Integrand size = 26, antiderivative size = 145 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac {5 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} d^4} \]
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Time = 0.05 (sec) , antiderivative size = 145, 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, 626, 635, 212} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=-\frac {5 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} d^4}+\frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3} \]
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Rule 212
Rule 626
Rule 635
Rule 698
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^2} \, dx}{12 c d^2} \\ & = -\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}+\frac {5 \int \sqrt {a+b x+c x^2} \, dx}{16 c^2 d^4} \\ & = \frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3 d^4} \\ & = \frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac {\left (5 \left (b^2-4 a c\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 )}{64 c^3 d^4} \\ & = \frac {5 (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3 d^4}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{6 c d^4 (b+2 c x)^3}-\frac {5 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} d^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(331\) vs. \(2(145)=290\).
Time = 2.58 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^4+80 b^3 c x+32 b c^2 x \left (-7 a+3 c x^2\right )+8 b^2 c \left (-5 a+16 c x^2\right )+16 c^2 \left (-2 a^2-14 a c x^2+3 c^2 x^4\right )\right )}{(b+2 c x)^3}+\frac {12 \sqrt {b^2-4 a c} \left (b^4-2 a b^2 c-8 a^2 c^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b^2-4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b^3}-15 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )+\frac {12 \sqrt {-b^2+4 a c} \left (b^4-2 a b^2 c-8 a^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{b^3}}{192 c^{7/2} d^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(328\) vs. \(2(123)=246\).
Time = 3.16 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.27
method | result | size |
risch | \(\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{64 c^{3} d^{4}}+\frac {-\frac {5 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+20 a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {\left (96 a^{2} c^{2}-48 a \,b^{2} c +6 b^{4}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{c \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {\left (128 c^{3} a^{3}-96 a^{2} b^{2} c^{2}+24 a \,b^{4} c -2 b^{6}\right ) \left (-\frac {4 c \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {32 c^{3} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{3 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )}\right )}{16 c^{4}}}{128 c^{3} d^{4}}\) | \(329\) |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{16 d^{4} c^{4}}\) | \(367\) |
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (123) = 246\).
Time = 0.64 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.65 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\left [-\frac {15 \, {\left (b^{5} - 4 \, a b^{3} c + 8 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \, {\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \, {\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}}, \frac {15 \, {\left (b^{5} - 4 \, a b^{3} c + 8 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + 12 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 6 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, c^{5} x^{4} + 96 \, b c^{4} x^{3} + 15 \, b^{4} c - 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3} + 32 \, {\left (4 \, b^{2} c^{3} - 7 \, a c^{4}\right )} x^{2} + 16 \, {\left (5 \, b^{3} c^{2} - 14 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}}\right ] \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (123) = 246\).
Time = 0.43 (sec) , antiderivative size = 612, normalized size of antiderivative = 4.22 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\frac {1}{64} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, x}{c^{2} d^{4}} + \frac {b}{c^{3} d^{4}}\right )} + \frac {5 \, {\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}} d^{4}} + \frac {36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{4} c^{2} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{2} c^{3} + 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} c^{4} + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{5} c^{\frac {3}{2}} - 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{3} c^{\frac {5}{2}} + 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b c^{\frac {7}{2}} + 66 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{6} c - 576 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{4} c^{2} + 1440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{3} - 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{3} c^{4} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{7} \sqrt {c} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{5} c^{\frac {3}{2}} + 864 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{3} c^{\frac {5}{2}} - 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} b c^{\frac {7}{2}} + 7 \, b^{8} - 82 \, a b^{6} c + 348 \, a^{2} b^{4} c^{2} - 640 \, a^{3} b^{2} c^{3} + 448 \, a^{4} c^{4}}{192 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3} c^{\frac {7}{2}} d^{4}} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4} \,d x \]
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