Integrand size = 26, antiderivative size = 170 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {5 c^2 d^{7/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c^2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{3/4}} \]
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Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {700, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {5 c^2 d^{7/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c^2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (5 c d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (5 c^2 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (5 c d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right ) \\ & = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (5 c d^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {\left (5 c^2 d^4\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 )}{\sqrt {b^2-4 a c}}-\frac {\left (5 c^2 d^4\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 )}{\sqrt {b^2-4 a c}} \\ & = -\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {5 c^2 d^{7/2} \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 )^{3/4}}-\frac {5 c^2 d^{7/2} \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 )^{3/4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.55 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{2}+\frac {i}{2}\right ) c^2 (d (b+2 c x))^{7/2} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (b^2+9 b c x+c \left (5 a+9 c x^2\right )\right )}{c^2 (b+2 c x)^3 (a+x (b+c x))^2}-\frac {5 i \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2}}+\frac {5 i \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2}}+\frac {5 i \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(142)=284\).
Time = 2.77 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(64 c^{2} d^{5} \left (\frac {-\frac {9 \left (2 c d x +b d \right )^{\frac {5}{2}}}{32}+16 \left (-\frac {5}{128} a c \,d^{2}+\frac {5}{512} b^{2} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(309\) |
default | \(64 c^{2} d^{5} \left (\frac {-\frac {9 \left (2 c d x +b d \right )^{\frac {5}{2}}}{32}+16 \left (-\frac {5}{128} a c \,d^{2}+\frac {5}{512} b^{2} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(309\) |
pseudoelliptic | \(-\frac {5 d^{3} \left (-\frac {\sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2} \ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )}{2}+\arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2}-\arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (\frac {9 c^{2} x^{2}}{5}+\left (\frac {9 b x}{5}+a \right ) c +\frac {b^{2}}{5}\right ) \sqrt {d \left (2 c x +b \right )}\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )^{2}}\) | \(366\) |
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 696, normalized size of antiderivative = 4.09 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} + 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) + 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (-i \, c^{2} x^{4} - 2 i \, b c x^{3} - 2 i \, a b x - i \, {\left (b^{2} + 2 \, a c\right )} x^{2} - i \, a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (i \, b^{2} - 4 i \, a c\right )}\right ) + 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (i \, c^{2} x^{4} + 2 i \, b c x^{3} + 2 i \, a b x + i \, {\left (b^{2} + 2 \, a c\right )} x^{2} + i \, a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (-i \, b^{2} + 4 i \, a c\right )}\right ) + {\left (9 \, c^{2} d^{3} x^{2} + 9 \, b c d^{3} x + {\left (b^{2} + 5 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (142) = 284\).
Time = 0.31 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.00 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {2 \, {\left (5 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{7} - 20 \, \sqrt {2 \, c d x + b d} a c^{3} d^{7} - 9 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2} d^{5}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]
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Time = 9.41 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.82 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (40\,a\,c^3\,d^7-10\,b^2\,c^2\,d^7\right )+18\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-\frac {5\,c^2\,d^{7/2}\,\mathrm {atan}\left (\frac {2000\,c^6\,d^{27/2}\,\sqrt {b\,d+2\,c\,d\,x}}{\left (\frac {2000\,b^2\,c^6\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}-\frac {8000\,a\,c^7\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{3/4}}-\frac {5\,c^2\,d^{7/2}\,\mathrm {atanh}\left (\frac {2000\,c^6\,d^{27/2}\,\sqrt {b\,d+2\,c\,d\,x}}{\left (\frac {2000\,b^2\,c^6\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}-\frac {8000\,a\,c^7\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{3/4}} \]
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