Integrand size = 24, antiderivative size = 134 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(2 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 134, 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, 79, 53, 65, 214} \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {(a d+2 b c) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}}+\frac {a}{2 b \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {a d+2 b c}{2 b \sqrt {c+d x^2} (b c-a d)^2} \]
[In]
[Out]
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)^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+a d) \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 b (b c-a d)} \\ & = \frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+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)^2} \\ & = \frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 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 )}{2 d (b c-a d)^2} \\ & = \frac {2 b c+a d}{2 b (b c-a d)^2 \sqrt {c+d x^2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(2 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {3 a c+2 b c x^2+a d x^2}{(b c-a d)^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(2 b c+a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{5/2}}\right ) \]
[In]
[Out]
Time = 3.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {d \,x^{2}+c}\, \left (b \,x^{2}+a \right ) \left (a d +2 b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2}+\frac {3 \sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {d \,x^{2}}{3}+c \right ) a +\frac {2 c b \,x^{2}}{3}\right )}{2}}{\left (b \,x^{2}+a \right ) \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}\, \sqrt {d \,x^{2}+c}}\) | \(125\) |
default | \(\text {Expression too large to display}\) | \(1922\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (114) = 228\).
Time = 0.32 (sec) , antiderivative size = 732, normalized size of antiderivative = 5.46 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (2 \, b^{2} c d + a b d^{2}\right )} x^{4} + 2 \, a b c^{2} + a^{2} c d + {\left (2 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + {\left (2 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{4} c^{4} - 3 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} - a^{4} b c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{4} + {\left (b^{5} c^{4} - 2 \, a b^{4} c^{3} d + 2 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{2}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c d + a b d^{2}\right )} x^{4} + 2 \, a b c^{2} + a^{2} c d + {\left (2 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + {\left (2 \, b^{3} c^{2} - a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{4} c^{4} - 3 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} - a^{4} b c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{4} + {\left (b^{5} c^{4} - 2 \, a b^{4} c^{3} d + 2 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x^{2}\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (2 \, b c d + a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (d x^{2} + c\right )} b c d - 2 \, b c^{2} d + {\left (d x^{2} + c\right )} a d^{2} + 2 \, a c d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{2} + c} b c + \sqrt {d x^{2} + c} a d\right )}}}{2 \, d} \]
[In]
[Out]
Time = 5.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {c}{a\,d-b\,c}+\frac {\left (d\,x^2+c\right )\,\left (a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^2}}{b\,{\left (d\,x^2+c\right )}^{3/2}+\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )\,\left (a\,d+2\,b\,c\right )}{2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}} \]
[In]
[Out]