Integrand size = 22, antiderivative size = 257 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \]
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Time = 0.22 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 654, 626, 635, 212} \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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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 )^{5/2}}{6 c}+\frac {\int \left (\frac {1}{2} \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )+\frac {7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (-\frac {7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2} \\ & = \frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 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 )}{512 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \\ \end{align*}
Time = 2.68 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)-48 b^2 c^2 \left (a e (50 d+9 e x)-c x \left (5 d^2+4 d e x+e^2 x^2\right )\right )-8 b^3 c \left (-95 a e^2+c \left (45 d^2+30 d e x+7 e^2 x^2\right )\right )+16 b c^2 \left (-81 a^2 e^2+6 a c \left (25 d^2+14 d e x+3 e^2 x^2\right )+4 c^2 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )+32 c^3 \left (3 a^2 e (32 d+5 e x)+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )\right )\right )+15 \left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{7680 c^{9/2}} \]
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Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.79
method | result | size |
risch | \(-\frac {\left (-1280 c^{5} e^{2} x^{5}-1664 b \,c^{4} e^{2} x^{4}-3072 c^{5} d e \,x^{4}-2240 a \,c^{4} e^{2} x^{3}-48 b^{2} c^{3} e^{2} x^{3}-4224 b \,c^{4} d e \,x^{3}-1920 c^{5} d^{2} x^{3}-288 a b \,c^{3} e^{2} x^{2}-6144 a \,c^{4} d e \,x^{2}+56 b^{3} c^{2} e^{2} x^{2}-192 b^{2} c^{3} d e \,x^{2}-2880 b \,c^{4} d^{2} x^{2}-480 a^{2} c^{3} e^{2} x +432 a \,b^{2} c^{2} e^{2} x -1344 a b \,c^{3} d e x -4800 a \,c^{4} d^{2} x -70 b^{4} c \,e^{2} x +240 b^{3} c^{2} d e x -240 b^{2} c^{3} d^{2} x +1296 a^{2} b \,c^{2} e^{2}-3072 a^{2} c^{3} d e -760 a \,b^{3} c \,e^{2}+2400 a \,b^{2} c^{2} d e -2400 a b \,c^{3} d^{2}+105 b^{5} e^{2}-360 b^{4} c d e +360 b^{3} c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}-\frac {\left (64 a^{3} c^{3} e^{2}-144 a^{2} b^{2} c^{2} e^{2}+384 a^{2} b \,c^{3} d e -384 a^{2} c^{4} d^{2}+60 a \,b^{4} c \,e^{2}-192 a \,b^{3} c^{2} d e +192 a \,b^{2} c^{3} d^{2}-7 b^{6} e^{2}+24 b^{5} c d e -24 b^{4} c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}\) | \(460\) |
default | \(d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(505\) |
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Time = 0.36 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.50 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1656 vs. \(2 (248) = 496\).
Time = 0.70 (sec) , antiderivative size = 1656, normalized size of antiderivative = 6.44 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (231) = 462\).
Time = 0.29 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.81 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e^{2} x + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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