Integrand size = 23, antiderivative size = 323 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {d} (b c-9 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {425, 541, 539, 429, 422} \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d} \sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 b x (b c-3 a d)}{3 a^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)} \]
[In]
[Out]
Rule 422
Rule 425
Rule 429
Rule 539
Rule 541
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}-\frac {\int \frac {-2 b c+3 a d-3 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a (b c-a d)} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\int \frac {a d (b c+3 a d)+2 b d (b c-3 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^2} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(b d (b c-9 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a (b c-a d)^3}+\frac {\left (d \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^3} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {d} (b c-9 a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.40 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (3 a^4 d^3+6 a^3 b d^3 x^2-2 b^4 c^2 x^2 \left (c+d x^2\right )+a^2 b^2 d \left (8 c^2+8 c d x^2+3 d^2 x^4\right )+a b^3 c \left (-3 c^2+4 c d x^2+7 d^2 x^4\right )\right )+i b c \left (-2 b^2 c^2+7 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i b c \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c (-b c+a d)^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 6.71 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.74
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a \left (a d -b c \right )^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) b x \left (7 a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (b d \,x^{2}+a d \right ) d^{2} x}{c \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {b d}{3 \left (a d -b c \right )^{2} a}-\frac {b \left (7 a d -2 b c \right )}{3 \left (a d -b c \right )^{2} a^{2}}-\frac {b^{2} c \left (7 a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )^{3}}+\frac {d^{2}}{\left (a d -b c \right )^{2} c}-\frac {a \,d^{3}}{c \left (a d -b c \right )^{3}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (-\frac {b^{2} d \left (7 a d -2 b c \right )}{3 \left (a d -b c \right )^{3} a^{2}}-\frac {b \,d^{3}}{\left (a d -b c \right )^{3} c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(561\) |
default | \(-\frac {-3 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} d^{3} x^{5}-7 \sqrt {-\frac {b}{a}}\, a \,b^{3} c \,d^{2} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{2} d \,x^{5}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c \,d^{2} x^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{2} d \,x^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{3} x^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c \,d^{2} x^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{2} d \,x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{3} x^{2}-6 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{3} x^{3}-8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{2} x^{3}-4 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d \,x^{3}+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} x^{3}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3}-3 \sqrt {-\frac {b}{a}}\, a^{4} d^{3} x -8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d x +3 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} x}{3 \sqrt {d \,x^{2}+c}\, \left (a d -b c \right )^{3} a^{2} \sqrt {-\frac {b}{a}}\, c \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(964\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (305) = 610\).
Time = 0.13 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.65 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, a^{2} b^{3} c^{3} - 7 \, a^{3} b^{2} c^{2} d - 3 \, a^{4} b c d^{2} + {\left (2 \, b^{5} c^{2} d - 7 \, a b^{4} c d^{2} - 3 \, a^{2} b^{3} d^{3}\right )} x^{6} + {\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d - 17 \, a^{2} b^{3} c d^{2} - 6 \, a^{3} b^{2} d^{3}\right )} x^{4} + {\left (4 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d - 13 \, a^{3} b^{2} c d^{2} - 3 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a^{2} b^{3} c^{3} + {\left (2 \, b^{5} c^{2} d + {\left (a^{2} b^{3} - 7 \, a b^{4}\right )} c d^{2} - 3 \, {\left (3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d^{3}\right )} x^{6} + {\left (2 \, b^{5} c^{3} + {\left (a^{2} b^{3} - 3 \, a b^{4}\right )} c^{2} d - {\left (7 \, a^{3} b^{2} + 17 \, a^{2} b^{3}\right )} c d^{2} - 6 \, {\left (3 \, a^{4} b + a^{3} b^{2}\right )} d^{3}\right )} x^{4} + {\left (a^{4} b - 7 \, a^{3} b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{5} + a^{4} b\right )} c d^{2} + {\left (4 \, a b^{4} c^{3} + 2 \, {\left (a^{3} b^{2} - 6 \, a^{2} b^{3}\right )} c^{2} d - {\left (17 \, a^{4} b + 13 \, a^{3} b^{2}\right )} c d^{2} - 3 \, {\left (3 \, a^{5} + a^{4} b\right )} d^{3}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{4} c^{2} d - 7 \, a^{2} b^{3} c d^{2} - 3 \, a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (a b^{4} c^{3} - 2 \, a^{2} b^{3} c^{2} d - 4 \, a^{3} b^{2} c d^{2} - 3 \, a^{4} b d^{3}\right )} x^{3} + {\left (3 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d - 3 \, a^{5} d^{3}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{3} c^{5} - 3 \, a^{6} b^{2} c^{4} d + 3 \, a^{7} b c^{3} d^{2} - a^{8} c^{2} d^{3} + {\left (a^{3} b^{5} c^{4} d - 3 \, a^{4} b^{4} c^{3} d^{2} + 3 \, a^{5} b^{3} c^{2} d^{3} - a^{6} b^{2} c d^{4}\right )} x^{6} + {\left (a^{3} b^{5} c^{5} - a^{4} b^{4} c^{4} d - 3 \, a^{5} b^{3} c^{3} d^{2} + 5 \, a^{6} b^{2} c^{2} d^{3} - 2 \, a^{7} b c d^{4}\right )} x^{4} + {\left (2 \, a^{4} b^{4} c^{5} - 5 \, a^{5} b^{3} c^{4} d + 3 \, a^{6} b^{2} c^{3} d^{2} + a^{7} b c^{2} d^{3} - a^{8} c d^{4}\right )} x^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
[In]
[Out]