Integrand size = 38, antiderivative size = 211 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}+\frac {\left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{a+b x} \]
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Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1014, 828, 857, 635, 212, 738} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 a d e+4 b c d-b e^2\right ) \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 d^{3/2} (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{a+b x}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)} \]
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1014
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2}}{x} \, dx}{2 a b+2 b^2 x} \\ & = \frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {-8 a b c d-b \left (4 a d e+b \left (4 c d-e^2\right )\right ) x}{x \sqrt {c+e x+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )} \\ & = \frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}+\frac {\left (2 a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{x \sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x}+\frac {\left (b \left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )} \\ & = \frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}-\frac {\left (4 a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {2 c+e x}{\sqrt {c+e x+d x^2}}\right )}{2 a b+2 b^2 x}+\frac {\left (b \left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{2 d \left (2 a b+2 b^2 x\right )} \\ & = \frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}+\frac {\left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{a+b x} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\frac {\sqrt {(a+b x)^2} \left (\left (4 b c d+4 a d e-b e^2\right ) \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+x (e+d x)}}\right )+2 \sqrt {d} \left (\sqrt {c+x (e+d x)} (4 a d+b (e+2 d x))+8 a \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {d} x-\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )\right )\right )}{8 d^{3/2} (a+b x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\operatorname {csgn}\left (b x +a \right ) \left (4 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} b x -8 \sqrt {c}\, d^{\frac {5}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a +8 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} a +2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} b e +4 d^{2} \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a e +4 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b c \,d^{2}-\ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b d \,e^{2}\right )}{8 d^{\frac {5}{2}}}\) | \(215\) |
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Time = 0.78 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\left [\frac {8 \, a \sqrt {c} d^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{16 \, d^{2}}, \frac {4 \, a \sqrt {c} d^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{8 \, d^{2}}, \frac {16 \, a \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{16 \, d^{2}}, \frac {8 \, a \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{8 \, d^{2}}\right ] \]
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\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\int \frac {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x}\, dx \]
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\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\int { \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x} \,d x \]
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