Integrand size = 35, antiderivative size = 147 \[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]
-1/4*(2*a^(1/2)*b^(1/2)-c)*arctan(2^(1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2 )/(a*x^2+b))*2^(1/2)/a^(3/4)/b^(3/4)-1/4*(2*a^(1/2)*b^(1/2)+c)*arctanh(2^( 1/2)*a^(1/4)*b^(1/4)*(a*x^3+b*x)^(1/2)/(a*x^2+b))*2^(1/2)/a^(3/4)/b^(3/4)
Time = 1.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=-\frac {\sqrt {x} \sqrt {b+a x^2} \left (\left (2 \sqrt {a} \sqrt {b}-c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )+\left (2 \sqrt {a} \sqrt {b}+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {b+a x^2}}\right )\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x \left (b+a x^2\right )}} \]
-1/2*(Sqrt[x]*Sqrt[b + a*x^2]*((2*Sqrt[a]*Sqrt[b] - c)*ArcTan[(Sqrt[2]*a^( 1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]] + (2*Sqrt[a]*Sqrt[b] + c)*ArcTanh[( Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]]))/(Sqrt[2]*a^(3/4)*b^(3/ 4)*Sqrt[x*(b + a*x^2)])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.29 (sec) , antiderivative size = 460, normalized size of antiderivative = 3.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
| \(\displaystyle \int \frac {a x^2+b+c x}{\left (a x^2-b\right ) \sqrt {a x^3+b x}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2+b} \int -\frac {a x^2+c x+b}{\sqrt {x} \left (b-a x^2\right ) \sqrt {a x^2+b}}dx}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2+b} \int \frac {a x^2+c x+b}{\sqrt {x} \left (b-a x^2\right ) \sqrt {a x^2+b}}dx}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \int \frac {a x^2+c x+b}{\left (b-a x^2\right ) \sqrt {a x^2+b}}d\sqrt {x}}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \int \left (\frac {2 b+c x}{\left (b-a x^2\right ) \sqrt {a x^2+b}}-\frac {1}{\sqrt {a x^2+b}}\right )d\sqrt {x}}{\sqrt {a x^3+b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2+b} \left (\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{3/4} b^{3/4} \sqrt {a x^2+b}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{3/4} b^{3/4} \sqrt {a x^2+b}}+\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^2+b}}\right )}{\sqrt {a x^3+b x}}\) |
(-2*Sqrt[x]*Sqrt[b + a*x^2]*(((2*Sqrt[a]*Sqrt[b] - c)*ArcTan[(Sqrt[2]*a^(1 /4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x^2]])/(4*Sqrt[2]*a^(3/4)*b^(3/4)) + ((2*S qrt[a]*Sqrt[b] + c)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[b + a*x ^2]])/(4*Sqrt[2]*a^(3/4)*b^(3/4)) - ((Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2 )/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[b + a*x^2]) + ((2*Sqrt[a]*Sqrt[b] - c)*(Sqrt [b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*Arc Tan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(8*a^(3/4)*b^(3/4)*Sqrt[b + a*x^2]) + ((2*Sqrt[a]*Sqrt[b] + c)*(Sqrt[b] + Sqrt[a]*x)*Sqrt[(b + a*x^2)/(Sqrt[b] + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[(a^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(8*a ^(3/4)*b^(3/4)*Sqrt[b + a*x^2])))/Sqrt[b*x + a*x^3]
3.21.50.3.1 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 = 0.53 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) c \sqrt {a b}+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) c \sqrt {a b}+2 \ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) a b -4 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) a b \right )}{8 a b \left (a b \right )^{\frac {1}{4}}}\) | \(197\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) c \sqrt {a b}+2 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) c \sqrt {a b}+2 \ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {\left (a \,x^{2}+b \right ) x}}\right ) a b -4 \arctan \left (\frac {\sqrt {\left (a \,x^{2}+b \right ) x}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right ) a b \right )}{8 a b \left (a b \right )^{\frac {1}{4}}}\) | \(197\) |
| elliptic | \(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\) | \(691\) |
-1/8*2^(1/2)*(ln((-2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2))/(2^(1/2)*(a* b)^(1/4)*x-((a*x^2+b)*x)^(1/2)))*c*(a*b)^(1/2)+2*arctan(1/2*((a*x^2+b)*x)^ (1/2)/x*2^(1/2)/(a*b)^(1/4))*c*(a*b)^(1/2)+2*ln((-2^(1/2)*(a*b)^(1/4)*x-(( a*x^2+b)*x)^(1/2))/(2^(1/2)*(a*b)^(1/4)*x-((a*x^2+b)*x)^(1/2)))*a*b-4*arct an(1/2*((a*x^2+b)*x)^(1/2)/x*2^(1/2)/(a*b)^(1/4))*a*b)/a/b/(a*b)^(1/4)
Leaf count of result is larger than twice the leaf count of optimal. 1557 vs. \(2 (107) = 214\).
Time = 0.57 (sec) , antiderivative size = 1557, normalized size of antiderivative = 10.59 \[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\text {Too large to display} \]
1/8*sqrt(1/2)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4 *c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 + 6*(16 *a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4*a^3*b^ 2*c + a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 + a^2*b^2*c^2)*x - 2*(a^4*b^3*x^2 - a^ 3*b^3*c*x + a^3*b^4)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))*sqrt( a*x^3 + b*x)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) + 4* c)/(a*b)) + 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^3*b^4 - a^2*b^3*c^2)*x )*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b ^2)) - 1/8*sqrt(1/2)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^ 3)) + 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)*x^2 - 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2*c^3 + (4 *a^3*b^2*c + a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 + a^2*b^2*c^2)*x - 2*(a^4*b^3*x ^2 - a^3*b^3*c*x + a^3*b^4)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)) )*sqrt(a*x^3 + b*x)*sqrt((a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3 )) + 4*c)/(a*b)) + 4*((4*a^4*b^3 - a^3*b^2*c^2)*x^3 + (4*a^3*b^4 - a^2*b^3 *c^2)*x)*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4)/(a^3*b^3)))/(a^2*x^4 - 2*a*b* x^2 + b^2)) + 1/8*sqrt(1/2)*sqrt(-(a*b*sqrt((16*a^2*b^2 + 8*a*b*c^2 + c^4) /(a^3*b^3)) - 4*c)/(a*b))*log(-(16*a^2*b^4 - b^2*c^4 + (16*a^4*b^2 - a^2*c ^4)*x^4 + 6*(16*a^3*b^3 - a*b*c^4)*x^2 + 4*sqrt(1/2)*(4*a^2*b^3*c + a*b^2* c^3 + (4*a^3*b^2*c + a^2*b*c^3)*x^2 - 4*(4*a^3*b^3 + a^2*b^2*c^2)*x + 2...
\[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\int \frac {a x^{2} + b + c x}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} - b\right )}\, dx \]
\[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\int { \frac {a x^{2} + c x + b}{\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}} \,d x } \]
\[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\int { \frac {a x^{2} + c x + b}{\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}} \,d x } \]
Timed out. \[ \int \frac {b+c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx=\text {Hanged} \]