Integrand size = 24, antiderivative size = 157 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {a^{3/2} \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}-\frac {\left (b^2 c^2+4 a b c d-8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 d^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {489, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {a^{3/2} \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}-\frac {\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 d^{3/2}}+\frac {x \sqrt {c+d x^2} (b c-4 a d)}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b} \]
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 489
Rule 537
Rule 596
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {c+d x^2}}{4 b}-\frac {\int \frac {x^2 \left (3 a c+(-b c+4 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 b} \\ & = \frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {\int \frac {-a c (b c-4 a d)+\left (-b^2 c^2-4 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 b^2 d} \\ & = \frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {\left (a^2 (b c-a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}-\frac {\left (b^2 c^2+4 a b c d-8 a^2 d^2\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^3 d} \\ & = \frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {\left (a^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^3}-\frac {\left (b^2 c^2+4 a b c d-8 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^3 d} \\ & = \frac {(b c-4 a d) x \sqrt {c+d x^2}}{8 b^2 d}+\frac {x^3 \sqrt {c+d x^2}}{4 b}+\frac {a^{3/2} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}-\frac {\left (b^2 c^2+4 a b c d-8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 d^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(395\) vs. \(2(157)=314\).
Time = 1.91 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.52 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\frac {x \sqrt {c+d x^2} \left (b c-4 a d+2 b d x^2\right )}{8 b^2 d}+\frac {\sqrt {a} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^3 d}+\frac {\sqrt {a} \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{b^3 d}+\frac {\left (-b^2 c^2-4 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{4 b^3 d^{3/2}} \]
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Time = 3.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+4 a d -b c \right ) x}{4 d}-\frac {\left (8 a^{2} d^{2}-4 a b c d -b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{4 d^{\frac {3}{2}}}+\frac {2 \left (a d -b c \right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{3}}\) | \(136\) |
risch | \(-\frac {x \left (-2 b d \,x^{2}+4 a d -b c \right ) \sqrt {d \,x^{2}+c}}{8 d \,b^{2}}+\frac {\frac {\left (8 a^{2} d^{2}-4 a b c d -b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}+\frac {4 a^{2} d \left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {4 a^{2} d \left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 d \,b^{2}}\) | \(425\) |
default | \(\frac {\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4 d}-\frac {c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4 d}}{b}-\frac {a \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{b^{2}}+\frac {a^{2} \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} \sqrt {-a b}}-\frac {a^{2} \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} \sqrt {-a b}}\) | \(767\) |
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Time = 0.37 (sec) , antiderivative size = 857, normalized size of antiderivative = 5.46 \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\left [\frac {4 \, \sqrt {-a b c + a^{2} d} a d^{2} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d^{2}}, \frac {2 \, \sqrt {-a b c + a^{2} d} a d^{2} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d^{2}}, \frac {8 \, \sqrt {a b c - a^{2} d} a d^{2} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d^{2}}, \frac {4 \, \sqrt {a b c - a^{2} d} a d^{2} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + {\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} d^{2} x^{3} + {\left (b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d^{2}}\right ] \]
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\[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int \frac {x^{4} \sqrt {c + d x^{2}}}{a + b x^{2}}\, dx \]
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\[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} x^{4}}{b x^{2} + a} \,d x } \]
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Exception generated. \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 \sqrt {c+d x^2}}{a+b x^2} \, dx=\int \frac {x^4\,\sqrt {d\,x^2+c}}{b\,x^2+a} \,d x \]
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