Integrand size = 28, antiderivative size = 253 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^4 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 46} \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {e^2}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
[In]
[Out]
Rule 46
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^5 (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^4 (b d-a e) (a+b x)^5}-\frac {e}{b^4 (b d-a e)^2 (a+b x)^4}+\frac {e^2}{b^4 (b d-a e)^3 (a+b x)^3}-\frac {e^3}{b^4 (b d-a e)^4 (a+b x)^2}+\frac {e^4}{b^4 (b d-a e)^5 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {e^3}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2}{2 (b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^4 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-\left ((b d-a e) \left (-25 a^3 e^3+a^2 b e^2 (23 d-52 e x)+a b^2 e \left (-13 d^2+20 d e x-42 e^2 x^2\right )+b^3 \left (3 d^3-4 d^2 e x+6 d e^2 x^2-12 e^3 x^3\right )\right )\right )+12 e^4 (a+b x)^4 \log (a+b x)-12 e^4 (a+b x)^4 \log (d+e x)}{12 (b d-a e)^5 (a+b x)^3 \sqrt {(a+b x)^2}} \]
[In]
[Out]
Time = 2.64 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.42
method | result | size |
default | \(-\frac {\left (12 \ln \left (b x +a \right ) b^{4} e^{4} x^{4}-12 \ln \left (e x +d \right ) x^{4} b^{4} e^{4}+48 \ln \left (b x +a \right ) x^{3} a \,b^{3} e^{4}-48 \ln \left (e x +d \right ) a \,b^{3} e^{4} x^{3}+72 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{4}-72 \ln \left (e x +d \right ) x^{2} a^{2} b^{2} e^{4}-12 x^{3} a \,b^{3} e^{4}+12 x^{3} b^{4} d \,e^{3}+48 \ln \left (b x +a \right ) x \,a^{3} b \,e^{4}-48 \ln \left (e x +d \right ) x \,a^{3} b \,e^{4}-42 x^{2} a^{2} b^{2} e^{4}+48 x^{2} a \,b^{3} d \,e^{3}-6 x^{2} b^{4} d^{2} e^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}-12 \ln \left (e x +d \right ) a^{4} e^{4}-52 x \,a^{3} b \,e^{4}+72 x \,a^{2} b^{2} d \,e^{3}-24 x a \,b^{3} d^{2} e^{2}+4 x \,b^{4} d^{3} e -25 e^{4} a^{4}+48 b \,e^{3} d \,a^{3}-36 b^{2} e^{2} d^{2} a^{2}+16 a \,b^{3} d^{3} e -3 b^{4} d^{4}\right ) \left (b x +a \right )}{12 \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(359\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b^{3} e^{3} x^{3}}{e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {\left (7 a e -b d \right ) b^{2} e^{2} x^{2}}{2 e^{4} a^{4}-8 b \,e^{3} d \,a^{3}+12 b^{2} e^{2} d^{2} a^{2}-8 a \,b^{3} d^{3} e +2 b^{4} d^{4}}+\frac {e b \left (13 a^{2} e^{2}-5 a b d e +b^{2} d^{2}\right ) x}{3 e^{4} a^{4}-12 b \,e^{3} d \,a^{3}+18 b^{2} e^{2} d^{2} a^{2}-12 a \,b^{3} d^{3} e +3 b^{4} d^{4}}+\frac {25 a^{3} e^{3}-23 a^{2} b d \,e^{2}+13 a \,b^{2} d^{2} e -3 b^{3} d^{3}}{12 e^{4} a^{4}-48 b \,e^{3} d \,a^{3}+72 b^{2} e^{2} d^{2} a^{2}-48 a \,b^{3} d^{3} e +12 b^{4} d^{4}}\right )}{\left (b x +a \right )^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{4} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(508\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (181) = 362\).
Time = 0.29 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.60 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{5} d^{5} - 5 \, a^{5} b^{4} d^{4} e + 10 \, a^{6} b^{3} d^{3} e^{2} - 10 \, a^{7} b^{2} d^{2} e^{3} + 5 \, a^{8} b d e^{4} - a^{9} e^{5} + {\left (b^{9} d^{5} - 5 \, a b^{8} d^{4} e + 10 \, a^{2} b^{7} d^{3} e^{2} - 10 \, a^{3} b^{6} d^{2} e^{3} + 5 \, a^{4} b^{5} d e^{4} - a^{5} b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{8} d^{5} - 5 \, a^{2} b^{7} d^{4} e + 10 \, a^{3} b^{6} d^{3} e^{2} - 10 \, a^{4} b^{5} d^{2} e^{3} + 5 \, a^{5} b^{4} d e^{4} - a^{6} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} d^{5} - 5 \, a^{3} b^{6} d^{4} e + 10 \, a^{4} b^{5} d^{3} e^{2} - 10 \, a^{5} b^{4} d^{2} e^{3} + 5 \, a^{6} b^{3} d e^{4} - a^{7} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} d^{5} - 5 \, a^{4} b^{5} d^{4} e + 10 \, a^{5} b^{4} d^{3} e^{2} - 10 \, a^{6} b^{3} d^{2} e^{3} + 5 \, a^{7} b^{2} d e^{4} - a^{8} b e^{5}\right )} x\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (181) = 362\).
Time = 0.28 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {b e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {e^{5} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b^{4} d^{4} - 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} - 48 \, a^{3} b d e^{3} + 25 \, a^{4} e^{4} - 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 18 \, a^{2} b^{2} d e^{3} - 13 \, a^{3} b e^{4}\right )} x}{12 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{4} \mathrm {sgn}\left (b x + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]