Integrand size = 24, antiderivative size = 150 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {d}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {478, 596, 537, 223, 212, 385, 211} \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {a} (3 b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {d}}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {x \sqrt {c+d x^2}}{b^2} \]
[In]
[Out]
Rule 211
Rule 212
Rule 223
Rule 385
Rule 478
Rule 537
Rule 596
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (3 c+4 d x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b} \\ & = \frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {\int \frac {4 a c d-2 d (b c-4 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 b^2 d} \\ & = \frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {(b c-4 a d) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 b^3}-\frac {(a (3 b c-4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b^3} \\ & = \frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {(b c-4 a d) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 b^3}-\frac {(a (3 b c-4 a d)) \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^3} \\ & = \frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {d}} \\ \end{align*}
Time = 10.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {b x \left (2 a+b x^2\right ) \sqrt {c+d x^2}}{a+b x^2}+\frac {\sqrt {a} (-3 b c+4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {b c-a d}}+\frac {(b c-4 a d) \log \left (d x+\sqrt {d} \sqrt {c+d x^2}\right )}{\sqrt {d}}}{2 b^3} \]
[In]
[Out]
Time = 3.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(-\frac {\frac {-\sqrt {d \,x^{2}+c}\, b x \sqrt {d}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) a d -\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) b c}{\sqrt {d}}+a \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 b^{3}}\) | \(145\) |
risch | \(\frac {x \sqrt {d \,x^{2}+c}}{2 b^{2}}-\frac {\frac {\left (4 a d -b c \right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {\left (a d -b c \right ) a \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}-\frac {\left (a d -b c \right ) a \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}+\frac {a \left (5 a d -3 b c \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 )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {a \left (5 a d -3 b c \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 )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{2 b^{2}}\) | \(907\) |
default | \(\text {Expression too large to display}\) | \(2002\) |
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.68 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {2 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \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 ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, -\frac {4 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \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 ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, \frac {{\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, -\frac {2 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - 2 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} x^{4}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (124) = 248\).
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.89 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} x}{2 \, b^{2}} + \frac {{\left (3 \, a b c \sqrt {d} - 4 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{3}} - \frac {{\left (b c - 4 \, a d\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3} \sqrt {d}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^2} \,d x \]
[In]
[Out]