Integrand size = 24, antiderivative size = 258 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-8 a d) (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}} \]
[Out]
Time = 0.34 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {478, 595, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac {\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}}-\frac {\sqrt {a} (3 b c-8 a d) (b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^5}+\frac {d x^3 \sqrt {c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]
[In]
[Out]
Rule 211
Rule 212
Rule 223
Rule 385
Rule 478
Rule 537
Rule 595
Rule 596
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (c+d x^2\right )^{3/2} \left (3 c+8 d x^2\right )}{a+b x^2} \, dx}{2 b} \\ & = \frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \sqrt {c+d x^2} \left (6 c (3 b c-4 a d)+6 d (7 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{12 b^2} \\ & = \frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (6 c \left (12 b^2 c^2-37 a b c d+24 a^2 d^2\right )+6 d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^3} \\ & = \frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\int \frac {6 a c d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right )-6 d \left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{96 b^4 d} \\ & = \frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (a (3 b c-8 a d) (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 b^5} \\ & = \frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (a (3 b c-8 a d) (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 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 b^5} \\ & = \frac {\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^4}+\frac {d (7 b c-8 a d) x^3 \sqrt {c+d x^2}}{8 b^3}+\frac {2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac {x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-8 a d) (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 b^5}+\frac {\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^5 \sqrt {d}} \\ \end{align*}
Time = 10.20 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {b x \sqrt {c+d x^2} \left (33 b^2 c^2-108 a b c d+72 a^2 d^2+2 b d (13 b c-12 a d) x^2+8 b^2 d^2 x^4+\frac {24 a (b c-a d)^2}{a+b x^2}\right )+24 \sqrt {a} (b c-a d)^{3/2} (-3 b c+8 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )+\frac {3 \left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \log \left (d x+\sqrt {d} \sqrt {c+d x^2}\right )}{\sqrt {d}}}{48 b^5} \]
[In]
[Out]
Time = 3.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(-\frac {-\frac {b \sqrt {d \,x^{2}+c}\, \left (8 b^{2} d^{2} x^{4}-24 x^{2} a b \,d^{2}+26 x^{2} b^{2} c d +72 a^{2} d^{2}-108 a b c d +33 b^{2} c^{2}\right ) x}{12}+\frac {\left (64 a^{3} d^{3}-120 a^{2} b c \,d^{2}+60 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{4 \sqrt {d}}+2 \left (a d -b c \right )^{2} a \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (8 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{4 b^{5}}\) | \(219\) |
risch | \(\text {Expression too large to display}\) | \(1101\) |
default | \(\text {Expression too large to display}\) | \(5359\) |
[In]
[Out]
none
Time = 1.69 (sec) , antiderivative size = 1697, normalized size of antiderivative = 6.58 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (222) = 444\).
Time = 0.35 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.00 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, d^{2} x^{2}}{b^{2}} + \frac {13 \, b^{12} c d^{5} - 12 \, a b^{11} d^{6}}{b^{14} d^{4}}\right )} x^{2} + \frac {3 \, {\left (11 \, b^{12} c^{2} d^{4} - 36 \, a b^{11} c d^{5} + 24 \, a^{2} b^{10} d^{6}\right )}}{b^{14} d^{4}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (3 \, a b^{3} c^{3} \sqrt {d} - 14 \, a^{2} b^{2} c^{2} d^{\frac {3}{2}} + 19 \, a^{3} b c d^{\frac {5}{2}} - 8 \, a^{4} d^{\frac {7}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{5}} - \frac {{\left (5 \, b^{3} c^{3} - 60 \, a b^{2} c^{2} d + 120 \, a^{2} b c d^{2} - 64 \, a^{3} d^{3}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{32 \, b^{5} \sqrt {d}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{4} d^{\frac {7}{2}} - a b^{3} c^{4} \sqrt {d} + 2 \, a^{2} b^{2} c^{3} d^{\frac {3}{2}} - a^{3} b c^{2} d^{\frac {5}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{5}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,{\left (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^2} \,d x \]
[In]
[Out]