Integrand size = 26, antiderivative size = 165 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=-\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{256 c^{5/2} \left (b^2-4 a c\right )^{3/2} d^7} \]
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Time = 0.08 (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 = {698, 707, 702, 211} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{256 c^{5/2} d^7 \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 d^7 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6} \]
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Rule 211
Rule 698
Rule 702
Rule 707
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx}{8 c d^2} \\ & = -\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{128 c^2 d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{256 c^2 \left (b^2-4 a c\right ) d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{64 c \left (b^2-4 a c\right ) d^6} \\ & = -\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{256 c^{5/2} \left (b^2-4 a c\right )^{3/2} d^7} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\frac {2 (a+x (b+c x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{5 \left (b^2-4 a c\right )^4 d^7} \]
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Time = 2.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {-\frac {3 \left (2 c x +b \right )^{6} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{256}+\left (\frac {c^{2} x^{2}}{4}+\left (\frac {b x}{4}+a \right ) c -\frac {3 b^{2}}{16}\right ) \left (\frac {3 c^{2} x^{2}}{2}+\left (\frac {3 b x}{2}+a \right ) c +\frac {b^{2}}{8}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \sqrt {c \,x^{2}+b x +a}}{12 \sqrt {4 c^{2} a -b^{2} c}\, d^{7} \left (2 c x +b \right )^{6} c^{2} \left (-\frac {b^{2}}{4}+a c \right )}\) | \(157\) |
default | \(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {6 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{128 d^{7} c^{7}}\) | \(415\) |
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Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (141) = 282\).
Time = 3.34 (sec) , antiderivative size = 1064, normalized size of antiderivative = 6.45 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\left [\frac {3 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, {\left (3 \, b^{6} c - 4 \, a b^{4} c^{2} - 160 \, a^{2} b^{2} c^{3} + 512 \, a^{3} c^{4} - 48 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} - 96 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} - 16 \, {\left (b^{4} c^{3} + 10 \, a b^{2} c^{4} - 56 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (b^{5} c^{2} - 11 \, a b^{3} c^{3} + 28 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1536 \, {\left (64 \, {\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{7} x + {\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{7}\right )}}, -\frac {3 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3 \, b^{6} c - 4 \, a b^{4} c^{2} - 160 \, a^{2} b^{2} c^{3} + 512 \, a^{3} c^{4} - 48 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} - 96 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} - 16 \, {\left (b^{4} c^{3} + 10 \, a b^{2} c^{4} - 56 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (b^{5} c^{2} - 11 \, a b^{3} c^{3} + 28 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, {\left (64 \, {\left (b^{4} c^{9} - 8 \, a b^{2} c^{10} + 16 \, a^{2} c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{5} c^{8} - 8 \, a b^{3} c^{9} + 16 \, a^{2} b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{6} c^{7} - 8 \, a b^{4} c^{8} + 16 \, a^{2} b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{7} c^{6} - 8 \, a b^{5} c^{7} + 16 \, a^{2} b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{8} c^{5} - 8 \, a b^{6} c^{6} + 16 \, a^{2} b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{9} c^{4} - 8 \, a b^{7} c^{5} + 16 \, a^{2} b^{5} c^{6}\right )} d^{7} x + {\left (b^{10} c^{3} - 8 \, a b^{8} c^{4} + 16 \, a^{2} b^{6} c^{5}\right )} d^{7}\right )}}\right ] \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1312 vs. \(2 (141) = 282\).
Time = 0.40 (sec) , antiderivative size = 1312, normalized size of antiderivative = 7.95 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \]
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