Integrand size = 26, antiderivative size = 133 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 d^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 133, 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, 104, 163, 65, 223, 212, 95, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 d^{3/2}}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d} \]
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Rule 65
Rule 95
Rule 104
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 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}+\frac {\text {Subst}\left (\int \frac {a^2 d-\frac {1}{2} b (b c-3 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{2 d} \\ & = \frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}+\frac {1}{2} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )-\frac {(b (b c-3 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d} \\ & = \frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}+a^2 \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )-\frac {(b c-3 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 d} \\ & = \frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {(b c-3 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 d} \\ & = \frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 d}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{2 d^{3/2}} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2}}{d}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}-\frac {\sqrt {b} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{d^{3/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(101)=202\).
Time = 3.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.83
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 d}+\frac {3 a b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{4 \sqrt {b d}}-\frac {b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c}{4 d \sqrt {b d}}-\frac {a^{2} \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{2 \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(243\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {b d}\, \ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a^{2} d -3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b d +\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c -2 \sqrt {b d}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {a c}\, b \right )}{4 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, \sqrt {a c}\, d}\) | \(251\) |
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Time = 0.72 (sec) , antiderivative size = 918, normalized size of antiderivative = 6.90 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\left [\frac {2 \, a d \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 ) - {\left (b c - 3 \, a d\right )} \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 ) + 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{8 \, d}, \frac {a d \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 ) + {\left (b c - 3 \, a d\right )} \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 ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{4 \, d}, \frac {4 \, a d \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 ) - {\left (b c - 3 \, a d\right )} \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 ) + 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{8 \, d}, \frac {2 \, a d \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 ) + {\left (b c - 3 \, a d\right )} \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 ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} b}{4 \, d}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x\,\sqrt {d\,x^2+c}} \,d x \]
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