Integrand size = 24, antiderivative size = 99 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {457, 79, 65, 214} \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {c+d x^2}}{2 b \left (a+b x^2\right ) (b c-a d)}-\frac {(2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \]
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Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {a \sqrt {c+d x^2}}{2 b (b c-a d) \left (a+b x^2\right )}+\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 b (b c-a d)} \\ & = \frac {a \sqrt {c+d x^2}}{2 b (b c-a d) \left (a+b x^2\right )}+\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 b d (b c-a d)} \\ & = \frac {a \sqrt {c+d x^2}}{2 b (b c-a d) \left (a+b x^2\right )}-\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{3/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\frac {a \sqrt {b} \sqrt {c+d x^2}}{(b c-a d) \left (a+b x^2\right )}-\frac {(2 b c-a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{2 b^{3/2}} \]
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Time = 3.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-\frac {a \sqrt {d \,x^{2}+c}}{b \,x^{2}+a}+\frac {\left (a d -2 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{2 \left (a d -b c \right ) b}\) | \(83\) |
default | \(-\frac {\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 b^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {\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 b^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {\sqrt {-a b}\, \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a 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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{3}}-\frac {\sqrt {-a b}\, \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a 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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 b^{3}}\) | \(816\) |
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 450, normalized size of antiderivative = 4.55 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [\frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2}\right )}}, -\frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\frac {\sqrt {d x^{2} + c} a d^{2}}{{\left (b^{2} c - a b d\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}}}{2 \, d} \]
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Time = 5.69 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}}{\sqrt {a\,d-b\,c}}\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {a\,d\,\sqrt {d\,x^2+c}}{2\,b\,\left (a\,d-b\,c\right )\,\left (b\,\left (d\,x^2+c\right )+a\,d-b\,c\right )} \]
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