Integrand size = 24, antiderivative size = 117 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 117, 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 = {481, 541, 12, 385, 211} \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {a^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}}+\frac {x (b c-4 a d)}{3 d \sqrt {c+d x^2} (b c-a d)^2}-\frac {c x}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 481
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {a c+(b c-3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 d (b c-a d)} \\ & = -\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt {c+d x^2}}+\frac {\int \frac {3 a^2 c d}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c d (b c-a d)^2} \\ & = -\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2} \\ & = -\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a^2 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{(b c-a d)^2} \\ & = -\frac {c x}{3 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(b c-4 a d) x}{3 d (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(351\) vs. \(2(117)=234\).
Time = 1.84 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.00 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {-3 a c x+b c x^3-4 a d x^3}{3 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {a} \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{d (-b c+a d)^3}+\frac {\sqrt {a} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (b c-a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{d (-b c+a d)^3} \]
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Time = 2.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}}-x \sqrt {\left (a d -b c \right ) a}\, \left (\left (\frac {4 d \,x^{2}}{3}+c \right ) a -\frac {c b \,x^{2}}{3}\right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) a}\, \left (a d -b c \right )^{2}}\) | \(107\) |
default | \(\text {Expression too large to display}\) | \(1502\) |
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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (99) = 198\).
Time = 0.48 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.48 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left ({\left (b c - 4 \, a d\right )} x^{3} - 3 \, a c x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a d^{2} x^{4} + 2 \, a c d x^{2} + a c^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - 2 \, {\left ({\left (b c - 4 \, a d\right )} x^{3} - 3 \, a c x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{4}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (99) = 198\).
Time = 0.31 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.60 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {a^{2} \sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\frac {{\left (b^{3} c^{4} d - 6 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} - 4 \, a^{3} c d^{4}\right )} x^{2}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}} - \frac {3 \, {\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^4}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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