Integrand size = 22, antiderivative size = 157 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\frac {\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1675, 654, 626, 635, 212} \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) \left (-4 a c C+16 A c^2+5 b^2 C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c C+16 A c^2+5 b^2 C\right )}{64 c^3}-\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1675
Rubi steps \begin{align*} \text {integral}& = \frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\int \left (4 A c-a C-\frac {5 b C x}{2}\right ) \sqrt {a+b x+c x^2} \, dx}{4 c} \\ & = -\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\left (\frac {5 b^2 C}{2}+2 c (4 A c-a C)\right ) \int \sqrt {a+b x+c x^2} \, dx}{8 c^2} \\ & = \frac {\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3} \\ & = \frac {\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^3} \\ & = \frac {\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}-\frac {5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (48 A c^2 (b+2 c x)+C \left (15 b^3-10 b^2 c x+24 c^2 x \left (a+2 c x^2\right )+b \left (-52 a c+8 c^2 x^2\right )\right )\right )-3 \left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{192 c^{7/2}} \]
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Time = 0.70 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {\left (48 c^{3} C \,x^{3}+8 b \,c^{2} C \,x^{2}+96 A \,c^{3} x +24 a \,c^{2} C x -10 C \,b^{2} c x +48 A b \,c^{2}-52 C a b c +15 C \,b^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{3}}+\frac {\left (64 A a \,c^{3}-16 A \,b^{2} c^{2}-16 C \,a^{2} c^{2}+24 C a \,b^{2} c -5 C \,b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}\) | \(151\) |
default | \(A \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+C \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )\) | \(253\) |
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Time = 0.29 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.26 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\left [-\frac {3 \, {\left (5 \, C b^{4} - 24 \, C a b^{2} c - 64 \, A a c^{3} + 16 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (48 \, C c^{4} x^{3} + 8 \, C b c^{3} x^{2} + 15 \, C b^{3} c - 52 \, C a b c^{2} + 48 \, A b c^{3} - 2 \, {\left (5 \, C b^{2} c^{2} - 12 \, C a c^{3} - 48 \, A c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{4}}, \frac {3 \, {\left (5 \, C b^{4} - 24 \, C a b^{2} c - 64 \, A a c^{3} + 16 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, C c^{4} x^{3} + 8 \, C b c^{3} x^{2} + 15 \, C b^{3} c - 52 \, C a b c^{2} + 48 \, A b c^{3} - 2 \, {\left (5 \, C b^{2} c^{2} - 12 \, C a c^{3} - 48 \, A c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (151) = 302\).
Time = 0.43 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.96 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {C b x^{2}}{24 c} + \frac {C x^{3}}{4} + \frac {x \left (A c + \frac {C a}{4} - \frac {5 C b^{2}}{48 c}\right )}{2 c} + \frac {A b - \frac {C a b}{12 c} - \frac {3 b \left (A c + \frac {C a}{4} - \frac {5 C b^{2}}{48 c}\right )}{4 c}}{c}\right ) + \left (A a - \frac {a \left (A c + \frac {C a}{4} - \frac {5 C b^{2}}{48 c}\right )}{2 c} - \frac {b \left (A b - \frac {C a b}{12 c} - \frac {3 b \left (A c + \frac {C a}{4} - \frac {5 C b^{2}}{48 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (- \frac {2 C a \left (a + b x\right )^{\frac {5}{2}}}{5 b^{2}} + \frac {C \left (a + b x\right )^{\frac {7}{2}}}{7 b^{2}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A b^{2} + C a^{2}\right )}{3 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\\sqrt {a} \left (A x + \frac {C x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, C x + \frac {C b}{c}\right )} x - \frac {5 \, C b^{2} c - 12 \, C a c^{2} - 48 \, A c^{3}}{c^{3}}\right )} x + \frac {15 \, C b^{3} - 52 \, C a b c + 48 \, A b c^{2}}{c^{3}}\right )} + \frac {{\left (5 \, C b^{4} - 24 \, C a b^{2} c + 16 \, C a^{2} c^{2} + 16 \, A b^{2} c^{2} - 64 \, A a c^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}}} \]
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Time = 13.75 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.53 \[ \int \sqrt {a+b x+c x^2} \left (A+C x^2\right ) \, dx=A\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-\frac {C\,a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {A\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}-\frac {5\,C\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {C\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \]
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