Integrand size = 19, antiderivative size = 144 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=-\frac {c d (a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {a c^2 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{5/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {745, 735, 739, 212} \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=-\frac {a c^2 d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}}-\frac {c d \sqrt {a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]
[In]
[Out]
Rule 212
Rule 735
Rule 739
Rule 745
Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}+\frac {(c d) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{c d^2+a e^2} \\ & = -\frac {c d (a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}+\frac {\left (a c^2 d\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^2} \\ & = -\frac {c d (a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {\left (a c^2 d\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^2} \\ & = -\frac {c d (a e-c d x) \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {e \left (a+c x^2\right )^{3/2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {a c^2 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{5/2}} \\ \end{align*}
Time = 10.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {c d^2+a e^2} \sqrt {a+c x^2} \left (-2 a^2 e^3+c^2 d^2 x (3 d+e x)-a c e \left (5 d^2+3 d e x+2 e^2 x^2\right )\right )+3 a c^2 d (d+e x)^3 \log (d+e x)-3 a c^2 d (d+e x)^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{6 \left (c d^2+a e^2\right )^{5/2} (d+e x)^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(987\) vs. \(2(128)=256\).
Time = 2.18 (sec) , antiderivative size = 988, normalized size of antiderivative = 6.86
method | result | size |
default | \(\frac {-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {c d e \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {c d e \left (-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{2} a +c \,d^{2}}\right )}{2 e^{2} a +2 c \,d^{2}}+\frac {c \,e^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 e^{2} a +2 c \,d^{2}}\right )}{e^{2} a +c \,d^{2}}}{e^{4}}\) | \(988\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (129) = 258\).
Time = 0.51 (sec) , antiderivative size = 861, normalized size of antiderivative = 5.98 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=\left [\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + a c^{2} d^{4}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (5 \, a c^{2} d^{4} e + 7 \, a^{2} c d^{2} e^{3} + 2 \, a^{3} e^{5} - {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} - a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{3} d^{9} + 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} + 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e + 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\right )}}, -\frac {3 \, {\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + a c^{2} d^{4}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (5 \, a c^{2} d^{4} e + 7 \, a^{2} c d^{2} e^{3} + 2 \, a^{3} e^{5} - {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3} - 2 \, a^{2} c e^{5}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} - a^{2} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{3} d^{9} + 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} + a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} + 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} + a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e + 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} + a^{3} d^{2} e^{7}\right )} x\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.67 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=-\frac {a c^{2} d \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{4} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{5} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{2} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{4} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{2} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{4} + a^{3} c^{\frac {5}{2}} d^{2} e^{3} - 2 \, a^{4} c^{\frac {3}{2}} e^{5}}{3 \, {\left (c^{2} d^{4} e^{2} + 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^4} \,d x \]
[In]
[Out]