3.32.44 \(\int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} (c+b x^2)^2} \, dx\) [3144]

3.32.44.1 Optimal result

Integrand size = 42, antiderivative size = 1196 \[ \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 )^2} \, dx=\frac {\left (-a^2 b c-a b^2 c+b^3 c+a b^3 c-b^4 x+a^3 c x-a^2 b c x-a^2 b^2 c x+a b^3 c x\right ) \sqrt {c+b x+a x^2}}{2 c \left (b^3+a^2 c-2 a b c+b^2 c\right ) \left (c+b x^2\right )}-\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^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-a b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-2 b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-3 a b^5 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+a^3 b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-5 a^2 b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-4 a^3 b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+7 a^2 b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )-a b^4 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{5/2} b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^{3/2} b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \sqrt {a} b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+6 a^{3/2} b^4 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a^{7/2} c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^{5/2} b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+8 a^{7/2} b c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-12 a^{5/2} b^2 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a^{3/2} b^3 c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+b^5 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^3 b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 b^2 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 b^3 c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a b^4 c \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 ]}{8 c \left (b^3+a^2 c-2 a b c+b^2 c\right )} \]

output

Unintegrable
 

3.32.44.2 Mathematica [A] (verified)

Time = 3.40 (sec) , antiderivative size = 1991, normalized size of antiderivative = 1.66 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx =\text {Too large to display} \]

input

Integrate[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)^2), 
x]
 

output

(-4*RootSum[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^3*Log[x]) + 4*a^2*c*Log[x] - a^2*b*c*Log[x] - 4*a*b^2*c*Log[x 
] + b^3*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] - 4*a^2*c*Log[-Sqrt[c 
] + Sqrt[c + b*x + a*x^2] - x*#1] + a^2*b*c*Log[-Sqrt[c] + Sqrt[c + b*x + 
a*x^2] - x*#1] + 4*a*b^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + 
2*b^2*Sqrt[c]*Log[x]*#1 - 2*b^2*Sqrt[c]*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^ 
2] - x*#1]*#1 + a*b*c*Log[x]*#1^2 - a*b*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) & 
] + (4*Sqrt[c]*Sqrt[c + x*(b + a*x)]*(a^3*c*x + b^3*(c - b*x) + a*b^2*c*(- 
1 + b + b*x) - a^2*b*c*(1 + x + b*x)) + (c + b*x^2)*RootSum[b^3 + a^2*c - 
4*b^2*Sqrt[c]*#1 - 2*a*c*#1^2 + 4*b*c*#1^2 + c*#1^4 & , (-3*b^6*Log[x] + 2 
*a^3*b^2*c*Log[x] + 16*a^2*b^3*c*Log[x] + 7*a*b^4*c*Log[x] - 2*a^2*b^4*c*L 
og[x] - 4*b^5*c*Log[x] - 17*a*b^5*c*Log[x] + 14*a^4*c^2*Log[x] - 30*a^3*b* 
c^2*Log[x] + 16*a^2*b^2*c^2*Log[x] - 14*a^3*b^2*c^2*Log[x] + 30*a^2*b^3*c^ 
2*Log[x] - 16*a*b^4*c^2*Log[x] + 3*b^6*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2 
] - x*#1] - 2*a^3*b^2*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] - 16* 
a^2*b^3*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] - 7*a*b^4*c*Log[-Sq 
rt[c] + Sqrt[c + b*x + a*x^2] - x*#1] + 2*a^2*b^4*c*Log[-Sqrt[c] + Sqrt[c 
+ b*x + a*x^2] - x*#1] + 4*b^5*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x* 
#1] + 17*a*b^5*c*Log[-Sqrt[c] + Sqrt[c + b*x + a*x^2] - x*#1] - 14*a^4*...
 

3.32.44.3 Rubi [A] (warning: unable to verify)

Time = 33.85 (sec) , antiderivative size = 1707, normalized size of antiderivative = 1.43, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2136, 27, 1369, 1363, 778}

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.

\(\displaystyle \int \frac {a^2 x^2+a b c+b^2 (-x)}{\left (b x^2+c\right )^2 \sqrt {a x^2+b x+c}} \, dx\)

\(\Big \downarrow \) 2136

\(\displaystyle \frac {\int \frac {b c \left (c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-b \left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) x\right )}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{4 b c \left ((a c-b c)^2+b^3 c\right )}-\frac {\left (b c \left (a^2+a (1-b) b-b^2\right )+x \left (-a^2 c (a-b)+a b^2 c (a-b)+b^4\right )\right ) \sqrt {a x^2+b x+c}}{2 \left ((a c-b c)^2+b^3 c\right ) \left (b x^2+c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-b \left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) x}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{4 \left ((a c-b c)^2+b^3 c\right )}-\frac {\left (b c \left (a^2+a (1-b) b-b^2\right )+x \left (-a^2 c (a-b)+a b^2 c (a-b)+b^4\right )\right ) \sqrt {a x^2+b x+c}}{2 \left ((a c-b c)^2+b^3 c\right ) \left (b x^2+c\right )}\)

\(\Big \downarrow \) 1369

\(\displaystyle \frac {\frac {\int \frac {c \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )-b \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}-\frac {\int \frac {c \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )-b \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) x}{\sqrt {a x^2+b x+c} \left (b x^2+c\right )}dx}{2 \sqrt {c} \sqrt {a^2 c-2 a b c+b^3+b^2 c}}}{4 \left ((a c-b c)^2+b^3 c\right )}-\frac {\left (b c \left (a^2+a (1-b) b-b^2\right )+x \left (-a^2 c (a-b)+a b^2 c (a-b)+b^4\right )\right ) \sqrt {a x^2+b x+c}}{2 \left ((a c-b c)^2+b^3 c\right ) \left (b x^2+c\right )}\)

\(\Big \downarrow \) 1363

\(\displaystyle \frac {\frac {b c^{3/2} \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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}{\frac {b^3 c^3 \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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}{a x^2+b x+c}-2 b^2 c^3 \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )}d\left (-\frac {b c \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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}}\right )}{\sqrt {b^3+c b^2-2 a c b+a^2 c}}-\frac {b c^{3/2} \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\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}{\frac {b^3 c^3 \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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}{a x^2+b x+c}-2 b^2 c^3 \left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right ) \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )\right )}d\left (-\frac {b c \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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}}\right )}{\sqrt {b^3+c b^2-2 a c b+a^2 c}}}{4 \left (c b^3+(a c-b c)^2\right )}-\frac {\left (b \left (a^2+(1-b) b a-b^2\right ) c+\left (b^4+a (a-b) c b^2-a^2 (a-b) c\right ) x\right ) \sqrt {a x^2+b x+c}}{2 \left (c b^3+(a c-b c)^2\right ) \left (b x^2+c\right )}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {\frac {\sqrt {b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \text {arctanh}\left (\frac {\sqrt {b} \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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 {2} \sqrt {b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}}-\frac {\sqrt {\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \text {arctanh}\left (\frac {\sqrt {b} \left (b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )+\left (\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\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 {2} \sqrt {\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) b^2+\left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right ) \left (a c-b c-\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {b c \left (b^4+a (3 b+2 c+1) b^3+a^2 (-6 c b+b+2 c) b-2 a^3 (1-2 b) c\right )-\left (b^4-a c b^3-a (a+2) c b^2+3 a^2 c b+a^3 c\right ) \left (a c-b c+\sqrt {b^3+c b^2-2 a c b+a^2 c} \sqrt {c}\right )} \sqrt {a x^2+b x+c}}\right )}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {b^3+c b^2-2 a c b+a^2 c}}}{4 \left (c b^3+(a c-b c)^2\right )}-\frac {\left (b \left (a^2+(1-b) b a-b^2\right ) c+\left (b^4+a (a-b) c b^2-a^2 (a-b) c\right ) x\right ) \sqrt {a x^2+b x+c}}{2 \left (c b^3+(a c-b c)^2\right ) \left (b x^2+c\right )}\)

input

Int[(a*b*c - b^2*x + a^2*x^2)/(Sqrt[c + b*x + a*x^2]*(c + b*x^2)^2),x]
 

output

-1/2*((b*(a^2 + a*(1 - b)*b - b^2)*c + (b^4 - a^2*(a - b)*c + a*(a - b)*b^ 
2*c)*x)*Sqrt[c + b*x + a*x^2])/((b^3*c + (a*c - b*c)^2)*(c + b*x^2)) + (-( 
(Sqrt[b^2*(b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c) + (b^4 - 2 
*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c))*(a*c - 
 b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b*c*(b^4 - 2*a^3 
*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c)) - (b^4 + a 
^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c)*(a*c - b*c + Sqrt[c]*Sqrt[b^ 
3 + a^2*c - 2*a*b*c + b^2*c])]*ArcTanh[(Sqrt[b]*(b*c*(b^4 - 2*a^3*(1 - 2*b 
)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c)) - (b^4 + a^3*c + 3* 
a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c)*(a*c - b*c + Sqrt[c]*Sqrt[b^3 + a^2*c 
 - 2*a*b*c + b^2*c]) + (b^2*(b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a 
*b^3*c) + (b^4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2* 
c - 6*b*c))*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c]))*x)) 
/(Sqrt[2]*Sqrt[b^2*(b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c) + 
 (b^4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c 
))*(a*c - b*c - Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[b*c*(b^ 
4 - 2*a^3*(1 - 2*b)*c + a*b^3*(1 + 3*b + 2*c) + a^2*b*(b + 2*c - 6*b*c)) - 
 (b^4 + a^3*c + 3*a^2*b*c - a*(2 + a)*b^2*c - a*b^3*c)*(a*c - b*c + Sqrt[c 
]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])]*Sqrt[c + b*x + a*x^2])])/(Sqrt[2]* 
Sqrt[b]*Sqrt[c]*Sqrt[b^3 + a^2*c - 2*a*b*c + b^2*c])) + (Sqrt[b*c*(b^4 ...
 

3.32.44.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 1363

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]
 

rule 1369

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]
 

rule 2136

Int[(Px_)*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_ 
), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P 
x, x, 2]}, Simp[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a 
*c*e^2 + (c*d - a*f)^2)*(p + 1)))*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2* 
d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - 
 a*f)))*x), x] + Simp[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1))   Int[ 
(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^ 
2 - ((-a)*e)*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C* 
f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c 
^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + ((-a 
)*B)*(2*c^2*d + (-c)*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*(c*d - a*f) - 
a*(c*C*d - B*c*e - a*C*f)))*((-c)*e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - 
a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; Free 
Q[{a, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[a*c*e^2 + 
(c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]
 

3.32.44.4 Maple [N/A] (verified)

Time = 0.46 (sec) , antiderivative size = 1250, normalized size of antiderivative = 1.05

method	result	size
default	\(\text {Expression too large to display}\)	\(1250\)

input

int((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c)^2,x,method=_RETURN 
VERBOSE)
 

output

-1/4*a*(b^2+a)/(-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)*(a*(x-(-b*c)^(1/2)/b) 
^2+(-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))+1/4*a*(b^2+a)/(-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)*(a*(x+(-b*c)^(1/2)/b)^2-(-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))+1/4*(-a*b^2*c+(-b*c)^(1/2)*b^2+a^2*c)/c/b^2*(1/(-(-b*c)^(1/2)*b+a 
*c-b*c)*b/(x-(-b*c)^(1/2)/b)*(a*(x-(-b*c)^(1/2)/b)^2+(-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)- 
1/2*(-b*c)^(1/2)/c*(-(-b*c)^(1/2)*b+2*a*c)/(-(-b*c)^(1/2)*b+a*c-b*c)/(-(-( 
-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)*(a*(x-(-b*c)^(1/2)/b)^2+(-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)))+1/4*(-a*b^2*c-(-b*c)^(1/2)*b^2+a^2*c)/c/b^2*(1/((-b*c)^(1/2)*b+ 
a*c-b*c)*b/(x+(-b*c)^(1/2)/b)*(a*(x+(-b*c)^(1/2)/b)^2-(-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...
 

3.32.44.5 Fricas [F(-1)]

Timed out. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx=\text {Timed out} \]

input

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c)^2,x, algorit 
hm="fricas")
 

output

Timed out
 

3.32.44.6 Sympy [N/A]

Not integrable

Time = 24.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.03 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx=\int \frac {a^{2} x^{2} + a b c - b^{2} x}{\left (b x^{2} + c\right )^{2} \sqrt {a x^{2} + b x + c}}\, dx \]

input

integrate((a**2*x**2+a*b*c-b**2*x)/(a*x**2+b*x+c)**(1/2)/(b*x**2+c)**2,x)
 

output

Integral((a**2*x**2 + a*b*c - b**2*x)/((b*x**2 + c)**2*sqrt(a*x**2 + b*x + 
 c)), x)
 

3.32.44.7 Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.04 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx=\int { \frac {a^{2} x^{2} + a b c - b^{2} x}{\sqrt {a x^{2} + b x + c} {\left (b x^{2} + c\right )}^{2}} \,d x } \]

input

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c)^2,x, algorit 
hm="maxima")
 

output

integrate((a^2*x^2 + a*b*c - b^2*x)/(sqrt(a*x^2 + b*x + c)*(b*x^2 + c)^2), 
 x)
 

3.32.44.8 Giac [F(-1)]

Timed out. \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx=\text {Timed out} \]

input

integrate((a^2*x^2+a*b*c-b^2*x)/(a*x^2+b*x+c)^(1/2)/(b*x^2+c)^2,x, algorit 
hm="giac")
 

output

Timed out
 

3.32.44.9 Mupad [N/A]

Not integrable

Time = 8.65 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.04 \[ \int \frac {a b c-b^2 x+a^2 x^2}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )^2} \, dx=\int \frac {a^2\,x^2+c\,a\,b-b^2\,x}{{\left (b\,x^2+c\right )}^2\,\sqrt {a\,x^2+b\,x+c}} \,d x \]

input

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)^2*(c + b*x + a*x^2)^(1/2)),x)
 

output

int((a^2*x^2 - b^2*x + a*b*c)/((c + b*x^2)^2*(c + b*x + a*x^2)^(1/2)), x)