Integrand size = 26, antiderivative size = 133 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=-\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{16 c \left (b^2-4 a c\right ) d^5 (b+2 c x)^2}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{32 c^{3/2} \left (b^2-4 a c\right )^{3/2} d^5} \]
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Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {698, 707, 702, 211} \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{32 c^{3/2} d^5 \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{16 c d^5 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4} \]
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Rule 211
Rule 698
Rule 702
Rule 707
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}+\frac {\int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{16 c d^2} \\ & = -\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{16 c \left (b^2-4 a c\right ) d^5 (b+2 c x)^2}+\frac {\int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{32 c \left (b^2-4 a c\right ) d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{16 c \left (b^2-4 a c\right ) d^5 (b+2 c x)^2}+\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 )}{8 \left (b^2-4 a c\right ) d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{8 c d^5 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{16 c \left (b^2-4 a c\right ) d^5 (b+2 c x)^2}+\frac {\tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{32 c^{3/2} \left (b^2-4 a c\right )^{3/2} d^5} \\ \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 = 0.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\frac {2 (a+x (b+c x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{3 \left (b^2-4 a c\right )^3 d^5} \]
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Time = 2.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {-16 \left (\frac {c^{2} x^{2}}{2}+\left (\frac {b x}{2}+a \right ) c -\frac {b^{2}}{8}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \sqrt {c \,x^{2}+b x +a}+\left (2 c x +b \right )^{4} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{128 \sqrt {4 c^{2} a -b^{2} c}\, d^{5} \left (2 c x +b \right )^{4} c \left (-\frac {b^{2}}{4}+a c \right )}\) | \(135\) |
default | \(\frac {-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {2 c^{2} \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{32 d^{5} c^{5}}\) | \(295\) |
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (113) = 226\).
Time = 1.03 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\left [\frac {{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\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 (b^{4} c - 12 \, a b^{2} c^{2} + 32 \, a^{2} c^{3} - 4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} - 4 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{64 \, {\left (16 \, {\left (b^{4} c^{6} - 8 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} d^{5} x^{4} + 32 \, {\left (b^{5} c^{5} - 8 \, a b^{3} c^{6} + 16 \, a^{2} b c^{7}\right )} d^{5} x^{3} + 24 \, {\left (b^{6} c^{4} - 8 \, a b^{4} c^{5} + 16 \, a^{2} b^{2} c^{6}\right )} d^{5} x^{2} + 8 \, {\left (b^{7} c^{3} - 8 \, a b^{5} c^{4} + 16 \, a^{2} b^{3} c^{5}\right )} d^{5} x + {\left (b^{8} c^{2} - 8 \, a b^{6} c^{3} + 16 \, a^{2} b^{4} c^{4}\right )} d^{5}\right )}}, -\frac {{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\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 (b^{4} c - 12 \, a b^{2} c^{2} + 32 \, a^{2} c^{3} - 4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} - 4 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{32 \, {\left (16 \, {\left (b^{4} c^{6} - 8 \, a b^{2} c^{7} + 16 \, a^{2} c^{8}\right )} d^{5} x^{4} + 32 \, {\left (b^{5} c^{5} - 8 \, a b^{3} c^{6} + 16 \, a^{2} b c^{7}\right )} d^{5} x^{3} + 24 \, {\left (b^{6} c^{4} - 8 \, a b^{4} c^{5} + 16 \, a^{2} b^{2} c^{6}\right )} d^{5} x^{2} + 8 \, {\left (b^{7} c^{3} - 8 \, a b^{5} c^{4} + 16 \, a^{2} b^{3} c^{5}\right )} d^{5} x + {\left (b^{8} c^{2} - 8 \, a b^{6} c^{3} + 16 \, a^{2} b^{4} c^{4}\right )} d^{5}\right )}}\right ] \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\frac {\int \frac {\sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \]
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Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^5} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^5} \,d x \]
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