Integrand size = 24, antiderivative size = 115 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{2 a c x^2}+\frac {(2 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{3/2}}-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 \sqrt {b c-a d}} \]
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Time = 0.08 (sec) , antiderivative size = 115, 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, 105, 162, 65, 214} \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 \sqrt {b c-a d}}+\frac {(a d+2 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{3/2}}-\frac {\sqrt {c+d x^2}}{2 a c x^2} \]
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Rule 65
Rule 105
Rule 162
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {c+d x^2}}{2 a c x^2}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (2 b c+a d)+\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a c} \\ & = -\frac {\sqrt {c+d x^2}}{2 a c x^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{(a+b 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}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 c} \\ & = -\frac {\sqrt {c+d x^2}}{2 a c x^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b 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}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^2 c d} \\ & = -\frac {\sqrt {c+d x^2}}{2 a c x^2}+\frac {(2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{3/2}}-\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 \sqrt {b c-a d}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {-\frac {a \sqrt {c+d x^2}}{c x^2}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(2 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{2 a^2} \]
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Time = 3.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{\frac {3}{2}} b^{2} x^{2}+\sqrt {\left (a d -b c \right ) b}\, \left (-x^{2} \left (a d +2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{2}+c}\, a \sqrt {c}\right )}{2 \sqrt {\left (a d -b c \right ) b}\, c^{\frac {3}{2}} a^{2} x^{2}}\) | \(115\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}}{2 a c \,x^{2}}-\frac {-\frac {\left (a d +2 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a \sqrt {c}}+\frac {b c \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}+\frac {b c \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{a \sqrt {-\frac {a d -b c}{b}}}}{2 a c}\) | \(370\) |
default | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}}{a}+\frac {b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{a^{2} \sqrt {c}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-\frac {a d -b c}{b}}}\) | \(384\) |
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Time = 0.33 (sec) , antiderivative size = 734, normalized size of antiderivative = 6.38 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [\frac {b c^{2} x^{2} \sqrt {\frac {b}{b c - a 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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (2 \, b c + a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c^{2} x^{2}}, \frac {b c^{2} x^{2} \sqrt {\frac {b}{b c - a 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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, {\left (2 \, b c + a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c^{2} x^{2}}, \frac {2 \, b c^{2} x^{2} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + {\left (2 \, b c + a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {d x^{2} + c} a c}{4 \, a^{2} c^{2} x^{2}}, \frac {b c^{2} x^{2} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - {\left (2 \, b c + a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - \sqrt {d x^{2} + c} a c}{2 \, a^{2} c^{2} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} x^{3}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {b^{2} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2}} - \frac {{\left (2 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c} c} - \frac {\sqrt {d x^{2} + c}}{2 \, a c x^{2}} \]
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Time = 5.80 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.44 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\ln \left (\sqrt {d\,x^2+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}+b^6\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{2\,a^3\,d-2\,a^2\,b\,c}-\frac {\ln \left (\sqrt {d\,x^2+c}\,{\left (b^4\,c-a\,b^3\,d\right )}^{3/2}-b^6\,c^2-a^2\,b^4\,d^2+2\,a\,b^5\,c\,d\right )\,\sqrt {b^4\,c-a\,b^3\,d}}{2\,\left (a^3\,d-a^2\,b\,c\right )}-\frac {\sqrt {d\,x^2+c}}{2\,a\,c\,x^2}-\frac {\mathrm {atan}\left (\frac {b^4\,d^4\,\sqrt {d\,x^2+c}\,3{}\mathrm {i}}{2\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{2\,c}+\frac {5\,a\,b^3\,d^5}{4\,c^2}+\frac {a^2\,b^2\,d^6}{4\,c^3}\right )}+\frac {b^2\,d^6\,\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{4\,\sqrt {c^3}\,\left (\frac {5\,b^3\,d^5}{4\,a}+\frac {b^2\,d^6}{4\,c}+\frac {3\,b^4\,c\,d^4}{2\,a^2}\right )}+\frac {b^3\,d^5\,\sqrt {d\,x^2+c}\,5{}\mathrm {i}}{4\,\sqrt {c^3}\,\left (\frac {3\,b^4\,d^4}{2\,a}+\frac {5\,b^3\,d^5}{4\,c}+\frac {a\,b^2\,d^6}{4\,c^2}\right )}\right )\,\left (a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{2\,a^2\,\sqrt {c^3}} \]
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