Integrand size = 22, antiderivative size = 244 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}} \]
[Out]
Time = 0.58 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1665, 1668, 858, 223, 212, 739} \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (4 c d^2-a e^2\right )}{c^{3/2} e^5}-\frac {d^4 \left (5 a e^2+4 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^5 \left (a e^2+c d^2\right )^{3/2}}+\frac {\sqrt {a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac {5 d \sqrt {a+c x^2} (d+e x)}{3 c e^4}+\frac {\sqrt {a+c x^2} (d+e x)^2}{3 c e^4} \]
[In]
[Out]
Rule 212
Rule 223
Rule 739
Rule 858
Rule 1665
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {-\frac {a d^4}{e^3}+\frac {d^3 \left (c d^2+a e^2\right ) x}{e^4}-\frac {d^2 \left (c d^2+a e^2\right ) x^2}{e^3}+d \left (a+\frac {c d^2}{e^2}\right ) x^3-\frac {\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = \frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-a d^2 e \left (c d^2-2 a e^2\right )+4 d \left (c d^2+a e^2\right )^2 x+2 e \left (c d^2+a e^2\right )^2 x^2+10 c d e^2 \left (c d^2+a e^2\right ) x^3}{(d+e x) \sqrt {a+c x^2}} \, dx}{3 c e^4 \left (c d^2+a e^2\right )} \\ & = \frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c d^2 e^4 \left (2 c d^2+a e^2\right )-2 c d e^3 \left (c d^2+a e^2\right )^2 x-2 c e^4 \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^2 e^7 \left (c d^2+a e^2\right )} \\ & = \frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\int \frac {-6 a c^2 d^2 e^6 \left (2 c d^2+a e^2\right )+6 c^2 d e^5 \left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{6 c^3 e^9 \left (c d^2+a e^2\right )} \\ & = \frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c e^5}+\frac {\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^5 \left (c d^2+a e^2\right )} \\ & = \frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {\left (d \left (4 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c e^5}-\frac {\left (d^4 \left (4 c d^2+5 a e^2\right )\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 )}{e^5 \left (c d^2+a e^2\right )} \\ & = \frac {\left (13 c d^2-2 a e^2\right ) \sqrt {a+c x^2}}{3 c^2 e^4}+\frac {d^5 \sqrt {a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac {5 d (d+e x) \sqrt {a+c x^2}}{3 c e^4}+\frac {(d+e x)^2 \sqrt {a+c x^2}}{3 c e^4}-\frac {d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^5}-\frac {d^4 \left (4 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (-2 a^2 e^4 (d+e x)+a c e^2 \left (7 d^3+4 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+c^2 d^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{c^2 \left (c d^2+a e^2\right ) (d+e x)}+\frac {6 d^4 \left (4 c d^2+5 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {3 \left (4 c d^3-a d e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{3 e^5} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(218)=436\).
Time = 0.48 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.80
method | result | size |
risch | \(-\frac {\left (-c \,e^{2} x^{2}+3 c d e x +2 e^{2} a -9 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{3 c^{2} e^{4}}+\frac {d \left (\frac {\left (e^{2} a -4 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {5 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {c \,d^{4} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\right )}{e^{4} c}\) | \(440\) |
default | \(\frac {\frac {x^{2} \sqrt {c \,x^{2}+a}}{3 c}-\frac {2 a \sqrt {c \,x^{2}+a}}{3 c^{2}}}{e^{2}}-\frac {4 d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{5} \sqrt {c}}-\frac {2 d \left (\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\right )}{e^{3}}+\frac {3 d^{2} \sqrt {c \,x^{2}+a}}{e^{4} c}-\frac {d^{5} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{7}}-\frac {5 d^{4} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{6} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(477\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (219) = 438\).
Time = 25.37 (sec) , antiderivative size = 2025, normalized size of antiderivative = 8.30 \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^{5}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {x^{5}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^5}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^5}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
[In]
[Out]