frac {2 b^3}{\sqrt {b^2+a^2 x^2} \left (b^3-a^3 x^3\right )}-\frac {1}{\sqrt {b^2+a^2 x^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3+b^2 x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2+b^2} \left (\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b}}+\frac {\left (1-\sqrt [3]{-1}\right ) (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^2+b^2}}+\frac {(a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {a} \sqrt {b} \sqrt {a^2 x^2+b^2}}-\frac {(a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^2+b^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a^2 x^3+b^2 x}}\) |
(-2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*((2*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[ b^2 + a^2*x^2]])/(3*Sqrt[a]*Sqrt[b]) + ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sq rt[x])/Sqrt[b^2 + a^2*x^2]]/(3*Sqrt[2]*Sqrt[a]*Sqrt[b]) - ((b + a*x)*Sqrt[ (b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]] , 1/2])/(3*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + ((b + a*x)*Sqrt[(b^2 + a ^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/ (3*(1 - (-1)^(1/3))*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2]) + ((1 - (-1)^(1/3 ))*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a] *Sqrt[x])/Sqrt[b]], 1/2])/(3*Sqrt[a]*Sqrt[b]*Sqrt[b^2 + a^2*x^2])))/Sqrt[b ^2*x + a^2*x^3]
3.19.11.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\) | \(69\) |
pseudoelliptic | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\) | \(69\) |
elliptic | \(\text {Expression too large to display}\) | \(875\) |
-1/3*(2^(1/2)*arctanh(1/2*(x*(a^2*x^2+b^2))^(1/2)/x*2^(1/2)/(a*b)^(1/2))-4 *arctan((x*(a^2*x^2+b^2))^(1/2)/x/(a*b)^(1/2)))/(a*b)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (95) = 190\).
Time = 0.35 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.44 \[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=\left [\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} + 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 12 \, a b^{3} x + b^{4} - 4 \, \sqrt {2} {\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {\frac {1}{a b}}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {-\frac {1}{a b}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {-a b} \log \left (\frac {a^{4} x^{4} - 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {-a b}}{a^{4} x^{4} + 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}\right ] \]
[1/12*(sqrt(2)*a*b*sqrt(1/(a*b))*log((a^4*x^4 + 12*a^3*b*x^3 + 6*a^2*b^2*x ^2 + 12*a*b^3*x + b^4 - 4*sqrt(2)*(a^3*b*x^2 + 2*a^2*b^2*x + a*b^3)*sqrt(a ^2*x^3 + b^2*x)*sqrt(1/(a*b)))/(a^4*x^4 - 4*a^3*b*x^3 + 6*a^2*b^2*x^2 - 4* a*b^3*x + b^4)) + 8*sqrt(a*b)*arctan(1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2)*sqrt(a*b)/(a^3*b*x^3 + a*b^3*x)))/(a*b), 1/6*(sqrt(2)*a*b*sqr t(-1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a*b*sqrt(-1/(a*b))/(a^2 *x^2 + 2*a*b*x + b^2)) - 2*sqrt(-a*b)*log((a^4*x^4 - 6*a^3*b*x^3 + 3*a^2*b ^2*x^2 - 6*a*b^3*x + b^4 - 4*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2) *sqrt(-a*b))/(a^4*x^4 + 2*a^3*b*x^3 + 3*a^2*b^2*x^2 + 2*a*b^3*x + b^4)))/( a*b)]
\[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=\int \frac {\left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}\, dx \]
Integral((a*x + b)*(a**2*x**2 - a*b*x + b**2)/(sqrt(x*(a**2*x**2 + b**2))* (a*x - b)*(a**2*x**2 + a*b*x + b**2)), x)
\[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=\int { \frac {a^{3} x^{3} + b^{3}}{{\left (a^{3} x^{3} - b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
\[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=\int { \frac {a^{3} x^{3} + b^{3}}{{\left (a^{3} x^{3} - b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
Timed out. \[ \int \frac {b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (-b^3+a^3 x^3\right )} \, dx=\text {Hanged} \]