Integrand size = 36, antiderivative size = 161 \[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {\left (-2 \sqrt {a} \sqrt {b}-c\right ) \arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )}{4 a^{3/4} b^{3/4}}+\frac {\left (-2 \sqrt {a} \sqrt {b}+c\right ) \text {arctanh}\left (\frac {-b+2 \sqrt {a} \sqrt {b} x+a x^2}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\right )}{4 a^{3/4} b^{3/4}} \]
1/4*(-2*a^(1/2)*b^(1/2)-c)*arctan(2*a^(1/4)*b^(1/4)*(a*x^3-b*x)^(1/2)/(-b- 2*a^(1/2)*b^(1/2)*x+a*x^2))/a^(3/4)/b^(3/4)+1/4*(-2*a^(1/2)*b^(1/2)+c)*arc tanh(1/2*(-b+2*a^(1/2)*b^(1/2)*x+a*x^2)/a^(1/4)/b^(1/4)/(a*x^3-b*x)^(1/2)) /a^(3/4)/b^(3/4)
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.99 \[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {x} \sqrt {-b+a x^2} \left (\left (2 i \sqrt {a} \sqrt {b}-c\right ) \arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+\left (2 \sqrt {a} \sqrt {b}-i c\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{a^{3/4} b^{3/4} \sqrt {-b x+a x^3}} \]
((1/4 + I/4)*Sqrt[x]*Sqrt[-b + a*x^2]*(((2*I)*Sqrt[a]*Sqrt[b] - c)*ArcTan[ ((1 + I)*a^(1/4)*b^(1/4)*Sqrt[x])/Sqrt[-b + a*x^2]] + (2*Sqrt[a]*Sqrt[b] - I*c)*ArcTan[((1/2 + I/2)*Sqrt[-b + a*x^2])/(a^(1/4)*b^(1/4)*Sqrt[x])]))/( a^(3/4)*b^(3/4)*Sqrt[-(b*x) + a*x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.46 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.50, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, 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 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {a x^2-b} \int -\frac {-a x^2-c x+b}{\sqrt {x} \sqrt {a x^2-b} \left (a x^2+b\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {a x^2-b} \int \frac {-a x^2-c x+b}{\sqrt {x} \sqrt {a x^2-b} \left (a x^2+b\right )}dx}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2-b} \int \frac {-a x^2-c x+b}{\sqrt {a x^2-b} \left (a x^2+b\right )}d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2-b} \int \left (\frac {2 b-c x}{\sqrt {a x^2-b} \left (a x^2+b\right )}-\frac {1}{\sqrt {a x^2-b}}\right )d\sqrt {x}}{\sqrt {a x^3-b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {a x^2-b} \left (\frac {\left (2 a \sqrt {b}-\sqrt {-a} c\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a x^2-b}}+\frac {\left (2 a \sqrt {b}+\sqrt {-a} c\right ) \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{4 a^{5/4} \sqrt [4]{b} \sqrt {a x^2-b}}-\frac {\sqrt [4]{b} \sqrt {1-\frac {a x^2}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right ),-1\right )}{\sqrt [4]{a} \sqrt {a x^2-b}}+\frac {\left (2 \sqrt {-a} \sqrt {b}-c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{4 \sqrt {2} (-a)^{3/4} b^{3/4}}+\frac {\left (2 \sqrt {-a} \sqrt {b}+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{4 \sqrt {2} (-a)^{3/4} b^{3/4}}\right )}{\sqrt {a x^3-b x}}\) |
(-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*Sqrt[-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)) - (b^(1/4)*Sqrt[1 - (a* x^2)/b]*EllipticF[ArcSin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(a^(1/4)*Sqrt[-b + a*x^2]) + ((2*a*Sqrt[b] - Sqrt[-a]*c)*Sqrt[1 - (a*x^2)/b]*EllipticF[Arc Sin[(a^(1/4)*Sqrt[x])/b^(1/4)], -1])/(4*a^(5/4)*b^(1/4)*Sqrt[-b + a*x^2]) + ((2*a*Sqrt[b] + Sqrt[-a]*c)*Sqrt[1 - (a*x^2)/b]*EllipticF[ArcSin[(a^(1/4 )*Sqrt[x])/b^(1/4)], -1])/(4*a^(5/4)*b^(1/4)*Sqrt[-b + a*x^2])))/Sqrt[-(b* x) + a*x^3]
3.22.76.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(119)=238\).
Time = 0.67 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.63
| method | result | size |
| default | \(\frac {\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) c \sqrt {a b}}{4}+\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) a b}{2}+\left (a b +\frac {\sqrt {a b}\, c}{2}\right ) \left (\arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-\arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{2 \left (a b \right )^{\frac {1}{4}} b a}\) | \(263\) |
| pseudoelliptic | \(\frac {\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) c \sqrt {a b}}{4}+\frac {\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) a b}{2}+\left (a b +\frac {\sqrt {a b}\, c}{2}\right ) \left (\arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-\arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{2 \left (a b \right )^{\frac {1}{4}} b a}\) | \(263\) |
| 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 )}\) | \(664\) |
1/2/(a*b)^(1/4)*(1/4*ln((a*x^2+2*x*(a*b)^(1/2)+2*(a*b)^(1/4)*(x*(a*x^2-b)) ^(1/2)-b)/(a*x^2+2*x*(a*b)^(1/2)-2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/2)-b))*c*( a*b)^(1/2)+1/2*ln((a*x^2+2*x*(a*b)^(1/2)-2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/2) -b)/(a*x^2+2*x*(a*b)^(1/2)+2*(a*b)^(1/4)*(x*(a*x^2-b))^(1/2)-b))*a*b+(a*b+ 1/2*(a*b)^(1/2)*c)*(arctan((x*(a*b)^(1/4)+(x*(a*x^2-b))^(1/2))/(a*b)^(1/4) /x)-arctan((x*(a*b)^(1/4)-(x*(a*x^2-b))^(1/2))/(a*b)^(1/4)/x)))/b/a
Leaf count of result is larger than twice the leaf count of optimal. 1593 vs. \(2 (119) = 238\).
Time = 0.52 (sec) , antiderivative size = 1593, normalized size of antiderivative = 9.89 \[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\text {Too large to display} \]
-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)))*s qrt(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...
\[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int \frac {a x^{2} - b + c x}{\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} + b\right )}\, dx \]
\[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + c x - b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
\[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\int { \frac {a x^{2} + c x - b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}} \,d x } \]
Timed out. \[ \int \frac {-b+c x+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx=\text {Hanged} \]