Optimal. Leaf size=65 \[ -\frac {(b d-a e)^2}{4 e^3 (d+e x)^4}+\frac {2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac {b^2}{2 e^3 (d+e x)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45}
\begin {gather*} \frac {2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac {(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac {b^2}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^5} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^5} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^5}-\frac {2 b (b d-a e)}{e^2 (d+e x)^4}+\frac {b^2}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{4 e^3 (d+e x)^4}+\frac {2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac {b^2}{2 e^3 (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 0.85 \begin {gather*} -\frac {3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 71, normalized size = 1.09
method | result | size |
gosper | \(-\frac {6 b^{2} x^{2} e^{2}+8 a b \,e^{2} x +4 b^{2} d e x +3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(62\) |
risch | \(\frac {-\frac {b^{2} x^{2}}{2 e}-\frac {b \left (2 a e +b d \right ) x}{3 e^{2}}-\frac {3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(63\) |
default | \(-\frac {2 b \left (a e -b d \right )}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b^{2}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{4 e^{3} \left (e x +d \right )^{4}}\) | \(71\) |
norman | \(\frac {-\frac {b^{2} x^{2}}{2 e}-\frac {\left (2 a b \,e^{2}+b^{2} d e \right ) x}{3 e^{3}}-\frac {3 a^{2} e^{3}+2 a b d \,e^{2}+b^{2} d^{2} e}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 92, normalized size = 1.42 \begin {gather*} -\frac {6 \, b^{2} x^{2} e^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.34, size = 89, normalized size = 1.37 \begin {gather*} -\frac {b^{2} d^{2} + {\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} e^{2} + 2 \, {\left (2 \, b^{2} d x + a b d\right )} e}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 104, normalized size = 1.60 \begin {gather*} \frac {- 3 a^{2} e^{2} - 2 a b d e - b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (- 8 a b e^{2} - 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.00, size = 96, normalized size = 1.48 \begin {gather*} -\frac {1}{12} \, {\left (\frac {6 \, b^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {8 \, b^{2} d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, b^{2} d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac {8 \, a b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {6 \, a b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {3 \, a^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 96, normalized size = 1.48 \begin {gather*} -\frac {\frac {3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2}{12\,e^3}+\frac {b^2\,x^2}{2\,e}+\frac {b\,x\,\left (2\,a\,e+b\,d\right )}{3\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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