Optimal. Leaf size=65 \[ -\frac {(b d-a e)^2}{6 e^3 (d+e x)^6}+\frac {2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac {b^2}{4 e^3 (d+e x)^4} \]
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Rubi [A]
time = 0.02, 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)}{5 e^3 (d+e x)^5}-\frac {(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac {b^2}{4 e^3 (d+e x)^4} \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)^7} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^7} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^7}-\frac {2 b (b d-a e)}{e^2 (d+e x)^6}+\frac {b^2}{e^2 (d+e x)^5}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{6 e^3 (d+e x)^6}+\frac {2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac {b^2}{4 e^3 (d+e x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 0.85 \begin {gather*} -\frac {10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 71, normalized size = 1.09
method | result | size |
gosper | \(-\frac {15 b^{2} x^{2} e^{2}+24 a b \,e^{2} x +6 b^{2} d e x +10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}}{60 e^{3} \left (e x +d \right )^{6}}\) | \(62\) |
risch | \(\frac {-\frac {b^{2} x^{2}}{4 e}-\frac {b \left (4 a e +b d \right ) x}{10 e^{2}}-\frac {10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}}{60 e^{3}}}{\left (e x +d \right )^{6}}\) | \(63\) |
default | \(-\frac {2 b \left (a e -b d \right )}{5 e^{3} \left (e x +d \right )^{5}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{6 e^{3} \left (e x +d \right )^{6}}-\frac {b^{2}}{4 e^{3} \left (e x +d \right )^{4}}\) | \(71\) |
norman | \(\frac {-\frac {b^{2} x^{2}}{4 e}-\frac {\left (4 a b \,e^{4}+b^{2} d \,e^{3}\right ) x}{10 e^{5}}-\frac {10 a^{2} e^{5}+4 a b d \,e^{4}+b^{2} d^{2} e^{3}}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 112, normalized size = 1.72 \begin {gather*} -\frac {15 \, b^{2} x^{2} e^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \, {\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \, {\left (x^{6} e^{9} + 6 \, d x^{5} e^{8} + 15 \, d^{2} x^{4} e^{7} + 20 \, d^{3} x^{3} e^{6} + 15 \, d^{4} x^{2} e^{5} + 6 \, d^{5} x e^{4} + d^{6} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.92, size = 110, normalized size = 1.69 \begin {gather*} -\frac {b^{2} d^{2} + {\left (15 \, b^{2} x^{2} + 24 \, a b x + 10 \, a^{2}\right )} e^{2} + 2 \, {\left (3 \, b^{2} d x + 2 \, a b d\right )} e}{60 \, {\left (x^{6} e^{9} + 6 \, d x^{5} e^{8} + 15 \, d^{2} x^{4} e^{7} + 20 \, d^{3} x^{3} e^{6} + 15 \, d^{4} x^{2} e^{5} + 6 \, d^{5} x e^{4} + d^{6} e^{3}\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. 128 vs.
\(2 (56) = 112\).
time = 0.78, size = 128, normalized size = 1.97 \begin {gather*} \frac {- 10 a^{2} e^{2} - 4 a b d e - b^{2} d^{2} - 15 b^{2} e^{2} x^{2} + x \left (- 24 a b e^{2} - 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.99, size = 60, normalized size = 0.92 \begin {gather*} -\frac {{\left (15 \, b^{2} x^{2} e^{2} + 6 \, b^{2} d x e + b^{2} d^{2} + 24 \, a b x e^{2} + 4 \, a b d e + 10 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 118, normalized size = 1.82 \begin {gather*} -\frac {\frac {10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2}{60\,e^3}+\frac {b^2\,x^2}{4\,e}+\frac {b\,x\,\left (4\,a\,e+b\,d\right )}{10\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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