Integrand size = 26, antiderivative size = 181 \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
[Out]
Time = 0.11 (sec) , antiderivative size = 181, 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 = {700, 706, 708, 335, 304, 209, 212} \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {44}{3} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2} \]
[In]
[Out]
Rule 209
Rule 212
Rule 304
Rule 335
Rule 700
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c d^2\right ) \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx \\ & = \frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx \\ & = \frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx \\ & = \frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac {1}{2} \left (11 \left (b^2-4 a c\right )^2 d^5\right ) \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 ) \\ & = \frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\left (11 \left (b^2-4 a c\right )^2 d^5\right ) \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 ) \\ & = \frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}-\left (22 c \left (b^2-4 a c\right )^2 d^7\right ) \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 (22 c \left (b^2-4 a c\right )^2 d^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = \frac {44}{3} c \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{3/2}+\frac {44}{7} c d^3 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-22 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.60 \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\left (\frac {1}{21}+\frac {i}{21}\right ) c (d (b+2 c x))^{13/2} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (77 b^4-616 a b^2 c+1232 a^2 c^2-44 b^2 (b+2 c x)^2+176 a c (b+2 c x)^2-12 (b+2 c x)^4\right )}{c (b+2 c x)^5 (a+x (b+c x))}-\frac {231 \left (b^2-4 a c\right )^{7/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}+\frac {231 \left (b^2-4 a c\right )^{7/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}-\frac {231 \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{13/2}}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(151)=302\).
Time = 3.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.06
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+d^{4} \left (\frac {\left (-a^{2} c^{2}+\frac {1}{2} a \,b^{2} c -\frac {1}{16} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {\left (44 a^{2} c^{2}-22 a \,b^{2} c +\frac {11}{4} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\right )\) | \(373\) |
default | \(16 c \,d^{3} \left (-\frac {8 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+d^{4} \left (\frac {\left (-a^{2} c^{2}+\frac {1}{2} a \,b^{2} c -\frac {1}{16} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {\left (44 a^{2} c^{2}-22 a \,b^{2} c +\frac {11}{4} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\right )\) | \(373\) |
pseudoelliptic | \(\frac {88 d^{3} \left (-\frac {2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (\frac {8 c^{2} x^{2}}{11}+\left (\frac {8 b x}{11}+a \right ) c -\frac {3 b^{2}}{44}\right ) d^{2} \left (-\frac {b^{2}}{4}+a c \right ) \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{3}+\left (\frac {2 \left (d \left (2 c x +b \right )\right )^{\frac {7}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{77}+\frac {\sqrt {2}\, d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )}{16}\right ) \left (c \,x^{2}+b x +a \right ) c \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (c \,x^{2}+b x +a \right )}\) | \(377\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1211, normalized size of antiderivative = 6.69 \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (151) = 302\).
Time = 0.31 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.57 \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {32}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c d^{5} - \frac {128}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{2} d^{5} + \frac {16}{7} \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c d^{3} - 11 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c^{2} d^{5}\right )} \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 ) - 11 \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c^{2} d^{5}\right )} \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 ) + \frac {11}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c^{2} d^{5}\right )} \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 ) - \frac {11}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} c d^{5} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c^{2} d^{5}\right )} \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 ) + \frac {4 \, {\left ({\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} c d^{7} - 8 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{7} + 16 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a^{2} c^{3} d^{7}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \]
[In]
[Out]
Time = 9.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.38 \[ \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {16\,c\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (64\,a^2\,c^3\,d^7-32\,a\,b^2\,c^2\,d^7+4\,b^4\,c\,d^7\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}+22\,c\,d^{13/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}-\frac {32\,c\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (4\,a\,c-b^2\right )}{3}+c\,d^{13/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}\,1{}\mathrm {i}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}\,22{}\mathrm {i} \]
[In]
[Out]