Integrand size = 26, antiderivative size = 178 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac {3 c^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{7/4}} \]
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Time = 0.09 (sec) , antiderivative size = 178, 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 = {700, 701, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {3 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac {3 c^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 701
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (c d^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (3 c^2 d^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(3 c d) \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 )}{4 \left (b^2-4 a c\right )} \\ & = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(3 c d) \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 )}{2 \left (b^2-4 a c\right )} \\ & = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (3 c^2 d^2\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 (b^2-4 a c\right )^{3/2}}+\frac {\left (3 c^2 d^2\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 (b^2-4 a c\right )^{3/2}} \\ & = -\frac {d \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2}-\frac {c d \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {3 c^2 d^{3/2} \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 )^{7/4}}+\frac {3 c^2 d^{3/2} \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 )^{7/4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.34 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.47 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {i (d (b+2 c x))^{3/2} \left (i \left (b^2-4 a c\right )^{3/4} \sqrt {b+2 c x} \left (b^2+b c x+c \left (-3 a+c x^2\right )\right )+(3+3 i) c^2 (a+x (b+c x))^2 \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-(3+3 i) c^2 (a+x (b+c x))^2 \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-(3+3 i) c^2 (a+x (b+c x))^2 \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{2 \left (b^2-4 a c\right )^{7/4} (b+2 c x)^{3/2} (a+x (b+c x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(323\) vs. \(2(150)=300\).
Time = 2.95 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(64 c^{2} d^{5} \left (\frac {\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{32 d^{2} \left (4 a c -b^{2}\right )}-\frac {3 \sqrt {2 c d x +b d}}{32}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 d^{2} \left (4 a c -b^{2}\right ) \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(324\) |
default | \(64 c^{2} d^{5} \left (\frac {\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{32 d^{2} \left (4 a c -b^{2}\right )}-\frac {3 \sqrt {2 c d x +b d}}{32}}{\left (\left (2 c d x +b d \right )^{2}+4 a c \,d^{2}-b^{2} d^{2}\right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 d^{2} \left (4 a c -b^{2}\right ) \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(324\) |
pseudoelliptic | \(-\frac {3 d \left (-\frac {\sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2} \ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )}{2}+\arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2}-\arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right ) \sqrt {2}\, c^{2} d^{2} \left (c \,x^{2}+b x +a \right )^{2}+\left (-\frac {c^{2} x^{2}}{3}+\left (-\frac {b x}{3}+a \right ) c -\frac {b^{2}}{3}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}\right )}{8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )^{2} \left (-\frac {b^{2}}{4}+a c \right )}\) | \(375\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1345, normalized size of antiderivative = 7.56 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (150) = 300\).
Time = 0.31 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.19 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {3 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} - 8 \, \sqrt {2} a b^{2} c + 16 \, \sqrt {2} a^{2} c^{2}} + \frac {3 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} - 8 \, \sqrt {2} a b^{2} c + 16 \, \sqrt {2} a^{2} c^{2}} + \frac {3 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{4} - 8 \, \sqrt {2} a b^{2} c + 16 \, \sqrt {2} a^{2} c^{2}\right )}} - \frac {3 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{4} - 8 \, \sqrt {2} a b^{2} c + 16 \, \sqrt {2} a^{2} c^{2}\right )}} - \frac {2 \, {\left (3 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{5} - 12 \, \sqrt {2 \, c d x + b d} a c^{3} d^{5} + {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2} d^{3}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \]
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Time = 9.38 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.43 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {3\,c^2\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )}{{\left (b^2-4\,a\,c\right )}^{7/4}}-\frac {6\,c^2\,d^5\,\sqrt {b\,d+2\,c\,d\,x}-\frac {2\,c^2\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{4\,a\,c-b^2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}+\frac {3\,c^2\,d^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )}{{\left (b^2-4\,a\,c\right )}^{7/4}} \]
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