Optimal. Leaf size=329 \[ \frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 44, 53, 65,
214} \begin {gather*} \frac {63 b^2 e^2 (a+b x)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}+\frac {21 b e^2 (a+b x)}{4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac {63 e^2 (a+b x)}{20 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {1}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {9 e}{4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^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}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (9 b e \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}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{8 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 b e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 b^3 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 b^3 e \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 )}{4 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 241, normalized size = 0.73 \begin {gather*} \frac {e^2 (a+b x)^3 \left (\frac {-8 a^4 e^4+8 a^3 b e^3 (7 d+3 e x)-24 a^2 b^2 e^2 \left (12 d^2+17 d e x+7 e^2 x^2\right )-a b^3 e \left (85 d^3+831 d^2 e x+1239 d e^2 x^2+525 e^3 x^3\right )+b^4 \left (10 d^4-45 d^3 e x-483 d^2 e^2 x^2-735 d e^3 x^3-315 e^4 x^4\right )}{e^2 (-b d+a e)^5 (a+b x)^2 (d+e x)^{5/2}}-\frac {315 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right )}{20 \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(517\) vs.
\(2(231)=462\).
time = 0.68, size = 518, normalized size = 1.57
method | result | size |
default | \(-\frac {\left (315 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{5} e^{2} x^{2}+630 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{4} e^{2} x +315 \left (e x +d \right )^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{3} e^{2}+315 \sqrt {b \left (a e -b d \right )}\, b^{4} e^{4} x^{4}+525 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} e^{4} x^{3}+735 \sqrt {b \left (a e -b d \right )}\, b^{4} d \,e^{3} x^{3}+168 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} e^{4} x^{2}+1239 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d \,e^{3} x^{2}+483 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{2} e^{2} x^{2}-24 \sqrt {b \left (a e -b d \right )}\, a^{3} b \,e^{4} x +408 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d \,e^{3} x +831 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{2} e^{2} x +45 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{3} e x +8 \sqrt {b \left (a e -b d \right )}\, a^{4} e^{4}-56 \sqrt {b \left (a e -b d \right )}\, a^{3} b d \,e^{3}+288 \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{2} e^{2}+85 \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{3} e -10 \sqrt {b \left (a e -b d \right )}\, b^{4} d^{4}\right ) \left (b x +a \right )}{20 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(518\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 894 vs.
\(2 (241) = 482\).
time = 1.78, size = 1800, normalized size = 5.47 \begin {gather*} \left [-\frac {315 \, {\left ({\left (b^{4} x^{5} + 2 \, a b^{3} x^{4} + a^{2} b^{2} x^{3}\right )} e^{5} + 3 \, {\left (b^{4} d x^{4} + 2 \, a b^{3} d x^{3} + a^{2} b^{2} d x^{2}\right )} e^{4} + 3 \, {\left (b^{4} d^{2} x^{3} + 2 \, a b^{3} d^{2} x^{2} + a^{2} b^{2} d^{2} x\right )} e^{3} + {\left (b^{4} d^{3} x^{2} + 2 \, a b^{3} d^{3} x + a^{2} b^{2} d^{3}\right )} e^{2}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d + 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, {\left (10 \, b^{4} d^{4} - {\left (315 \, b^{4} x^{4} + 525 \, a b^{3} x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x + 8 \, a^{4}\right )} e^{4} - {\left (735 \, b^{4} d x^{3} + 1239 \, a b^{3} d x^{2} + 408 \, a^{2} b^{2} d x - 56 \, a^{3} b d\right )} e^{3} - 3 \, {\left (161 \, b^{4} d^{2} x^{2} + 277 \, a b^{3} d^{2} x + 96 \, a^{2} b^{2} d^{2}\right )} e^{2} - 5 \, {\left (9 \, b^{4} d^{3} x + 17 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{40 \, {\left (b^{7} d^{8} x^{2} + 2 \, a b^{6} d^{8} x + a^{2} b^{5} d^{8} - {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )} e^{8} + {\left (5 \, a^{4} b^{3} d x^{5} + 7 \, a^{5} b^{2} d x^{4} - a^{6} b d x^{3} - 3 \, a^{7} d x^{2}\right )} e^{7} - {\left (10 \, a^{3} b^{4} d^{2} x^{5} + 5 \, a^{4} b^{3} d^{2} x^{4} - 17 \, a^{5} b^{2} d^{2} x^{3} - 9 \, a^{6} b d^{2} x^{2} + 3 \, a^{7} d^{2} x\right )} e^{6} + {\left (10 \, a^{2} b^{5} d^{3} x^{5} - 10 \, a^{3} b^{4} d^{3} x^{4} - 35 \, a^{4} b^{3} d^{3} x^{3} - a^{5} b^{2} d^{3} x^{2} + 13 \, a^{6} b d^{3} x - a^{7} d^{3}\right )} e^{5} - 5 \, {\left (a b^{6} d^{4} x^{5} - 4 \, a^{2} b^{5} d^{4} x^{4} - 5 \, a^{3} b^{4} d^{4} x^{3} + 5 \, a^{4} b^{3} d^{4} x^{2} + 4 \, a^{5} b^{2} d^{4} x - a^{6} b d^{4}\right )} e^{4} + {\left (b^{7} d^{5} x^{5} - 13 \, a b^{6} d^{5} x^{4} + a^{2} b^{5} d^{5} x^{3} + 35 \, a^{3} b^{4} d^{5} x^{2} + 10 \, a^{4} b^{3} d^{5} x - 10 \, a^{5} b^{2} d^{5}\right )} e^{3} + {\left (3 \, b^{7} d^{6} x^{4} - 9 \, a b^{6} d^{6} x^{3} - 17 \, a^{2} b^{5} d^{6} x^{2} + 5 \, a^{3} b^{4} d^{6} x + 10 \, a^{4} b^{3} d^{6}\right )} e^{2} + {\left (3 \, b^{7} d^{7} x^{3} + a b^{6} d^{7} x^{2} - 7 \, a^{2} b^{5} d^{7} x - 5 \, a^{3} b^{4} d^{7}\right )} e\right )}}, -\frac {315 \, {\left ({\left (b^{4} x^{5} + 2 \, a b^{3} x^{4} + a^{2} b^{2} x^{3}\right )} e^{5} + 3 \, {\left (b^{4} d x^{4} + 2 \, a b^{3} d x^{3} + a^{2} b^{2} d x^{2}\right )} e^{4} + 3 \, {\left (b^{4} d^{2} x^{3} + 2 \, a b^{3} d^{2} x^{2} + a^{2} b^{2} d^{2} x\right )} e^{3} + {\left (b^{4} d^{3} x^{2} + 2 \, a b^{3} d^{3} x + a^{2} b^{2} d^{3}\right )} e^{2}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) + {\left (10 \, b^{4} d^{4} - {\left (315 \, b^{4} x^{4} + 525 \, a b^{3} x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x + 8 \, a^{4}\right )} e^{4} - {\left (735 \, b^{4} d x^{3} + 1239 \, a b^{3} d x^{2} + 408 \, a^{2} b^{2} d x - 56 \, a^{3} b d\right )} e^{3} - 3 \, {\left (161 \, b^{4} d^{2} x^{2} + 277 \, a b^{3} d^{2} x + 96 \, a^{2} b^{2} d^{2}\right )} e^{2} - 5 \, {\left (9 \, b^{4} d^{3} x + 17 \, a b^{3} d^{3}\right )} e\right )} \sqrt {x e + d}}{20 \, {\left (b^{7} d^{8} x^{2} + 2 \, a b^{6} d^{8} x + a^{2} b^{5} d^{8} - {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )} e^{8} + {\left (5 \, a^{4} b^{3} d x^{5} + 7 \, a^{5} b^{2} d x^{4} - a^{6} b d x^{3} - 3 \, a^{7} d x^{2}\right )} e^{7} - {\left (10 \, a^{3} b^{4} d^{2} x^{5} + 5 \, a^{4} b^{3} d^{2} x^{4} - 17 \, a^{5} b^{2} d^{2} x^{3} - 9 \, a^{6} b d^{2} x^{2} + 3 \, a^{7} d^{2} x\right )} e^{6} + {\left (10 \, a^{2} b^{5} d^{3} x^{5} - 10 \, a^{3} b^{4} d^{3} x^{4} - 35 \, a^{4} b^{3} d^{3} x^{3} - a^{5} b^{2} d^{3} x^{2} + 13 \, a^{6} b d^{3} x - a^{7} d^{3}\right )} e^{5} - 5 \, {\left (a b^{6} d^{4} x^{5} - 4 \, a^{2} b^{5} d^{4} x^{4} - 5 \, a^{3} b^{4} d^{4} x^{3} + 5 \, a^{4} b^{3} d^{4} x^{2} + 4 \, a^{5} b^{2} d^{4} x - a^{6} b d^{4}\right )} e^{4} + {\left (b^{7} d^{5} x^{5} - 13 \, a b^{6} d^{5} x^{4} + a^{2} b^{5} d^{5} x^{3} + 35 \, a^{3} b^{4} d^{5} x^{2} + 10 \, a^{4} b^{3} d^{5} x - 10 \, a^{5} b^{2} d^{5}\right )} e^{3} + {\left (3 \, b^{7} d^{6} x^{4} - 9 \, a b^{6} d^{6} x^{3} - 17 \, a^{2} b^{5} d^{6} x^{2} + 5 \, a^{3} b^{4} d^{6} x + 10 \, a^{4} b^{3} d^{6}\right )} e^{2} + {\left (3 \, b^{7} d^{7} x^{3} + a b^{6} d^{7} x^{2} - 7 \, a^{2} b^{5} d^{7} x - 5 \, a^{3} b^{4} d^{7}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs.
\(2 (241) = 482\).
time = 1.63, size = 487, normalized size = 1.48 \begin {gather*} \frac {63 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {15 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{2} - 17 \, \sqrt {x e + d} b^{4} d e^{2} + 17 \, \sqrt {x e + d} a b^{3} e^{3}}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} + \frac {2 \, {\left (30 \, {\left (x e + d\right )}^{2} b^{2} e^{2} + 5 \, {\left (x e + d\right )} b^{2} d e^{2} + b^{2} d^{2} e^{2} - 5 \, {\left (x e + d\right )} a b e^{3} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}}{5 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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