Integrand size = 21, antiderivative size = 199 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {c^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {390, 386, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\frac {c^2 (5 b c-6 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}}+\frac {c x \sqrt {c+d x^2} (5 b c-6 a d)}{16 a^3 \left (a+b x^2\right ) (b c-a d)}+\frac {x \left (c+d x^2\right )^{3/2} (5 b c-6 a d)}{24 a^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)} \]
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Rule 211
Rule 385
Rule 386
Rule 390
Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {(5 b c-6 a d) \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^3} \, dx}{6 a (b c-a d)} \\ & = \frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {(c (5 b c-6 a d)) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx}{8 a^2 (b c-a d)} \\ & = \frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {\left (c^2 (5 b c-6 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{16 a^3 (b c-a d)} \\ & = \frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {\left (c^2 (5 b c-6 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 )}{16 a^3 (b c-a d)} \\ & = \frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {c^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 15.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\frac {-\frac {\sqrt {a} x \sqrt {c+d x^2} \left (15 b^3 c^2 x^4+8 a b^2 c x^2 \left (5 c-d x^2\right )-6 a^3 d \left (5 c+2 d x^2\right )+a^2 b \left (33 c^2-22 c d x^2-4 d^2 x^4\right )\right )}{(-b c+a d) \left (a+b x^2\right )^3}+\frac {3 c^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}}}{48 a^{7/2}} \]
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Time = 2.48 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {5 \left (d \left (\frac {2 d \,x^{2}}{5}+c \right ) a^{3}-\frac {11 \left (-\frac {4}{33} d^{2} x^{4}-\frac {2}{3} c d \,x^{2}+c^{2}\right ) b \,a^{2}}{10}-\frac {4 \left (-\frac {d \,x^{2}}{5}+c \right ) x^{2} b^{2} c a}{3}-\frac {c^{2} b^{3} x^{4}}{2}\right ) x \sqrt {d \,x^{2}+c}\, \sqrt {\left (a d -b c \right ) a}+3 \left (b \,x^{2}+a \right )^{3} \left (a d -\frac {5 b c}{6}\right ) c^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{8 \sqrt {\left (a d -b c \right ) a}\, \left (a d -b c \right ) a^{3} \left (b \,x^{2}+a \right )^{3}}\) | \(179\) |
default | \(\text {Expression too large to display}\) | \(12958\) |
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (175) = 350\).
Time = 0.48 (sec) , antiderivative size = 972, normalized size of antiderivative = 4.88 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\left [-\frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} 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 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 ) - 4 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{192 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}, \frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \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 ) + 2 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (175) = 350\).
Time = 1.26 (sec) , antiderivative size = 919, normalized size of antiderivative = 4.62 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=-\frac {{\left (5 \, b c^{3} \sqrt {d} - 6 \, a c^{2} d^{\frac {3}{2}}\right )} \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 )}{16 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{5} c^{3} \sqrt {d} - 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b^{4} c^{2} d^{\frac {3}{2}} - 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{5} c^{4} \sqrt {d} + 240 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b^{4} c^{3} d^{\frac {3}{2}} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} b^{3} c^{2} d^{\frac {5}{2}} - 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{3} b^{2} c d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{4} b d^{\frac {9}{2}} + 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{5} c^{5} \sqrt {d} - 620 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b^{4} c^{4} d^{\frac {3}{2}} + 968 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} b^{3} c^{3} d^{\frac {5}{2}} - 720 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{3} b^{2} c^{2} d^{\frac {7}{2}} + 64 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{4} b c d^{\frac {9}{2}} + 128 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{5} d^{\frac {11}{2}} - 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{5} c^{6} \sqrt {d} + 600 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{4} c^{5} d^{\frac {3}{2}} - 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b^{3} c^{4} d^{\frac {5}{2}} + 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} b^{2} c^{3} d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{4} b c^{2} d^{\frac {9}{2}} + 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{5} c^{7} \sqrt {d} - 210 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{4} c^{6} d^{\frac {3}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{3} c^{5} d^{\frac {5}{2}} + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b^{2} c^{4} d^{\frac {7}{2}} - 15 \, b^{5} c^{8} \sqrt {d} + 8 \, a b^{4} c^{7} d^{\frac {3}{2}} + 4 \, a^{2} b^{3} c^{6} d^{\frac {5}{2}}}{24 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{3}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^4} \,d x \]
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