Optimal. Leaf size=1293 \[ \frac {\left (-x b^4+a c b^3+c b^3+a c x b^3-a c b^2-a^2 c x b^2-a^2 c b-a^2 c x b+a^3 c x\right ) \sqrt {a x^2+b x+c}}{2 c \left (b^3+c b^2-2 a c b+a^2 c\right ) \left (b x^2+c\right )}-\frac {\text {RootSum}\left [b \text {$\#$1}^4+4 a c \text {$\#$1}^2-2 b c \text {$\#$1}^2-4 \sqrt {a} b c \text {$\#$1}+b c^2+b^2 c\& ,\frac {\log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b^5-3 a c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^5-2 c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^5-a c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b^4-a c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^4-a c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^4+6 a^{3/2} c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^4+2 \sqrt {a} c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^4-a^2 c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b^3-2 a c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b^3+7 a^2 c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^3+2 a c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^3-a^2 c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^3+4 a^{3/2} c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^3+2 a^{3/2} c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^3+3 a^2 c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b^2-4 a^3 c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^2-5 a^2 c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b^2-12 a^{5/2} c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^2+2 a^{5/2} c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b^2+a^3 c \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}^2 b+a^3 c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) b+8 a^{7/2} c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b+4 a^{5/2} c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1} b-4 a^{7/2} c^2 \log \left (-\sqrt {a} x-\text {$\#$1}+\sqrt {a x^2+b x+c}\right ) \text {$\#$1}}{\text {$\#$1}^3 b^4-\sqrt {a} c b^4-c \text {$\#$1} b^4+c \text {$\#$1}^3 b^3-\sqrt {a} c^2 b^3-c^2 \text {$\#$1} b^3+2 a c \text {$\#$1} b^3-2 a c \text {$\#$1}^3 b^2+2 a^{3/2} c^2 b^2+4 a c^2 \text {$\#$1} b^2+a^2 c \text {$\#$1}^3 b-a^{5/2} c^2 b-5 a^2 c^2 \text {$\#$1} b+2 a^3 c^2 \text {$\#$1}}\& \right ]}{8 c} \]
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Rubi [B] time = 43.63, antiderivative size = 3187, normalized size of antiderivative = 2.46, number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1062, 1036, 1030, 208}
result too large to display
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 1030
Rule 1036
Rule 1062
Rubi steps
\begin {align*} \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 (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}+\frac {\int \frac {b c^2 \left (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)\right )-b^2 c \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{4 b c \left (b^3 c+(a c-b c)^2\right )}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}+\frac {\int \frac {b c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{8 b c^{5/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}-\frac {\int \frac {b c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2} \left (c+b x^2\right )} \, dx}{8 b c^{5/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}-\frac {\left (b^2 c^{3/2} \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b^4 c^5 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {-b^2 c^2 \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{4 \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}+\frac {\left (b^2 c^{3/2} \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b^4 c^5 \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b c x^2} \, dx,x,\frac {-b^2 c^2 \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )-b^2 c^2 \left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x}{\sqrt {c+b x+a x^2}}\right )}{4 \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ &=-\frac {\left (b \left (a^2+a (1-b) b-b^2\right ) c+\left (b^4-a^2 (a-b) c+a (a-b) b^2 c\right ) x\right ) \sqrt {c+b x+a x^2}}{2 \left (b^3 c+(a c-b c)^2\right ) \left (c+b x^2\right )}-\frac {\sqrt {2 b^5 c-a^4 c^2-b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3-2 b c+b^2 c+2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2+c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a \sqrt {c}+2 a^2 \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}-2 a \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}+6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)-a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^4 \left (2 a^2 c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \tanh ^{-1}\left (\frac {\sqrt {b} \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+\left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x\right )}{\sqrt {2} \sqrt {2 b^5 c-a^4 c^2-b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3-2 b c+b^2 c+2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2+c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a \sqrt {c}+2 a^2 \sqrt {c}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}-2 a \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}-\sqrt {b^3+a^2 c-2 a b c+b^2 c}+6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)-a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^4 \left (2 a^2 c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {c+b x+a x^2}}\right )}{4 \sqrt {2} \sqrt {b} c^{3/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}+\frac {\sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}-6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)+a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a^2 \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}+2 a \left (\sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^4 \left (2 a^2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {2 b^5 c-a^4 c^2+b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2-c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3+b^2 c-2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \left (2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \tanh ^{-1}\left (\frac {\sqrt {b} \left (b c \left (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)\right )-\left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right ) \left (a c-b c-\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+\left (b^2 \left (b^4+a^3 c+3 a^2 b c-a (2+a) b^2 c-a b^3 c\right )+\left (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)\right ) \left (a c-b c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right ) x\right )}{\sqrt {2} \sqrt {b^6-b^5 (c+4 a c)-2 a^3 c^{3/2} \left (a \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a^2 b^2 \sqrt {c} \left (2 a (1-5 c) \sqrt {c}-2 c^{3/2}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}-6 c \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+b^3 \left (a^2 c (3+8 c)+a \sqrt {c} (1+2 c) \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+2 a^2 b c^{3/2} \left (2 a^2 \sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}+2 a \left (\sqrt {c}+\sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )+b^4 \left (2 a^2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-a \left (2 c+2 c^2-3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {2 b^5 c-a^4 c^2+b^4 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}+a^2 b c \left (b^2+7 b c-6 b^2 c+3 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )-a^3 \left (4 b c^2-5 b^2 c^2-c^{3/2} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )+a b^2 c \left (3 b^3+b^2 c-2 \sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}-b \left (2 c+\sqrt {c} \sqrt {b^3+a^2 c-2 a b c+b^2 c}\right )\right )} \sqrt {c+b x+a x^2}}\right )}{4 \sqrt {2} \sqrt {b} c^{3/2} \left (b^3+a^2 c-2 a b c+b^2 c\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 8.57, size = 882, normalized size = 0.68 \begin {gather*} \frac {\frac {4 \sqrt {c} \sqrt {c+x (b+a x)} \left (c x a^3-b c (b x+x+1) a^2+b^2 c (x b+b-1) a+b^3 (c-b x)\right )}{b x^2+c}+\frac {\left (\sqrt {c} b^{9/2}-i (3 a c+c) b^4-a c^{3/2} b^{7/2}-i a c (2 c+1) b^3-a (a+2) c^{3/2} b^{5/2}+i a^2 c (6 c-1) b^2+3 a^2 c^{3/2} b^{3/2}-2 i a^2 (2 a+1) c^2 b+a^3 c^{3/2} \sqrt {b}+2 i a^3 c^2\right ) \log \left (-\frac {4 \left (-i b^{3/2}+\sqrt {c} b-a \sqrt {c}\right ) c \left (x b^{3/2}-i \sqrt {c} b+2 c \sqrt {b}-2 i a \sqrt {c} x+2 \sqrt {-i \sqrt {c} b^{3/2}+c b-a c} \sqrt {c+x (b+a x)}\right )}{\sqrt {-i \sqrt {c} b^{3/2}+c b-a c} \left (b^3-3 i a \sqrt {c} b^{5/2}+2 a c b^2-2 i a \sqrt {c} b^{3/2}-4 a^2 c b-i a^2 \sqrt {c} \sqrt {b}+2 a^2 c\right ) \left (\sqrt {c}-i \sqrt {b} x\right )}\right )}{\sqrt {-i \sqrt {c} b^{3/2}+c b-a c}}+\frac {\left (\sqrt {c} b^{9/2}+i (3 a c+c) b^4-a c^{3/2} b^{7/2}+i a c (2 c+1) b^3-a (a+2) c^{3/2} b^{5/2}-i a^2 c (6 c-1) b^2+3 a^2 c^{3/2} b^{3/2}+2 i a^2 (2 a+1) c^2 b+a^3 c^{3/2} \sqrt {b}-2 i a^3 c^2\right ) \log \left (-\frac {4 \left (i b^{3/2}+\sqrt {c} b-a \sqrt {c}\right ) c \left (x b^{3/2}+i \sqrt {c} b+2 c \sqrt {b}+2 i a \sqrt {c} x+2 \sqrt {i \sqrt {c} b^{3/2}+c b-a c} \sqrt {c+x (b+a x)}\right )}{\sqrt {i \sqrt {c} b^{3/2}+c b-a c} \left (b^3+3 i a \sqrt {c} b^{5/2}+2 a c b^2+2 i a \sqrt {c} b^{3/2}-4 a^2 c b+i a^2 \sqrt {c} \sqrt {b}+2 a^2 c\right ) \left (i \sqrt {b} x+\sqrt {c}\right )}\right )}{\sqrt {i \sqrt {c} b^{3/2}+c b-a c}}}{8 c^{3/2} \left (b^3+c b^2-2 a c b+a^2 c\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.31, size = 1571, normalized size = 1.22
result too large to display
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 1250, normalized size = 0.97
method | result | size |
default | \(\frac {\left (-a \,b^{2} c -\sqrt {-b c}\, b^{2}+a^{2} c \right ) \left (\frac {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}}}{\left (\sqrt {-b c}\, b +a c -b c \right ) \left (x +\frac {\sqrt {-b c}}{b}\right )}+\frac {\sqrt {-b c}\, \left (\sqrt {-b c}\, b +2 a 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 c \left (\sqrt {-b c}\, b +a c -b c \right ) \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}\right )}{4 c \,b^{2}}+\frac {\left (-a \,b^{2} c +\sqrt {-b c}\, b^{2}+a^{2} c \right ) \left (\frac {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}}}{\left (-\sqrt {-b c}\, b +a c -b c \right ) \left (x -\frac {\sqrt {-b c}}{b}\right )}-\frac {\sqrt {-b c}\, \left (-\sqrt {-b c}\, b +2 a 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 c \left (-\sqrt {-b c}\, b +a c -b c \right ) \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}\right )}{4 c \,b^{2}}-\frac {a \left (b^{2}+a \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 )}{4 \sqrt {-b c}\, b \sqrt {-\frac {-\sqrt {-b c}\, b +a c -b c}{b}}}+\frac {a \left (b^{2}+a \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 )}{4 \sqrt {-b c}\, b \sqrt {-\frac {\sqrt {-b c}\, b +a c -b c}{b}}}\) | \(1250\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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