Integrand size = 42, antiderivative size = 343 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=-\frac {a^{3/2} \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{b}+\frac {\text {RootSum}\left [b^2 c+b c^2-4 \sqrt {a} b c \text {$\#$1}+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2+b \text {$\#$1}^4\&,\frac {-a^2 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{5/2} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^{3/2} b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^3 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} b c+2 a c \text {$\#$1}-b c \text {$\#$1}+b \text {$\#$1}^3}\&\right ]}{2 b} \]
Time = 0.56 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.22 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=-\frac {4 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {c}-\sqrt {c+x (b+a x)}}\right )+\frac {\text {RootSum}\left [b^3+a^2 c-4 b^2 \sqrt {c} \text {$\#$1}-2 a c \text {$\#$1}^2+4 b c \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {-b^4 \log (x)+a^3 c \log (x)-a^2 b^2 c \log (x)+b^4 \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )-a^3 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+a^2 b^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+2 b^3 \sqrt {c} \log (x) \text {$\#$1}-2 b^3 \sqrt {c} \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}-a^2 c \log (x) \text {$\#$1}^2+a b^2 c \log (x) \text {$\#$1}^2+a^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a b^2 c \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{b^2+a \sqrt {c} \text {$\#$1}-2 b \sqrt {c} \text {$\#$1}-\sqrt {c} \text {$\#$1}^3}\&\right ]}{\sqrt {c}}}{2 b} \]
-1/2*(4*a^(3/2)*ArcTanh[(Sqrt[a]*x)/(Sqrt[c] - Sqrt[c + x*(b + a*x)])] + R ootSum[b^3 + a^2*c - 4*b^2*Sqrt[c]*#1 - 2*a*c*#1^2 + 4*b*c*#1^2 + c*#1^4 & , (-(b^4*Log[x]) + a^3*c*Log[x] - a^2*b^2*c*Log[x] + b^4*Log[-Sqrt[c] + S qrt[c + b*x + a*x^2] - x*#1] - a^3*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + a^2*b^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + 2*b^3*S qrt[c]*Log[x]*#1 - 2*b^3*Sqrt[c]*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x* #1]*#1 - a^2*c*Log[x]*#1^2 + a*b^2*c*Log[x]*#1^2 + a^2*c*Log[-Sqrt[c] + Sq rt[c + b*x + a*x^2] - x*#1]*#1^2 - a*b^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a *x^2] - x*#1]*#1^2)/(b^2 + a*Sqrt[c]*#1 - 2*b*Sqrt[c]*#1 - Sqrt[c]*#1^3) & ]/Sqrt[c])/b
Leaf count is larger than twice the leaf count of optimal. \(1106\) vs. \(2(343)=686\).
Time = 3.06 (sec) , antiderivative size = 1106, normalized size of antiderivative = 3.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2144, 25, 1092, 219, 1369, 25, 27, 1363, 218}
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.
| \(\displaystyle \int \frac {a^2 x^2+a b c+b^2 (-x)}{\left (b x^2+c\right ) \sqrt {a x^2+b x+c}} \, dx\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {a^2 \int \frac {1}{\sqrt {a x^2+b x+c}}dx}{b}+\frac {\int -\frac {x b^3+a \left (a-b^2\right ) c}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a^2 \int \frac {1}{\sqrt {a x^2+b x+c}}dx}{b}-\frac {\int \frac {x b^3+a \left (a-b^2\right ) c}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{b}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 a^2 \int \frac {1}{4 a-\frac {(b+2 a x)^2}{a x^2+b x+c}}d\frac {b+2 a x}{\sqrt {a x^2+b x+c}}}{b}-\frac {\int \frac {x b^3+a \left (a-b^2\right ) c}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\int \frac {x b^3+a \left (a-b^2\right ) c}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{b}\) |
| \(\Big \downarrow \) 1369 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\frac {\int -\frac {\sqrt {c} \left (\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}-\frac {\int -\frac {\sqrt {c} \left (\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\frac {\int \frac {\sqrt {c} \left (\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}-\frac {\int \frac {\sqrt {c} \left (\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\frac {\int \frac {\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {a^2 c-2 a b c+b^3+b^2 c}}-\frac {\int \frac {\left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) x b^2+\sqrt {c} \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {a^2 c-2 a b c+b^3+b^2 c}}}{b}\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\frac {b^2 c^{3/2} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \int \frac {1}{2 c^{5/2} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) b^3+\frac {c^2 \left (b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x\right )^2 b^3}{a x^2+b x+c}}d\frac {b \sqrt {c} \left (b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x\right )}{\sqrt {a x^2+b x+c}}}{\sqrt {b^3+c b^2-2 a c b+a^2 c}}-\frac {b^2 c^{3/2} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \int \frac {1}{2 c^{5/2} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right ) \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) b^3+\frac {c^2 \left (b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x\right )^2 b^3}{a x^2+b x+c}}d\frac {b \sqrt {c} \left (b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x\right )}{\sqrt {a x^2+b x+c}}}{\sqrt {b^3+c b^2-2 a c b+a^2 c}}}{b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a^{3/2} \text {arctanh}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{b}-\frac {\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \arctan \left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c-a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (\sqrt {c} b+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {b^4-a c b^3+\left (c a^2+\sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c} a\right ) b^2+a^2 c b-a^2 \sqrt {c} \left (\sqrt {c} a+\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )}}-\frac {\sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \arctan \left (\frac {b \sqrt {c} \left (\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )\right )-\left (b^4-a \left (a-b^2\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {2} \sqrt [4]{c} \sqrt {b^4-a c b^3+a^2 c b^2+a^2 c b-a^3 c+a \left (a-b^2\right ) \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}} \sqrt {\sqrt {c} a^2+(1-b) b \sqrt {c} a-b \left (b \sqrt {c}-\sqrt {b^3+c b^2-2 a c b+a^2 c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {b^4-a c b^3+\left (a^2 c-a \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}\right ) b^2+a^2 c b-a^3 c+a^2 \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}}}}{b}\) |
-((-((Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a ^2*c - 2*a*b*c + b^2*c])]*(b^4 - a*(a - b^2)*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*ArcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)*b *Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) - (b^4 - a *(a - b^2)*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/( Sqrt[2]*c^(1/4)*Sqrt[b^4 - a^3*c + a^2*b*c + a^2*b^2*c - a*b^3*c + a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]]*Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] - Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt [c + b*x + a*x^2])])/(Sqrt[2]*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]* Sqrt[b^4 - a^3*c + a^2*b*c - a*b^3*c + a^2*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a* b*c + b^2*c] + b^2*(a^2*c - a*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]) ])) + (Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*(b^4 - a*(a - b^2)*(a*c - b*c + Sqrt[c]*Sqrt[b^ 3 + a^2*c - 2*a*b*c + b^2*c]))*ArcTan[(b*Sqrt[c]*(a^2*Sqrt[c] + a*(1 - b)* b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) - (b^4 - a*(a - b^2)*(a*c - b*c + Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)/ (Sqrt[2]*c^(1/4)*Sqrt[b^4 - a^3*c + a^2*b*c + a^2*b^2*c - a*b^3*c - a*(a - b^2)*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]]*Sqrt[a^2*Sqrt[c] + a*(1 - b)*b*Sqrt[c] - b*(b*Sqrt[c] + Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqr t[c + b*x + a*x^2])])/(Sqrt[2]*c^(1/4)*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2...
3.30.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp [1/(2*q) Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c *e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[ Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.32 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(\frac {a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{b}-\frac {\left (-a \,b^{2} c -\sqrt {-b c}\, b^{2}+a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (\sqrt {-b c}\, b +a c -b c \right )}{b}-\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-b c}}{b}\right )^{2} a -\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}{x +\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}-\frac {\left (a \,b^{2} c -\sqrt {-b c}\, b^{2}-a^{2} c \right ) \ln \left (\frac {-\frac {2 \left (-\sqrt {-b c}\, b +a c -b c \right )}{b}+\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}+2 \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-b c}}{b}\right )^{2} a +\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}{b c}-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}{x -\frac {\sqrt {-b c}}{b}}\right )}{2 \sqrt {-b c}\, b \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}\) | \(520\) |
a^(3/2)/b*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+b*x+c)^(1/2))-1/2*(-a*b^2*c-(-b*c) ^(1/2)*b^2+a^2*c)/(-b*c)^(1/2)/b/(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((- 2*((-b*c)^(1/2)*b+a*c-b*c)/b-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+(- b*c)^(1/2)/b)+2*(-((-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*((x+(-b*c)^(1/2)/b)^2* a-(-b*c)^(1/2)/b/c*((-b*c)^(1/2)*b+2*a*c)*(x+(-b*c)^(1/2)/b)-((-b*c)^(1/2) *b+a*c-b*c)/b)^(1/2))/(x+(-b*c)^(1/2)/b))-1/2*(a*b^2*c-(-b*c)^(1/2)*b^2-a^ 2*c)/(-b*c)^(1/2)/b/(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*ln((-2*(-(-b*c)^( 1/2)*b+a*c-b*c)/b+(-b*c)^(1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2) /b)+2*(-(-(-b*c)^(1/2)*b+a*c-b*c)/b)^(1/2)*((x-(-b*c)^(1/2)/b)^2*a+(-b*c)^ (1/2)/b/c*(-(-b*c)^(1/2)*b+2*a*c)*(x-(-b*c)^(1/2)/b)-(-(-b*c)^(1/2)*b+a*c- b*c)/b)^(1/2))/(x-(-b*c)^(1/2)/b))
Timed out. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 2.73 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.11 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int \frac {a^{2} x^{2} + a b c - b^{2} x}{\left (b x^{2} + c\right ) \sqrt {a x^{2} + b x + c}}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int { \frac {a^{2} x^{2} + a b c - b^{2} x}{\sqrt {a x^{2} + b x + c} {\left (b x^{2} + c\right )}} \,d x } \]
Exception generated. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Not integrable
Time = 7.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.12 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx=\int \frac {a^2\,x^2+c\,a\,b-b^2\,x}{\left (b\,x^2+c\right )\,\sqrt {a\,x^2+b\,x+c}} \,d x \]