Integrand size = 26, antiderivative size = 172 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {10 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2}}+\frac {10 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {701, 707, 708, 335, 304, 209, 212} \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {10 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}+\frac {10 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4}}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \sqrt {b d+2 c d x}}-\frac {20 c}{d \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 701
Rule 707
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {(5 c) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c} \\ & = -\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {(5 c) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 \left (b^2-4 a c\right )^2 d^3} \\ & = -\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {5 \text {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^2 d^3} \\ & = -\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}+\frac {(10 c) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^2 d}-\frac {(10 c) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^2 d} \\ & = -\frac {20 c}{\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )}-\frac {10 c \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2}}+\frac {10 c \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {(1+i) c \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x) \left (b^2+20 b c x+4 c \left (4 a+5 c x^2\right )\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {5 (b+2 c x)^{3/2} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac {5 (b+2 c x)^{3/2} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4}}+\frac {5 (b+2 c x)^{3/2} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{9/4}}\right )}{(d (b+2 c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(148)=296\).
Time = 2.82 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.88
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (-\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{4} \left (4 a c -b^{2}\right )^{2}}-\frac {1}{d^{4} \left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}}\right )\) | \(323\) |
default | \(16 c \,d^{3} \left (-\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{4} \left (4 a c -b^{2}\right )^{2}}-\frac {1}{d^{4} \left (4 a c -b^{2}\right )^{2} \sqrt {2 c d x +b d}}\right )\) | \(323\) |
pseudoelliptic | \(-\frac {5 \left (c \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d \left (2 c x +b \right )}+\frac {32 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (\frac {5 c^{2} x^{2}}{4}+\left (\frac {5 b x}{4}+a \right ) c +\frac {b^{2}}{16}\right )}{5}\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, d \left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right )^{2}}\) | \(341\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1884, normalized size of antiderivative = 10.95 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (148) = 296\).
Time = 0.30 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.78 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} + \frac {5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} + \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt {2} b^{6} d^{3} - 12 \, \sqrt {2} a b^{4} c d^{3} + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d^{3} - 64 \, \sqrt {2} a^{3} c^{3} d^{3}} - \frac {4 \, {\left (4 \, b^{2} c d^{2} - 16 \, a c^{2} d^{2} - 5 \, {\left (2 \, c d x + b d\right )}^{2} c\right )}}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} {\left (\sqrt {2 \, c d x + b d} b^{2} d^{2} - 4 \, \sqrt {2 \, c d x + b d} a c d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )}} \]
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Time = 9.38 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {16\,c\,d}{4\,a\,c-b^2}+\frac {20\,c\,{\left (b\,d+2\,c\,d\,x\right )}^2}{d\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {b\,d+2\,c\,d\,x}\,\left (b^2\,d^2-4\,a\,c\,d^2\right )-{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}-\frac {10\,c\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}}-\frac {c\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,10{}\mathrm {i}}{d^{3/2}\,{\left (b^2-4\,a\,c\right )}^{9/4}} \]
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