Optimal. Leaf size=307 \[ \frac {4 b e^3}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 (b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b e^4 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 46}
\begin {gather*} \frac {e^4 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^5}+\frac {5 b e^4 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {5 b e^4 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {4 b e^3}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {2 b e}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\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)^2} \, 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^3 (b d-a e)^2 (a+b x)^5}-\frac {2 e}{b^3 (b d-a e)^3 (a+b x)^4}+\frac {3 e^2}{b^3 (b d-a e)^4 (a+b x)^3}-\frac {4 e^3}{b^3 (b d-a e)^5 (a+b x)^2}+\frac {5 e^4}{b^3 (b d-a e)^6 (a+b x)}-\frac {e^5}{b^5 (b d-a e)^5 (d+e x)^2}-\frac {5 e^5}{b^4 (b d-a e)^6 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {4 b e^3}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 (b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x)}{(b d-a e)^5 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^4 (a+b x) \log (a+b x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b e^4 (a+b x) \log (d+e x)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 167, normalized size = 0.54 \begin {gather*} \frac {8 b e (b d-a e)^3-\frac {3 b (b d-a e)^4}{a+b x}-18 b e^2 (b d-a e)^2 (a+b x)+48 b e^3 (b d-a e) (a+b x)^2+\frac {12 e^4 (b d-a e) (a+b x)^3}{d+e x}+60 b e^4 (a+b x)^3 \log (a+b x)-60 b e^4 (a+b x)^3 \log (d+e x)}{12 (b d-a e)^6 \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. \(650\) vs.
\(2(224)=448\).
time = 0.71, size = 651, normalized size = 2.12
method | result | size |
default | \(\frac {\left (180 a \,b^{4} d \,e^{4} x^{3}-40 a \,b^{4} d^{3} e^{2} x -240 \ln \left (e x +d \right ) a \,b^{4} d \,e^{4} x^{3}-60 \ln \left (e x +d \right ) a^{4} b \,e^{5} x -240 \ln \left (e x +d \right ) a^{3} b^{2} e^{5} x^{2}+60 \ln \left (b x +a \right ) a^{4} b d \,e^{4}+240 \ln \left (b x +a \right ) a^{3} b^{2} e^{5} x^{2}+60 \ln \left (b x +a \right ) a^{4} b \,e^{5} x -3 b^{5} d^{5}-60 \ln \left (e x +d \right ) a^{4} b d \,e^{4}+150 a^{2} b^{3} d \,e^{4} x^{2}+120 a \,b^{4} d^{2} e^{3} x^{2}-20 a^{3} b^{2} d \,e^{4} x +180 a^{2} b^{3} d^{2} e^{3} x -12 a^{5} e^{5}-360 \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{4} x^{2}-240 \ln \left (e x +d \right ) a^{3} b^{2} d \,e^{4} x +240 \ln \left (b x +a \right ) a \,b^{4} d \,e^{4} x^{3}+360 \ln \left (b x +a \right ) a^{2} b^{3} d \,e^{4} x^{2}+240 \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{4} x -240 \ln \left (e x +d \right ) a \,b^{4} e^{5} x^{4}-60 \ln \left (e x +d \right ) b^{5} d \,e^{4} x^{4}+360 \ln \left (b x +a \right ) a^{2} b^{3} e^{5} x^{3}+240 \ln \left (b x +a \right ) a \,b^{4} e^{5} x^{4}+60 \ln \left (b x +a \right ) b^{5} d \,e^{4} x^{4}-60 a \,b^{4} e^{5} x^{4}+60 b^{5} d \,e^{4} x^{4}-210 a^{2} b^{3} e^{5} x^{3}+30 b^{5} d^{2} e^{3} x^{3}-260 a^{3} b^{2} e^{5} x^{2}-10 b^{5} d^{3} e^{2} x^{2}-125 a^{4} b \,e^{5} x +5 b^{5} d^{4} e x -360 \ln \left (e x +d \right ) a^{2} b^{3} e^{5} x^{3}+60 \ln \left (b x +a \right ) b^{5} e^{5} x^{5}-60 \ln \left (e x +d \right ) b^{5} e^{5} x^{5}-65 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}-60 a^{2} b^{3} d^{3} e^{2}+20 a \,b^{4} d^{4} e \right ) \left (b x +a \right )}{12 \left (e x +d \right ) \left (a e -b d \right )^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(651\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {5 b^{4} e^{4} x^{4}}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {5 b^{3} \left (7 a e +b d \right ) e^{3} x^{3}}{2 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 b^{2} e^{2} \left (26 a^{2} e^{2}+11 a b d e -b^{2} d^{2}\right ) x^{2}}{6 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {5 \left (25 e^{3} a^{3}+29 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) b e x}{12 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {12 e^{4} a^{4}+77 a^{3} b d \,e^{3}-43 a^{2} b^{2} d^{2} e^{2}+17 a \,b^{3} d^{3} e -3 b^{4} d^{4}}{12 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\right )}{\left (b x +a \right )^{5} \left (e x +d \right )}+\frac {5 \sqrt {\left (b x +a \right )^{2}}\, e^{4} b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}-\frac {5 \sqrt {\left (b x +a \right )^{2}}\, e^{4} b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 d^{4} b^{4} e^{2} a^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\) | \(732\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1021 vs.
\(2 (229) = 458\).
time = 2.36, size = 1021, normalized size = 3.33 \begin {gather*} -\frac {3 \, b^{5} d^{5} + {\left (60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5}\right )} e^{5} - 5 \, {\left (12 \, b^{5} d x^{4} + 36 \, a b^{4} d x^{3} + 30 \, a^{2} b^{3} d x^{2} - 4 \, a^{3} b^{2} d x - 13 \, a^{4} b d\right )} e^{4} - 30 \, {\left (b^{5} d^{2} x^{3} + 4 \, a b^{4} d^{2} x^{2} + 6 \, a^{2} b^{3} d^{2} x + 4 \, a^{3} b^{2} d^{2}\right )} e^{3} + 10 \, {\left (b^{5} d^{3} x^{2} + 4 \, a b^{4} d^{3} x + 6 \, a^{2} b^{3} d^{3}\right )} e^{2} - 5 \, {\left (b^{5} d^{4} x + 4 \, a b^{4} d^{4}\right )} e - 60 \, {\left ({\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} e^{5} + {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} e^{5} + {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \log \left (x e + d\right )}{12 \, {\left (b^{10} d^{7} x^{4} + 4 \, a b^{9} d^{7} x^{3} + 6 \, a^{2} b^{8} d^{7} x^{2} + 4 \, a^{3} b^{7} d^{7} x + a^{4} b^{6} d^{7} + {\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )} e^{7} - {\left (6 \, a^{5} b^{5} d x^{5} + 23 \, a^{6} b^{4} d x^{4} + 32 \, a^{7} b^{3} d x^{3} + 18 \, a^{8} b^{2} d x^{2} + 2 \, a^{9} b d x - a^{10} d\right )} e^{6} + 3 \, {\left (5 \, a^{4} b^{6} d^{2} x^{5} + 18 \, a^{5} b^{5} d^{2} x^{4} + 22 \, a^{6} b^{4} d^{2} x^{3} + 8 \, a^{7} b^{3} d^{2} x^{2} - 3 \, a^{8} b^{2} d^{2} x - 2 \, a^{9} b d^{2}\right )} e^{5} - 5 \, {\left (4 \, a^{3} b^{7} d^{3} x^{5} + 13 \, a^{4} b^{6} d^{3} x^{4} + 12 \, a^{5} b^{5} d^{3} x^{3} - 2 \, a^{6} b^{4} d^{3} x^{2} - 8 \, a^{7} b^{3} d^{3} x - 3 \, a^{8} b^{2} d^{3}\right )} e^{4} + 5 \, {\left (3 \, a^{2} b^{8} d^{4} x^{5} + 8 \, a^{3} b^{7} d^{4} x^{4} + 2 \, a^{4} b^{6} d^{4} x^{3} - 12 \, a^{5} b^{5} d^{4} x^{2} - 13 \, a^{6} b^{4} d^{4} x - 4 \, a^{7} b^{3} d^{4}\right )} e^{3} - 3 \, {\left (2 \, a b^{9} d^{5} x^{5} + 3 \, a^{2} b^{8} d^{5} x^{4} - 8 \, a^{3} b^{7} d^{5} x^{3} - 22 \, a^{4} b^{6} d^{5} x^{2} - 18 \, a^{5} b^{5} d^{5} x - 5 \, a^{6} b^{4} d^{5}\right )} e^{2} + {\left (b^{10} d^{6} x^{5} - 2 \, a b^{9} d^{6} x^{4} - 18 \, a^{2} b^{8} d^{6} x^{3} - 32 \, a^{3} b^{7} d^{6} x^{2} - 23 \, a^{4} b^{6} d^{6} x - 6 \, a^{5} b^{5} d^{6}\right )} e\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 )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{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. 523 vs.
\(2 (229) = 458\).
time = 0.71, size = 523, normalized size = 1.70 \begin {gather*} \frac {5 \, b^{2} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} b e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {5 \, b e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{7} \mathrm {sgn}\left (b x + a\right )} - \frac {3 \, b^{5} d^{5} - 20 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 65 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} - 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (b^{5} d^{3} e^{2} - 12 \, a b^{4} d^{2} e^{3} - 15 \, a^{2} b^{3} d e^{4} + 26 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 8 \, a b^{4} d^{3} e^{2} + 36 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{12 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{4} {\left (x e + d\right )} \mathrm {sgn}\left (b x + a\right )} \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 )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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