Optimal. Leaf size=137 \[ -\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712}
\begin {gather*} -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {c^2}{3 e^5 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^8}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^7}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^6}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^5}+\frac {c^2}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 117, normalized size = 0.85 \begin {gather*} -\frac {2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 143, normalized size = 1.04
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {c \left (3 b e +2 c d \right ) x^{3}}{6 e^{2}}-\frac {\left (2 b^{2} e^{2}+3 b c d e +2 d^{2} c^{2}\right ) x^{2}}{10 e^{3}}-\frac {d \left (2 b^{2} e^{2}+3 b c d e +2 d^{2} c^{2}\right ) x}{30 e^{4}}-\frac {d^{2} \left (2 b^{2} e^{2}+3 b c d e +2 d^{2} c^{2}\right )}{210 e^{5}}}{\left (e x +d \right )^{7}}\) | \(131\) |
gosper | \(-\frac {70 c^{2} x^{4} e^{4}+105 b c \,e^{4} x^{3}+70 c^{2} d \,e^{3} x^{3}+42 b^{2} e^{4} x^{2}+63 b c d \,e^{3} x^{2}+42 c^{2} d^{2} e^{2} x^{2}+14 b^{2} d \,e^{3} x +21 b c \,d^{2} e^{2} x +14 c^{2} d^{3} e x +2 d^{2} e^{2} b^{2}+3 d^{3} e b c +2 c^{2} d^{4}}{210 e^{5} \left (e x +d \right )^{7}}\) | \(141\) |
default | \(-\frac {c \left (b e -2 c d \right )}{2 e^{5} \left (e x +d \right )^{4}}+\frac {d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{3 e^{5} \left (e x +d \right )^{6}}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{2} e^{2}-6 b c d e +6 d^{2} c^{2}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{3 e^{5} \left (e x +d \right )^{3}}\) | \(143\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {\left (3 e^{3} b c +2 d \,e^{2} c^{2}\right ) x^{3}}{6 e^{4}}-\frac {\left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right ) x^{2}}{10 e^{5}}-\frac {d \left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right ) x}{30 e^{6}}-\frac {d^{2} \left (2 e^{4} b^{2}+3 d \,e^{3} b c +2 d^{2} e^{2} c^{2}\right )}{210 e^{7}}}{\left (e x +d \right )^{7}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 192, normalized size = 1.40 \begin {gather*} -\frac {70 \, c^{2} x^{4} e^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \, {\left (x^{7} e^{12} + 7 \, d x^{6} e^{11} + 21 \, d^{2} x^{5} e^{10} + 35 \, d^{3} x^{4} e^{9} + 35 \, d^{4} x^{3} e^{8} + 21 \, d^{5} x^{2} e^{7} + 7 \, d^{6} x e^{6} + d^{7} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.30, size = 190, normalized size = 1.39 \begin {gather*} -\frac {2 \, c^{2} d^{4} + 7 \, {\left (10 \, c^{2} x^{4} + 15 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} e^{4} + 7 \, {\left (10 \, c^{2} d x^{3} + 9 \, b c d x^{2} + 2 \, b^{2} d x\right )} e^{3} + {\left (42 \, c^{2} d^{2} x^{2} + 21 \, b c d^{2} x + 2 \, b^{2} d^{2}\right )} e^{2} + {\left (14 \, c^{2} d^{3} x + 3 \, b c d^{3}\right )} e}{210 \, {\left (x^{7} e^{12} + 7 \, d x^{6} e^{11} + 21 \, d^{2} x^{5} e^{10} + 35 \, d^{3} x^{4} e^{9} + 35 \, d^{4} x^{3} e^{8} + 21 \, d^{5} x^{2} e^{7} + 7 \, d^{6} x e^{6} + d^{7} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 99.49, size = 221, normalized size = 1.61 \begin {gather*} \frac {- 2 b^{2} d^{2} e^{2} - 3 b c d^{3} e - 2 c^{2} d^{4} - 70 c^{2} e^{4} x^{4} + x^{3} \left (- 105 b c e^{4} - 70 c^{2} d e^{3}\right ) + x^{2} \left (- 42 b^{2} e^{4} - 63 b c d e^{3} - 42 c^{2} d^{2} e^{2}\right ) + x \left (- 14 b^{2} d e^{3} - 21 b c d^{2} e^{2} - 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 133, normalized size = 0.97 \begin {gather*} -\frac {{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 2 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{210 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 197, normalized size = 1.44 \begin {gather*} -\frac {\frac {x^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{10\,e^3}+\frac {c^2\,x^4}{3\,e}+\frac {d^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{210\,e^5}+\frac {c\,x^3\,\left (3\,b\,e+2\,c\,d\right )}{6\,e^2}+\frac {d\,x\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{30\,e^4}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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