Integrand size = 24, antiderivative size = 176 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {\left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {5 c (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2}} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {479, 594, 597, 12, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {5 c (b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2}}-\frac {c \sqrt {c+d x^2} (5 b c-3 a d)}{6 a^2 b x^3}+\frac {\sqrt {c+d x^2} \left (3 a^2 d^2-20 a b c d+15 b^2 c^2\right )}{6 a^3 b x}+\frac {\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]
[In]
[Out]
Rule 12
Rule 211
Rule 385
Rule 479
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}-\frac {\int \frac {\sqrt {c+d x^2} \left (-c (5 b c-3 a d)-2 b c d x^2\right )}{x^4 \left (a+b x^2\right )} \, dx}{2 a b} \\ & = -\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}-\frac {\int \frac {c \left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right )+2 b c d (5 b c-6 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 b} \\ & = -\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {\left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {\int \frac {15 b c^2 (b c-a d)^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 b c} \\ & = -\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {\left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {\left (5 c (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3} \\ & = -\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {\left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {\left (5 c (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^3} \\ & = -\frac {c (5 b c-3 a d) \sqrt {c+d x^2}}{6 a^2 b x^3}+\frac {\left (15 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 b x}+\frac {(b c-a d) \left (c+d x^2\right )^{3/2}}{2 a b x^3 \left (a+b x^2\right )}+\frac {5 c (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (15 b^2 c^2 x^4+10 a b c x^2 \left (c-2 d x^2\right )+a^2 \left (-2 c^2-14 c d x^2+3 d^2 x^4\right )\right )}{6 a^3 x^3 \left (a+b x^2\right )}-\frac {5 c (b c-a d)^{3/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2}} \]
[In]
[Out]
Time = 3.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {2 \sqrt {\left (a d -b c \right ) a}\, \left (\left (-\frac {3}{2} d^{2} x^{4}+7 c d \,x^{2}+c^{2}\right ) a^{2}-5 b c \,x^{2} \left (-2 d \,x^{2}+c \right ) a -\frac {15 b^{2} c^{2} x^{4}}{2}\right ) \sqrt {d \,x^{2}+c}-15 c \,x^{3} \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{6 \sqrt {\left (a d -b c \right ) a}\, a^{3} x^{3} \left (b \,x^{2}+a \right )}\) | \(155\) |
risch | \(\text {Expression too large to display}\) | \(1002\) |
default | \(\text {Expression too large to display}\) | \(5508\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.74 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\left [-\frac {15 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{5} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, b^{2} c^{2} - 20 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \, {\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{5} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (15 \, b^{2} c^{2} - 20 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} + 2 \, {\left (5 \, a b c^{2} - 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{x^{4} \left (a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{2} x^{4}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (152) = 304\).
Time = 1.10 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.82 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {5 \, {\left (b^{2} c^{3} \sqrt {d} - 2 \, a b c^{2} d^{\frac {3}{2}} + a^{2} c d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a^{3}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} - b^{3} c^{4} \sqrt {d} + 2 \, a b^{2} c^{3} d^{\frac {3}{2}} - a^{2} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a^{3} b} - \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c^{3} \sqrt {d} - 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a c^{2} d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{4} \sqrt {d} + 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c^{3} d^{\frac {3}{2}} + 6 \, b c^{5} \sqrt {d} - 7 \, a c^{4} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{x^4 \left (a+b x^2\right )^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{x^4\,{\left (b\,x^2+a\right )}^2} \,d x \]
[In]
[Out]