Integrand size = 24, antiderivative size = 88 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=-\frac {a \sqrt {c+d x^2}}{b^2}+\frac {\left (c+d x^2\right )^{3/2}}{3 b d}+\frac {a \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 81, 52, 65, 214} \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {a \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{5/2}}-\frac {a \sqrt {c+d x^2}}{b^2}+\frac {\left (c+d x^2\right )^{3/2}}{3 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x \sqrt {c+d x}}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {\left (c+d x^2\right )^{3/2}}{3 b d}-\frac {a \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{2 b} \\ & = -\frac {a \sqrt {c+d x^2}}{b^2}+\frac {\left (c+d x^2\right )^{3/2}}{3 b d}-\frac {(a (b c-a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b^2} \\ & = -\frac {a \sqrt {c+d x^2}}{b^2}+\frac {\left (c+d x^2\right )^{3/2}}{3 b d}-\frac {(a (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 )}{b^2 d} \\ & = -\frac {a \sqrt {c+d x^2}}{b^2}+\frac {\left (c+d x^2\right )^{3/2}}{3 b d}+\frac {a \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {\sqrt {c+d x^2} \left (-3 a d+b \left (c+d x^2\right )\right )}{3 b^2 d}+\frac {a \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{5/2}} \]
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Time = 3.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-b d \,x^{2}+3 a d -b c \right )}{3}+\frac {a d \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{d \,b^{2}}\) | \(83\) |
risch | \(-\frac {\left (-b d \,x^{2}+3 a d -b c \right ) \sqrt {d \,x^{2}+c}}{3 d \,b^{2}}+\frac {a \left (a d -b c \right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}\) | \(347\) |
default | \(\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3 b d}-\frac {a \left (\sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}-\frac {a \left (\sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}\) | \(666\) |
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Time = 0.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.35 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\left [\frac {3 \, a d \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (b d x^{2} + b c - 3 \, a d\right )} \sqrt {d x^{2} + c}}{12 \, b^{2} d}, \frac {3 \, a d \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (b d x^{2} + b c - 3 \, a d\right )} \sqrt {d x^{2} + c}}{6 \, b^{2} d}\right ] \]
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Time = 2.45 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\begin {cases} \frac {2 \left (- \frac {a d^{2} \sqrt {c + d x^{2}}}{2 b^{2}} + \frac {a d^{2} \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{2 b^{3} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x^{2}\right )^{\frac {3}{2}}}{6 b}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\sqrt {c} \left (- \frac {a \left (\begin {cases} \frac {x^{2}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x^{2} \right )}}{b} & \text {otherwise} \end {cases}\right )}{2 b} + \frac {x^{2}}{2 b}\right ) & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=-\frac {{\left (a b c - a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2} - 3 \, \sqrt {d x^{2} + c} a b d^{3}}{3 \, b^{3} d^{3}} \]
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Time = 5.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {{\left (d\,x^2+c\right )}^{3/2}}{3\,b\,d}-\frac {a\,\sqrt {d\,x^2+c}}{b^2}+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^2+c}\,\sqrt {a\,d-b\,c}}{a^2\,d-a\,b\,c}\right )\,\sqrt {a\,d-b\,c}}{b^{5/2}} \]
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