Optimal. Leaf size=58 \[ -\frac {(d+e x)^4}{5 (b d-a e) (a+b x)^5}+\frac {e (d+e x)^4}{20 (b d-a e)^2 (a+b x)^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 47, 37}
\begin {gather*} \frac {e (d+e x)^4}{20 (a+b x)^4 (b d-a e)^2}-\frac {(d+e x)^4}{5 (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^3}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^4}{5 (b d-a e) (a+b x)^5}-\frac {e \int \frac {(d+e x)^3}{(a+b x)^5} \, dx}{5 (b d-a e)}\\ &=-\frac {(d+e x)^4}{5 (b d-a e) (a+b x)^5}+\frac {e (d+e x)^4}{20 (b d-a e)^2 (a+b x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 97, normalized size = 1.67 \begin {gather*} -\frac {a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (3 d^2+10 d e x+10 e^2 x^2\right )+b^3 \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right )}{20 b^4 (a+b x)^5} \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. \(120\) vs.
\(2(54)=108\).
time = 0.63, size = 121, normalized size = 2.09
method | result | size |
default | \(-\frac {e^{3}}{2 b^{4} \left (b x +a \right )^{2}}+\frac {e^{2} \left (a e -b d \right )}{b^{4} \left (b x +a \right )^{3}}-\frac {-e^{3} a^{3}+3 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e +b^{3} d^{3}}{5 b^{4} \left (b x +a \right )^{5}}-\frac {3 e \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 b^{4} \left (b x +a \right )^{4}}\) | \(121\) |
norman | \(\frac {-\frac {e^{3} x^{3}}{2 b}+\frac {\left (-a b \,e^{3}-2 d \,e^{2} b^{2}\right ) x^{2}}{2 b^{3}}+\frac {\left (-a^{2} b \,e^{3}-2 a \,b^{2} d \,e^{2}-3 d^{2} e \,b^{3}\right ) x}{4 b^{4}}+\frac {-a^{3} b \,e^{3}-2 a^{2} b^{2} d \,e^{2}-3 d^{2} e a \,b^{3}-4 b^{4} d^{3}}{20 b^{5}}}{\left (b x +a \right )^{5}}\) | \(126\) |
risch | \(\frac {-\frac {e^{3} x^{3}}{2 b}-\frac {e^{2} \left (a e +2 b d \right ) x^{2}}{2 b^{2}}-\frac {e \left (a^{2} e^{2}+2 a b d e +3 b^{2} d^{2}\right ) x}{4 b^{3}}-\frac {e^{3} a^{3}+2 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +4 b^{3} d^{3}}{20 b^{4}}}{\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\) | \(128\) |
gosper | \(-\frac {10 e^{3} x^{3} b^{3}+10 a \,b^{2} e^{3} x^{2}+20 b^{3} d \,e^{2} x^{2}+5 a^{2} b \,e^{3} x +10 a \,b^{2} d \,e^{2} x +15 b^{3} d^{2} e x +e^{3} a^{3}+2 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +4 b^{3} d^{3}}{20 b^{4} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs.
\(2 (59) = 118\).
time = 0.28, size = 155, normalized size = 2.67 \begin {gather*} -\frac {10 \, b^{3} x^{3} e^{3} + 4 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 2 \, a^{2} b d e^{2} + a^{3} e^{3} + 10 \, {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \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. 149 vs.
\(2 (59) = 118\).
time = 2.61, size = 149, normalized size = 2.57 \begin {gather*} -\frac {4 \, b^{3} d^{3} + {\left (10 \, b^{3} x^{3} + 10 \, a b^{2} x^{2} + 5 \, a^{2} b x + a^{3}\right )} e^{3} + 2 \, {\left (10 \, b^{3} d x^{2} + 5 \, a b^{2} d x + a^{2} b d\right )} e^{2} + 3 \, {\left (5 \, b^{3} d^{2} x + a b^{2} d^{2}\right )} e}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs.
\(2 (46) = 92\).
time = 1.13, size = 172, normalized size = 2.97 \begin {gather*} \frac {- a^{3} e^{3} - 2 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 4 b^{3} d^{3} - 10 b^{3} e^{3} x^{3} + x^{2} \left (- 10 a b^{2} e^{3} - 20 b^{3} d e^{2}\right ) + x \left (- 5 a^{2} b e^{3} - 10 a b^{2} d e^{2} - 15 b^{3} d^{2} e\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 109, normalized size = 1.88 \begin {gather*} -\frac {10 \, b^{3} x^{3} e^{3} + 20 \, b^{3} d x^{2} e^{2} + 15 \, b^{3} d^{2} x e + 4 \, b^{3} d^{3} + 10 \, a b^{2} x^{2} e^{3} + 10 \, a b^{2} d x e^{2} + 3 \, a b^{2} d^{2} e + 5 \, a^{2} b x e^{3} + 2 \, a^{2} b d e^{2} + a^{3} e^{3}}{20 \, {\left (b x + a\right )}^{5} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 154, normalized size = 2.66 \begin {gather*} -\frac {\frac {a^3\,e^3+2\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+4\,b^3\,d^3}{20\,b^4}+\frac {e^3\,x^3}{2\,b}+\frac {e\,x\,\left (a^2\,e^2+2\,a\,b\,d\,e+3\,b^2\,d^2\right )}{4\,b^3}+\frac {e^2\,x^2\,\left (a\,e+2\,b\,d\right )}{2\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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