Integrand size = 20, antiderivative size = 161 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 635, 212} \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{128 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c} \]
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Rule 212
Rule 626
Rule 635
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{2 c} \\ & = \frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^2} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^3} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\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^3} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^4 e-10 b^3 c (3 d+e x)+8 b c^2 \left (25 a d+7 a e x+30 c d x^2+22 c e x^3\right )+4 b^2 c (-25 a e+c x (5 d+2 e x))+16 c^2 \left (8 a^2 e+2 c^2 x^3 (5 d+4 e x)+a c x (25 d+16 e x)\right )\right )-15 \left (b^2-4 a c\right )^2 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{640 c^{7/2}} \]
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Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.45
method | result | size |
risch | \(\frac {\left (128 c^{4} e \,x^{4}+176 b \,c^{3} e \,x^{3}+160 c^{4} d \,x^{3}+256 a \,c^{3} e \,x^{2}+8 b^{2} c^{2} e \,x^{2}+240 b \,c^{3} d \,x^{2}+56 a b \,c^{2} e x +400 a \,c^{3} d x -10 b^{3} c e x +20 b^{2} c^{2} d x +128 a^{2} c^{2} e -100 a \,b^{2} c e +200 a b \,c^{2} d +15 b^{4} e -30 b^{3} c d \right ) \sqrt {c \,x^{2}+b x +a}}{640 c^{3}}-\frac {3 \left (16 a^{2} b \,c^{2} e -32 a^{2} c^{3} d -8 a \,b^{3} c e +16 a \,b^{2} c^{2} d +b^{5} e -2 b^{4} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}\) | \(233\) |
default | \(d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(236\) |
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Time = 0.31 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.29 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (128 \, c^{5} e x^{4} + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} e\right )} x^{2} - 10 \, {\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d + {\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e + 2 \, {\left (10 \, {\left (b^{2} c^{3} + 20 \, a c^{4}\right )} d - {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2560 \, c^{4}}, -\frac {15 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (128 \, c^{5} e x^{4} + 16 \, {\left (10 \, c^{5} d + 11 \, b c^{4} e\right )} x^{3} + 8 \, {\left (30 \, b c^{4} d + {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} e\right )} x^{2} - 10 \, {\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d + {\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e + 2 \, {\left (10 \, {\left (b^{2} c^{3} + 20 \, a c^{4}\right )} d - {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1280 \, c^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (151) = 302\).
Time = 0.56 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.42 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {c e x^{4}}{5} + \frac {x^{3} \cdot \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + \frac {x^{2} \cdot \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{3 c} + \frac {x \left (2 a b e + 2 a c d - \frac {3 a \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + b^{2} d - \frac {5 b \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right )}{2 c} + \frac {a^{2} e + 2 a b d - \frac {2 a \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{3 c} - \frac {3 b \left (2 a b e + 2 a c d - \frac {3 a \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + b^{2} d - \frac {5 b \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right )}{4 c}}{c}\right ) + \left (a^{2} d - \frac {a \left (2 a b e + 2 a c d - \frac {3 a \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + b^{2} d - \frac {5 b \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 c}\right )}{2 c} - \frac {b \left (a^{2} e + 2 a b d - \frac {2 a \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{3 c} - \frac {3 b \left (2 a b e + 2 a c d - \frac {3 a \left (\frac {11 b c e}{10} + c^{2} d\right )}{4 c} + b^{2} d - \frac {5 b \left (\frac {6 a c e}{5} + b^{2} e + 2 b c d - \frac {7 b \left (\frac {11 b c e}{10} + c^{2} d\right )}{8 c}\right )}{6 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 {e \left (a + b x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- a e + b d\right )}{5 b}\right )}{b} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.55 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{640} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c e x + \frac {10 \, c^{5} d + 11 \, b c^{4} e}{c^{4}}\right )} x + \frac {30 \, b c^{4} d + b^{2} c^{3} e + 32 \, a c^{4} e}{c^{4}}\right )} x + \frac {10 \, b^{2} c^{3} d + 200 \, a c^{4} d - 5 \, b^{3} c^{2} e + 28 \, a b c^{3} e}{c^{4}}\right )} x - \frac {30 \, b^{3} c^{2} d - 200 \, a b c^{3} d - 15 \, b^{4} c e + 100 \, a b^{2} c^{2} e - 128 \, a^{2} c^{3} e}{c^{4}}\right )} - \frac {3 \, {\left (2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {7}{2}}} \]
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Time = 10.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.89 \[ \int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {e\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5\,c}+\frac {d\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )\,\left (3\,a\,c-\frac {3\,b^2}{4}\right )}{4\,c}-\frac {b\,e\,\left (\frac {3\,a\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{4}+\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4}+\frac {b\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{8\,c}-\frac {3\,b^2\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{16\,c}\right )}{2\,c}+\frac {d\,\left (\frac {b}{2}+c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \]
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