Integrand size = 27, antiderivative size = 666 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {2 \left (b^2 c e-2 a c^2 e-b^3 f-b c (c d-3 a f)-c \left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {f \left (c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )}}+\frac {f \left (c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}} \]
2*(b^2*c*e-2*a*c^2*e-b^3*f-b*c*(-3*a*f+c*d)-c*(-2*a*c*f+b^2*f-b*c*e+2*c^2* d)*x)/(-4*a*c+b^2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*x^2+b*x+a)^(1/2 )-1/2*f*arctanh(1/4*(4*a*f+2*x*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))-b*(e-(-4*d*f +e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b *f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))+f*( 2*a*f-b*(e+(-4*d*f+e^2)^(1/2))))/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))*2^(1 /2)/(-4*d*f+e^2)^(1/2)/(f*(2*a*f-b*(e-(-4*d*f+e^2)^(1/2)))+c*(e^2-2*d*f-e* (-4*d*f+e^2)^(1/2)))^(1/2)+1/2*f*arctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^(1/2 ))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2)/(c*x^2+b*x+a)^(1/2)/(c*e^2- 2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))*(f*(2*a*f-b*(e -(-4*d*f+e^2)^(1/2)))+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))/((-a*f+c*d)^2-(- a*e+b*d)*(-b*f+c*e))*2^(1/2)/(-4*d*f+e^2)^(1/2)/(c*(e^2-2*d*f+e*(-4*d*f+e^ 2)^(1/2))+f*(2*a*f-b*(e+(-4*d*f+e^2)^(1/2))))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 806, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {-2 \left (b^3 f+b^2 c (-e+f x)+b c (-3 a f+c (d-e x))+2 c^2 (c d x+a (e-f x))\right )+\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \text {RootSum}\left [c^2 d-b c e+b^2 f+2 \sqrt {a} c e \text {$\#$1}-4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2+b e \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-c^2 e^2 \log (x)+c^2 d f \log (x)+2 b c e f \log (x)-b^2 f^2 \log (x)-a c f^2 \log (x)+c^2 e^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-c^2 d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 b c e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+b^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a c f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 \sqrt {a} c e f \log (x) \text {$\#$1}+2 \sqrt {a} b f^2 \log (x) \text {$\#$1}+2 \sqrt {a} c e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \sqrt {a} b f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+c e^2 \log (x) \text {$\#$1}^2-c d f \log (x) \text {$\#$1}^2-b e f \log (x) \text {$\#$1}^2+a f^2 \log (x) \text {$\#$1}^2-c e^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+b e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {a} c e-2 \sqrt {a} b f-2 c d \text {$\#$1}+b e \text {$\#$1}+4 a f \text {$\#$1}-3 \sqrt {a} e \text {$\#$1}^2+2 d \text {$\#$1}^3}\&\right ]}{\left (b^2-4 a c\right ) \left (c^2 d^2-b c d e+f \left (b^2 d-a b e+a^2 f\right )+a c \left (e^2-2 d f\right )\right ) \sqrt {a+x (b+c x)}} \]
(-2*(b^3*f + b^2*c*(-e + f*x) + b*c*(-3*a*f + c*(d - e*x)) + 2*c^2*(c*d*x + a*(e - f*x))) + (b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]*RootSum[c^2*d - b*c* e + b^2*f + 2*Sqrt[a]*c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4*a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(c^2*e^2*Log[x]) + c^2*d*f*Lo g[x] + 2*b*c*e*f*Log[x] - b^2*f^2*Log[x] - a*c*f^2*Log[x] + c^2*e^2*Log[-S qrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - c^2*d*f*Log[-Sqrt[a] + Sqrt[a + b *x + c*x^2] - x*#1] - 2*b*c*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*# 1] + b^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + a*c*f^2*Log[-S qrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*Sqrt[a]*c*e*f*Log[x]*#1 + 2*Sqr t[a]*b*f^2*Log[x]*#1 + 2*Sqrt[a]*c*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2 ] - x*#1]*#1 - 2*Sqrt[a]*b*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1 ]*#1 + c*e^2*Log[x]*#1^2 - c*d*f*Log[x]*#1^2 - b*e*f*Log[x]*#1^2 + a*f^2*L og[x]*#1^2 - c*e^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 + c*d *f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 + b*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2 - a*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2)/(Sqrt[a]*c*e - 2*Sqrt[a]*b*f - 2*c*d*#1 + b*e*#1 + 4 *a*f*#1 - 3*Sqrt[a]*e*#1^2 + 2*d*#1^3) & ])/((b^2 - 4*a*c)*(c^2*d^2 - b*c* d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*Sqrt[a + x*(b + c*x)] )
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.
| \(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {2 \int -\frac {\left (b^2-4 a c\right ) \left (f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x\right )}{2 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {f (b e-a f)-c \left (e^2-d f\right )-f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}+\frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-c x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )-b c (c d-3 a f)-2 a c^2 e+b^3 (-f)+b^2 c e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\int -\frac {c e^2-b f e+a f^2-c d f+f (c e-b f) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{(c d-a f)^2-(b d-a e) (c e-b f)}\) |
3.2.24.3.1 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_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1905\) vs. \(2(609)=1218\).
Time = 1.09 (sec) , antiderivative size = 1906, normalized size of antiderivative = 2.86
-1/(-4*d*f+e^2)^(1/2)*(2/(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2 *a*f^2-b*e*f-2*c*d*f+c*e^2)*f^2/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c+1/f* (-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*(-b*f *(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^ 2)^(1/2)-2*f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/(-b*f*(-4*d*f+e^2)^(1/2)+(-4* d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)*(2*c*(x+1/2*(e+(-4*d*f+e^2 )^(1/2))/f)+1/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e))/(2*c*(-b*f*(-4*d*f+e^2)^( 1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2-1/f^2*(-c*(-4 *d*f+e^2)^(1/2)+b*f-c*e)^2)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c+1/f*(-c* (-4*d*f+e^2)^(1/2)+b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*(-b*f*(-4 *d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^( 1/2)-2/(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d *f+c*e^2)*f^2*2^(1/2)/((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a *f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+ e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2) +b*f-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-b*f*(-4*d*f+e^2) ^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*( x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c+4/f*(-c*(-4*d*f+e^2)^(1/2)+b*f-c*e)*(x +1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*(-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/ 2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^...
Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for more deta
Exception generated. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \]