Integrand size = 24, antiderivative size = 291 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}} \]
[Out]
Time = 0.41 (sec) , antiderivative size = 291, 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 = {488, 595, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {a^{3/2} (b c-a d)^{5/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}+\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac {x \sqrt {c+d x^2} \left (-64 a^3 d^3+144 a^2 b c d^2-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac {\left (-128 a^4 d^4+320 a^3 b c d^3-240 a^2 b^2 c^2 d^2+40 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac {d x^5 \sqrt {c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \]
[In]
[Out]
Rule 211
Rule 212
Rule 223
Rule 385
Rule 488
Rule 537
Rule 595
Rule 596
Rubi steps \begin{align*} \text {integral}& = \frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \sqrt {c+d x^2} \left (c (8 b c-5 a d)+d (11 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{8 b} \\ & = \frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {x^4 \left (c \left (48 b^2 c^2-85 a b c d+40 a^2 d^2\right )+d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{48 b^2} \\ & = \frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}-\frac {\int \frac {x^2 \left (3 a c d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right )-3 d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{192 b^3 d} \\ & = \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\int \frac {-3 a c d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right )-3 d \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{384 b^4 d^2} \\ & = \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 b^5 d} \\ & = \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {\left (a^2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 b^5 d} \\ & = \frac {\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt {c+d x^2}}{128 b^4 d}+\frac {\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt {c+d x^2}}{192 b^3}+\frac {d (11 b c-8 a d) x^5 \sqrt {c+d x^2}}{48 b^2}+\frac {d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac {a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^5}-\frac {\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 b^5 d^{3/2}} \\ \end{align*}
Time = 2.61 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.78 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {b \sqrt {d} x \sqrt {c+d x^2} \left (-192 a^3 d^3+48 a^2 b d^2 \left (9 c+2 d x^2\right )-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )\right )+384 \sqrt {a} \sqrt {d} (b c-a d)^2 \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )+384 \sqrt {a} \sqrt {d} (b c-a d)^2 \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )+6 \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{384 b^5 d^{3/2}} \]
[In]
[Out]
Time = 3.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {\frac {b \sqrt {d \,x^{2}+c}\, \left (-48 b^{3} d^{3} x^{6}+64 a \,b^{2} d^{3} x^{4}-136 b^{3} c \,d^{2} x^{4}-96 x^{2} a^{2} b \,d^{3}+208 x^{2} a \,b^{2} c \,d^{2}-118 x^{2} b^{3} c^{2} d +192 a^{3} d^{3}-432 a^{2} b c \,d^{2}+264 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) x}{192 d}-\frac {\left (128 a^{4} d^{4}-320 a^{3} b c \,d^{3}+240 a^{2} b^{2} c^{2} d^{2}-40 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{64 d^{\frac {3}{2}}}+\frac {2 \left (a d -b c \right )^{3} a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{5}}\) | \(259\) |
risch | \(-\frac {x \left (-48 b^{3} d^{3} x^{6}+64 a \,b^{2} d^{3} x^{4}-136 b^{3} c \,d^{2} x^{4}-96 x^{2} a^{2} b \,d^{3}+208 x^{2} a \,b^{2} c \,d^{2}-118 x^{2} b^{3} c^{2} d +192 a^{3} d^{3}-432 a^{2} b c \,d^{2}+264 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {d \,x^{2}+c}}{384 d \,b^{4}}+\frac {\frac {\left (128 a^{4} d^{4}-320 a^{3} b c \,d^{3}+240 a^{2} b^{2} c^{2} d^{2}-40 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {64 a^{2} d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}+\frac {64 a^{2} d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{128 d \,b^{4}}\) | \(602\) |
default | \(\text {Expression too large to display}\) | \(2252\) |
[In]
[Out]
none
Time = 4.53 (sec) , antiderivative size = 1443, normalized size of antiderivative = 4.96 \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {x^{4} \left (c + d x^{2}\right )^{\frac {5}{2}}}{a + b x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{4}}{b x^{2} + a} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {x^4\,{\left (d\,x^2+c\right )}^{5/2}}{b\,x^2+a} \,d x \]
[In]
[Out]