Integrand size = 24, antiderivative size = 119 \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b} \sqrt {b c-a d}} \]
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Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 101, 162, 65, 214} \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b} \sqrt {b c-a d}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )} \]
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Rule 65
Rule 101
Rule 162
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {-c-\frac {d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a} \\ & = \frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}+\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2}-\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2} \\ & = \frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}+\frac {c \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 d}-\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 d} \\ & = \frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {a \sqrt {c+d x^2}}{a+b x^2}+\frac {(-2 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \]
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Time = 3.01 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\frac {\left (b \,x^{2}+a \right ) \left (a d -2 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )-2 \left (\sqrt {c}\, \left (b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )-\frac {\sqrt {d \,x^{2}+c}\, a}{2}\right ) \sqrt {\left (a d -b c \right ) b}}{2 \sqrt {\left (a d -b c \right ) b}\, a^{2} \left (b \,x^{2}+a \right )}\) | \(120\) |
default | \(\text {Expression too large to display}\) | \(2001\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (97) = 194\).
Time = 0.35 (sec) , antiderivative size = 1054, normalized size of antiderivative = 8.86 \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\left [-\frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {8 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 4 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x \left (a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} d}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{2}} + \frac {c \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} \]
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Time = 5.65 (sec) , antiderivative size = 996, normalized size of antiderivative = 8.37 \[ \int \frac {\sqrt {c+d x^2}}{x \left (a+b x^2\right )^2} \, dx=\frac {d\,\sqrt {d\,x^2+c}}{2\,a\,\left (b\,\left (d\,x^2+c\right )+a\,d-b\,c\right )}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )}{a^2}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}-\frac {\left (2\,a\,b^2\,c\,d^3-\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}+\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}+\frac {\left (2\,a\,b^2\,c\,d^3+\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}}{\frac {b^2\,c^2\,d^3-\frac {a\,b\,c\,d^4}{2}}{a^3}+\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}-\frac {\left (2\,a\,b^2\,c\,d^3-\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}-\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}+\frac {\left (2\,a\,b^2\,c\,d^3+\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )} \]
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