Integrand size = 30, antiderivative size = 435 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 b^{5/2} e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {3003 b^{5/2} e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}+\frac {3003 b^2 e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}+\frac {1001 b e^4 (a+b x)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}+\frac {3003 e^4 (a+b x)}{320 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}+\frac {429 e^3}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}-\frac {143 e^2}{96 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}+\frac {13 e}{24 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {1}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 214
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)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (13 b^3 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 (d+e x)^{7/2}} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (143 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{48 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (429 b e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 (d+e x)^{7/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^2 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 (b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3003 b^3 e^3 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 (b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {429 e^3}{64 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{4 (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 e}{24 (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {143 e^2}{96 (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 e^4 (a+b x)}{320 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1001 b e^4 (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3003 b^2 e^4 (a+b x)}{64 (b d-a e)^7 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3003 b^{5/2} e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 1.49 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {-384 a^6 e^6+128 a^5 b e^5 (31 d+13 e x)-128 a^4 b^2 e^4 \left (253 d^2+351 d e x+143 e^2 x^2\right )-a^3 b^3 e^3 \left (22155 d^3+196001 d^2 e x+285857 d e^2 x^2+119691 e^3 x^3\right )-a^2 b^4 e^2 \left (-7630 d^4+35945 d^3 e x+347919 d^2 e^2 x^2+517803 d e^3 x^3+219219 e^4 x^4\right )-a b^5 e \left (1960 d^5-5460 d^4 e x+25025 d^3 e^2 x^2+256971 d^2 e^3 x^3+387387 d e^4 x^4+165165 e^5 x^5\right )+b^6 \left (240 d^6-520 d^5 e x+1430 d^4 e^2 x^2-6435 d^3 e^3 x^3-69069 d^2 e^4 x^4-105105 d e^5 x^5-45045 e^6 x^6\right )}{e^4 (-b d+a e)^7 (a+b x)^4 (d+e x)^{5/2}}-\frac {45045 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{15/2}}\right )}{960 \left ((a+b x)^2\right )^{5/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(950\) vs. \(2(307)=614\).
Time = 2.57 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.19
method | result | size |
default | \(-\frac {\left (256971 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{2} e^{4} x^{3}+347919 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{2} e^{4} x^{2}+180180 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{6} e^{4} x^{3}+270270 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{5} e^{4} x^{2}+25025 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{3} e^{3} x^{2}+196001 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{2} e^{4} x +35945 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{3} e^{3} x -5460 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{4} e^{2} x +180180 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{4} e^{4} x +165165 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} e^{6} x^{5}+105105 \sqrt {\left (a e -b d \right ) b}\, b^{6} d \,e^{5} x^{5}+219219 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} e^{6} x^{4}+119691 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} e^{6} x^{3}+18304 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} e^{6} x^{2}-1664 \sqrt {\left (a e -b d \right ) b}\, a^{5} b \,e^{6} x -3968 \sqrt {\left (a e -b d \right ) b}\, a^{5} b d \,e^{5}+384 \sqrt {\left (a e -b d \right ) b}\, a^{6} e^{6}-240 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{6}+45045 \sqrt {\left (a e -b d \right ) b}\, b^{6} e^{6} x^{6}+45045 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{7} e^{4} x^{4}+45045 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b^{3} e^{4}+387387 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d \,e^{5} x^{4}+517803 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d \,e^{5} x^{3}+285857 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d \,e^{5} x^{2}+44928 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d \,e^{5} x +69069 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{2} e^{4} x^{4}+6435 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{3} e^{3} x^{3}-1430 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{4} e^{2} x^{2}+520 \sqrt {\left (a e -b d \right ) b}\, b^{6} d^{5} e x +32384 \sqrt {\left (a e -b d \right ) b}\, a^{4} b^{2} d^{2} e^{4}+22155 \sqrt {\left (a e -b d \right ) b}\, a^{3} b^{3} d^{3} e^{3}-7630 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{4} d^{4} e^{2}+1960 \sqrt {\left (a e -b d \right ) b}\, a \,b^{5} d^{5} e \right ) \left (b x +a \right )}{960 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(951\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (307) = 614\).
Time = 1.43 (sec) , antiderivative size = 3368, normalized size of antiderivative = 7.74 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (307) = 614\).
Time = 0.32 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {3003 \, b^{3} e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (225 \, {\left (e x + d\right )}^{2} b^{2} e^{4} + 25 \, {\left (e x + d\right )} b^{2} d e^{4} + 3 \, b^{2} d^{2} e^{4} - 25 \, {\left (e x + d\right )} a b e^{5} - 6 \, a b d e^{5} + 3 \, a^{2} e^{6}\right )}}{15 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} {\left (e x + d\right )}^{\frac {5}{2}}} + \frac {3249 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} e^{4} - 10633 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d e^{4} + 11767 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{2} e^{4} - 4431 \, \sqrt {e x + d} b^{6} d^{3} e^{4} + 10633 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} e^{5} - 23534 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d e^{5} + 13293 \, \sqrt {e x + d} a b^{5} d^{2} e^{5} + 11767 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{6} - 13293 \, \sqrt {e x + d} a^{2} b^{4} d e^{6} + 4431 \, \sqrt {e x + d} a^{3} b^{3} e^{7}}{192 \, {\left (b^{7} d^{7} \mathrm {sgn}\left (b x + a\right ) - 7 \, a b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, a^{3} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) - 21 \, a^{5} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{6} b d e^{6} \mathrm {sgn}\left (b x + a\right ) - a^{7} e^{7} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]