Integrand size = 26, antiderivative size = 136 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}-\frac {\sqrt {a} (3 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 c^{3/2}}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \]
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Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {457, 100, 163, 65, 223, 212, 95, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {\sqrt {a} (3 b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 c^{3/2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2} \]
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Rule 65
Rule 95
Rule 100
Rule 163
Rule 212
Rule 214
Rule 223
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}-\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} a (3 b c-a d)-b^2 c x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )+\frac {(a (3 b c-a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c} \\ & = -\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}+b \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 )+\frac {(a (3 b c-a d)) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{2 c} \\ & = -\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}-\frac {\sqrt {a} (3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 c^{3/2}}+b \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 ) \\ & = -\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 c x^2}-\frac {\sqrt {a} (3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 c^{3/2}}+\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{c x^2}+\frac {\sqrt {a} (-3 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d}}\right ) \]
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Time = 3.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.43
method | result | size |
risch | \(-\frac {a \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 c \,x^{2}}-\frac {\left (-\frac {b^{2} c \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{\sqrt {b d}}-\frac {a \left (a d -3 b c \right ) \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{2 c \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(194\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{2 \sqrt {b d}}-\frac {a \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 c \,x^{2}}+\frac {a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) d}{4 c \sqrt {a c}}-\frac {3 a b \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{4 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(251\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c \,x^{2} \sqrt {a c}+\ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} d \,x^{2} \sqrt {b d}-3 \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a b c \,x^{2} \sqrt {b d}-2 a \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\right )}{4 c \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(263\) |
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Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (104) = 208\).
Time = 0.57 (sec) , antiderivative size = 958, normalized size of antiderivative = 7.04 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\left [\frac {2 \, b c x^{2} \sqrt {\frac {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^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) - {\left (3 \, b c - a d\right )} x^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a}{8 \, c x^{2}}, -\frac {4 \, b c x^{2} \sqrt {-\frac {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 {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + {\left (3 \, b c - a d\right )} x^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) + 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a}{8 \, c x^{2}}, \frac {b c x^{2} \sqrt {\frac {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^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + {\left (3 \, b c - a d\right )} x^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a}{4 \, c x^{2}}, -\frac {2 \, b c x^{2} \sqrt {-\frac {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 {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) - {\left (3 \, b c - a d\right )} x^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a}{4 \, c x^{2}}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{3} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (104) = 208\).
Time = 0.35 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.66 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {{\left (\frac {\sqrt {b d} b \log \left ({\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{d} + \frac {{\left (3 \, \sqrt {b d} a b^{2} c - \sqrt {b d} a^{2} b d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c} + \frac {2 \, {\left (\sqrt {b d} a b^{4} c^{2} - 2 \, \sqrt {b d} a^{2} b^{3} c d + \sqrt {b d} a^{3} b^{2} d^{2} - \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{2} c - \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} c}\right )} b}{2 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^3 \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^3\,\sqrt {d\,x^2+c}} \,d x \]
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