Integrand size = 26, antiderivative size = 165 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (b^2-4 a c\right )^3 d^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {699, 706, 635, 212} \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {d^2 \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]
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Rule 212
Rule 635
Rule 699
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^2 \sqrt {a+b x+c x^2} \, dx}{8 c} \\ & = -\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{128 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\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 )}{128 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (b^2-4 a c\right )^3 d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}} \\ \end{align*}
Time = 1.97 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.93 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {d^2 \left (\sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (-3 b^4+8 b^3 c x+32 b c^2 x \left (7 a+8 c x^2\right )+8 b^2 c \left (4 a+17 c x^2\right )+16 c^2 \left (3 a^2+14 a c x^2+8 c^2 x^4\right )\right )+3 \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{384 c^{5/2}} \]
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Time = 2.67 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {\left (256 c^{5} x^{5}+640 b \,x^{4} c^{4}+448 a \,c^{4} x^{3}+528 b^{2} c^{3} x^{3}+672 a b \,c^{3} x^{2}+152 x^{2} b^{3} c^{2}+96 a^{2} c^{3} x +288 a \,c^{2} b^{2} x +2 c x \,b^{4}+48 a^{2} b \,c^{2}+32 a \,b^{3} c -3 b^{5}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{2}}{384 c^{2}}-\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d^{2}}{256 c^{\frac {5}{2}}}\) | \(193\) |
default | \(d^{2} \left (b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+4 c^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+4 b c \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\right )\) | \(510\) |
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Time = 0.30 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.87 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} d^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (256 \, c^{6} d^{2} x^{5} + 640 \, b c^{5} d^{2} x^{4} + 16 \, {\left (33 \, b^{2} c^{4} + 28 \, a c^{5}\right )} d^{2} x^{3} + 8 \, {\left (19 \, b^{3} c^{3} + 84 \, a b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (b^{4} c^{2} + 144 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{1536 \, c^{3}}, -\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (256 \, c^{6} d^{2} x^{5} + 640 \, b c^{5} d^{2} x^{4} + 16 \, {\left (33 \, b^{2} c^{4} + 28 \, a c^{5}\right )} d^{2} x^{3} + 8 \, {\left (19 \, b^{3} c^{3} + 84 \, a b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (b^{4} c^{2} + 144 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (153) = 306\).
Time = 0.74 (sec) , antiderivative size = 1042, normalized size of antiderivative = 6.32 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.37 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.56 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, c^{3} d^{2} x + 5 \, b c^{2} d^{2}\right )} x + \frac {33 \, b^{2} c^{6} d^{2} + 28 \, a c^{7} d^{2}}{c^{5}}\right )} x + \frac {19 \, b^{3} c^{5} d^{2} + 84 \, a b c^{6} d^{2}}{c^{5}}\right )} x + \frac {b^{4} c^{4} d^{2} + 144 \, a b^{2} c^{5} d^{2} + 48 \, a^{2} c^{6} d^{2}}{c^{5}}\right )} x - \frac {3 \, b^{5} c^{3} d^{2} - 32 \, a b^{3} c^{4} d^{2} - 48 \, a^{2} b c^{5} d^{2}}{c^{5}}\right )} - \frac {{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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