Integrand size = 31, antiderivative size = 84 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=-\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+2 b d \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {465, 99, 12, 65, 223, 212} \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+2 b d \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )-\frac {b \sqrt {d x-c} \sqrt {c+d x}}{x} \]
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Rule 12
Rule 65
Rule 99
Rule 212
Rule 223
Rule 465
Rubi steps \begin{align*} \text {integral}& = \frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+b \int \frac {\sqrt {-c+d x} \sqrt {c+d x}}{x^2} \, dx \\ & = -\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+b \int \frac {d^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx \\ & = -\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+\left (b d^2\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx \\ & = -\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+(2 b d) \text {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right ) \\ & = -\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+(2 b d) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \\ & = -\frac {b \sqrt {-c+d x} \sqrt {c+d x}}{x}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{3 c^2 x^3}+2 b d \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=-\frac {\sqrt {-c+d x} \sqrt {c+d x} \left (3 b c^2 x^2+a \left (c^2-d^2 x^2\right )\right )}{3 c^2 x^3}+2 b d \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right ) \]
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Time = 4.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.48
method | result | size |
risch | \(\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (-a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+c^{2} a \right )}{3 x^{3} c^{2} \sqrt {d x -c}}+\frac {b \,d^{2} \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {d^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(124\) |
default | \(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (-3 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{2} d \,x^{3}-\operatorname {csgn}\left (d \right ) a \,d^{2} x^{2} \sqrt {d^{2} x^{2}-c^{2}}+3 \,\operatorname {csgn}\left (d \right ) b \,c^{2} x^{2} \sqrt {d^{2} x^{2}-c^{2}}+\operatorname {csgn}\left (d \right ) a \,c^{2} \sqrt {d^{2} x^{2}-c^{2}}\right ) \operatorname {csgn}\left (d \right )}{3 \sqrt {d^{2} x^{2}-c^{2}}\, c^{2} x^{3}}\) | \(153\) |
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=-\frac {3 \, b c^{2} d x^{3} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right ) + {\left (3 \, b c^{2} d - a d^{3}\right )} x^{3} + {\left (a c^{2} + {\left (3 \, b c^{2} - a d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, c^{2} x^{3}} \]
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Exception generated. \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=\text {Exception raised: MellinTransformStripError} \]
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Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=b d \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right ) - \frac {\sqrt {d^{2} x^{2} - c^{2}} b}{x} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a}{3 \, c^{2} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (70) = 140\).
Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=-\frac {3 \, b d^{2} \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4}\right ) + \frac {16 \, {\left (3 \, b c^{2} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{8} - 3 \, a d^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{8} + 24 \, b c^{4} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 48 \, b c^{6} d^{2} - 16 \, a c^{4} d^{4}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3}}}{6 \, d} \]
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Time = 7.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.81 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^4} \, dx=\frac {b\,d+\frac {5\,b\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}}{\frac {4\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {-c}-\sqrt {d\,x-c}}+\frac {4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}}-4\,b\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )-\frac {\left (\frac {a\,\sqrt {c+d\,x}}{3}-\frac {a\,d^2\,x^2\,\sqrt {c+d\,x}}{3\,c^2}\right )\,\sqrt {d\,x-c}}{x^3}+\frac {b\,d\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{4\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )} \]
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