Integrand size = 24, antiderivative size = 170 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {b} (2 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 79, 53, 65, 214} \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {b} (3 a d+2 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}}+\frac {a}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 a d+2 b c}{2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {3 a d+2 b c}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]
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Rule 53
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 (c+d x)^{5/2}} \, dx,x,x^2\right ) \\ & = \frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+3 a d) \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{4 b (b c-a d)} \\ & = \frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+3 a d) \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 (b c-a d)^2} \\ & = \frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (2 b c+3 a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^3} \\ & = \frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (2 b c+3 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 d (b c-a d)^3} \\ & = \frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {b} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {2 a^2 d \left (2 c+3 d x^2\right )+2 b^2 c x^2 \left (4 c+3 d x^2\right )+a b \left (11 c^2+16 c d x^2+9 d^2 x^4\right )}{(b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {3 \sqrt {b} (2 b c+3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}\right ) \]
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Time = 3.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\frac {9 \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \left (a d +\frac {2 b c}{3}\right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{4}+\left (2 x^{2} \left (\frac {3 d \,x^{2}}{4}+c \right ) c \,b^{2}+\frac {11 \left (\frac {9}{11} d^{2} x^{4}+\frac {16}{11} c d \,x^{2}+c^{2}\right ) a b}{4}+d \,a^{2} \left (\frac {3 d \,x^{2}}{2}+c \right )\right ) \sqrt {\left (a d -b c \right ) b}\right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) b}\, \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{3}}\) | \(162\) |
default | \(\text {Expression too large to display}\) | \(3483\) |
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Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (146) = 292\).
Time = 0.36 (sec) , antiderivative size = 993, normalized size of antiderivative = 5.84 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a 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 (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.53 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {3 \, \sqrt {d x^{2} + c} a b d^{2}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac {3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} b c d + b c^{2} d + 3 \, {\left (d x^{2} + c\right )} a d^{2} - a c d^{2}\right )}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{6 \, d} \]
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Time = 5.66 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {\frac {\left (d\,x^2+c\right )\,\left (3\,a\,d+2\,b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^2}-\frac {c}{3\,\left (a\,d-b\,c\right )}+\frac {b\,{\left (d\,x^2+c\right )}^2\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^3}}{b\,{\left (d\,x^2+c\right )}^{5/2}+{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-b\,c\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^{7/2}}\right )\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^{7/2}} \]
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