Integrand size = 26, antiderivative size = 203 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {18 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{13/4} d^{7/2}}+\frac {18 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{13/4} d^{7/2}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {701, 707, 708, 335, 304, 209, 212} \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {18 c \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}+\frac {18 c \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{13/4}}-\frac {36 c}{d^3 \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac {36 c}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}} \]
[In]
[Out]
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 (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {(9 c) \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {(9 c) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2 d^2} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {(9 c) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^4} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {9 \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 )^3 d^5} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {9 \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 )^3 d^5} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {(18 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 )^3 d^3}-\frac {(18 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 )^3 d^3} \\ & = -\frac {36 c}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac {36 c}{\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x}}-\frac {1}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}-\frac {18 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 )^{13/4} d^{7/2}}+\frac {18 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 )^{13/4} d^{7/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\left (\frac {1}{5}+\frac {i}{5}\right ) c \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x) \left (4 b^4-32 a b^2 c+64 a^2 c^2+36 b^2 (b+2 c x)^2-144 a c (b+2 c x)^2-45 (b+2 c x)^4\right )}{c \left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {45 (b+2 c x)^{7/2} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}-\frac {45 (b+2 c x)^{7/2} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}+\frac {45 (b+2 c x)^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{13/4}}\right )}{(d (b+2 c x))^{7/2}} \]
[In]
[Out]
Time = 2.57 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.72
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (-\frac {1}{5 d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {2}{d^{6} \left (4 a c -b^{2}\right )^{3} \sqrt {2 c d x +b d}}+\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{6} \left (4 a c -b^{2}\right )^{3}}\right )\) | \(350\) |
default | \(16 c \,d^{3} \left (-\frac {1}{5 d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {2}{d^{6} \left (4 a c -b^{2}\right )^{3} \sqrt {2 c d x +b d}}+\frac {\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{16 a c \,d^{2}-4 b^{2} d^{2}+4 \left (2 c d x +b d \right )^{2}}+\frac {9 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{32 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{6} \left (4 a c -b^{2}\right )^{3}}\right )\) | \(350\) |
pseudoelliptic | \(8 d^{3} c \left (\frac {\left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{8 c \,d^{8} \left (c \,x^{2}+b x +a \right ) \left (4 a c -b^{2}\right )^{3}}+\frac {9 \ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right ) \sqrt {2}}{16 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} d^{6} \left (4 a c -b^{2}\right )^{3}}+\frac {9 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} d^{6} \left (4 a c -b^{2}\right )^{3}}-\frac {9 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} d^{6} \left (4 a c -b^{2}\right )^{3}}-\frac {2}{5 d^{4} \left (4 a c -b^{2}\right )^{2} \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}+\frac {4}{d^{6} \left (4 a c -b^{2}\right )^{3} \sqrt {d \left (2 c x +b \right )}}\right )\) | \(426\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 3285, normalized size of antiderivative = 16.18 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (175) = 350\).
Time = 0.31 (sec) , antiderivative size = 779, normalized size of antiderivative = 3.84 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {9 \, \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^{8} d^{5} - 16 \, a b^{6} c d^{5} + 96 \, a^{2} b^{4} c^{2} d^{5} - 256 \, a^{3} b^{2} c^{3} d^{5} + 256 \, a^{4} c^{4} d^{5}} + \frac {9 \, \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^{8} d^{5} - 16 \, a b^{6} c d^{5} + 96 \, a^{2} b^{4} c^{2} d^{5} - 256 \, a^{3} b^{2} c^{3} d^{5} + 256 \, a^{4} c^{4} d^{5}} - \frac {9 \, {\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^{8} d^{5} - 16 \, \sqrt {2} a b^{6} c d^{5} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{5} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{5} + 256 \, \sqrt {2} a^{4} c^{4} d^{5}} + \frac {9 \, {\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^{8} d^{5} - 16 \, \sqrt {2} a b^{6} c d^{5} + 96 \, \sqrt {2} a^{2} b^{4} c^{2} d^{5} - 256 \, \sqrt {2} a^{3} b^{2} c^{3} d^{5} + 256 \, \sqrt {2} a^{4} c^{4} d^{5}} + \frac {4 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c}{{\left (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}\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}} - \frac {16 \, {\left (b^{2} c d^{2} - 4 \, a c^{2} d^{2} + 10 \, {\left (2 \, c d x + b d\right )}^{2} c\right )}}{5 \, {\left (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}\right )} {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \]
[In]
[Out]
Time = 9.70 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {36\,c\,{\left (b\,d+2\,c\,d\,x\right )}^4}{{\left (b^2\,d-4\,a\,c\,d\right )}^3}+\frac {16\,c\,d}{5\,\left (4\,a\,c-b^2\right )}-\frac {144\,c\,{\left (b\,d+2\,c\,d\,x\right )}^2}{5\,d\,{\left (4\,a\,c-b^2\right )}^2}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (b^2\,d^2-4\,a\,c\,d^2\right )-{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}-\frac {18\,c\,\mathrm {atan}\left (\frac {b^6\,\sqrt {b\,d+2\,c\,d\,x}-64\,a^3\,c^3\,\sqrt {b\,d+2\,c\,d\,x}+48\,a^2\,b^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-12\,a\,b^4\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{13/4}}\right )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{13/4}}-\frac {c\,\mathrm {atan}\left (\frac {b^6\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a^3\,c^3\,\sqrt {b\,d+2\,c\,d\,x}\,64{}\mathrm {i}+a^2\,b^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,48{}\mathrm {i}-a\,b^4\,c\,\sqrt {b\,d+2\,c\,d\,x}\,12{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{13/4}}\right )\,18{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{13/4}} \]
[In]
[Out]