Integrand size = 22, antiderivative size = 191 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}} \]
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Time = 0.15 (sec) , antiderivative size = 191, 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 = {756, 654, 626, 635, 212} \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{128 c^{7/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{64 c^3}+\frac {5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\int \left (\frac {1}{2} \left (8 c d^2-2 e \left (\frac {3 b d}{2}+a e\right )\right )+\frac {5}{2} e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2} \, dx}{4 c} \\ & = \frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\left (-\frac {5}{2} b e (2 c d-b e)+c \left (8 c d^2-2 e \left (\frac {3 b d}{2}+a e\right )\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{8 c^2} \\ & = \frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3} \\ & = \frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\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 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\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 = 1.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^3 e^2-2 b^2 c e (24 d+5 e x)+4 b c \left (-13 a e^2+2 c \left (6 d^2+4 d e x+e^2 x^2\right )\right )+8 c^2 \left (a e (16 d+3 e x)+2 c x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )\right )-3 \left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{192 c^{7/2}} \]
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Time = 0.36 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {\left (-48 e^{2} c^{3} x^{3}-8 b \,c^{2} e^{2} x^{2}-128 c^{3} d e \,x^{2}-24 a \,c^{2} e^{2} x +10 c \,e^{2} b^{2} x -32 b \,c^{2} d e x -96 c^{3} d^{2} x +52 a b c \,e^{2}-128 a \,c^{2} d e -15 b^{3} e^{2}+48 b^{2} c d e -48 b \,c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{3}}-\frac {\left (16 a^{2} c^{2} e^{2}-24 a \,b^{2} c \,e^{2}+64 a b \,c^{2} d e -64 a \,c^{3} d^{2}+5 b^{4} e^{2}-16 c e \,b^{3} d +16 c^{2} b^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}\) | \(229\) |
default | \(d^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\) | \(349\) |
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Time = 0.33 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.64 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\left [\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e + {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{4}}, \frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e + {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (182) = 364\).
Time = 0.64 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.48 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {e^{2} x^{3}}{4} + \frac {x^{2} \left (\frac {b e^{2}}{8} + 2 c d e\right )}{3 c} + \frac {x \left (\frac {a e^{2}}{4} + 2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{2 c} + \frac {2 a d e - \frac {2 a \left (\frac {b e^{2}}{8} + 2 c d e\right )}{3 c} + b d^{2} - \frac {3 b \left (\frac {a e^{2}}{4} + 2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}}{c}\right ) + \left (a d^{2} - \frac {a \left (\frac {a e^{2}}{4} + 2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{2 c} - \frac {b \left (2 a d e - \frac {2 a \left (\frac {b e^{2}}{8} + 2 c d e\right )}{3 c} + b d^{2} - \frac {3 b \left (\frac {a e^{2}}{4} + 2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {e^{2} \left (a + b x\right )^{\frac {7}{2}}}{7 b^{2}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 2 a e^{2} + 2 b d e\right )}{5 b^{2}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (a^{2} e^{2} - 2 a b d e + b^{2} d^{2}\right )}{3 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.24 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, e^{2} x + \frac {16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac {48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2} + 12 \, a c^{2} e^{2}}{c^{3}}\right )} x + \frac {48 \, b c^{2} d^{2} - 48 \, b^{2} c d e + 128 \, a c^{2} d e + 15 \, b^{3} e^{2} - 52 \, a b c e^{2}}{c^{3}}\right )} + \frac {{\left (16 \, b^{2} c^{2} d^{2} - 64 \, a c^{3} d^{2} - 16 \, b^{3} c d e + 64 \, a b c^{2} d e + 5 \, b^{4} e^{2} - 24 \, a b^{2} c e^{2} + 16 \, a^{2} c^{2} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}}} \]
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Time = 10.67 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.74 \[ \int (d+e x)^2 \sqrt {a+b x+c x^2} \, dx=d^2\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {e^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}-\frac {a\,e^2\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}-\frac {5\,b\,e^2\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {d\,e\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{12\,c^2}+\frac {d\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{8\,c^{5/2}} \]
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