Integrand size = 24, antiderivative size = 86 \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 \sqrt {b} d^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {455, 52, 65, 223, 212} \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 \sqrt {b} d^{3/2}} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d} \\ & = \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{2 b d} \\ & = \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{2 b d} \\ & = \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 \sqrt {b} d^{3/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 \sqrt {b} d^{3/2}} \]
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Time = 3.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 d}+\frac {\left (a d -b c \right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{4 d \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(124\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (a \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) d -b \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) c +2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\right )}{4 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, d \sqrt {b d}}\) | \(170\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {a \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{4 \sqrt {b d}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 d}-\frac {b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 d \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(178\) |
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Time = 0.34 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.01 \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\left [\frac {4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b d - {\left (b c - a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right )}{8 \, b d^{2}}, \frac {2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b d + {\left (b c - a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{4 \, b d^{2}}\right ] \]
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\[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x \sqrt {a + b x^{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.23 \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {b {\left (\frac {{\left (b c - a d\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d} + \frac {\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a}}{b d}\right )}}{2 \, {\left | b \right |}} \]
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Time = 9.51 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.26 \[ \int \frac {x \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (a\,d+b\,c\right )}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}+\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (c\,b^2+a\,d\,b\right )}{d^3\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {4\,\sqrt {a}\,b\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}+\frac {b^2}{d^2}-\frac {2\,b\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}}+\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )}{\sqrt {b}\,d^{3/2}} \]
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