Optimal. Leaf size=132 \[ -\frac {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 132, 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}{3 e^5 (d+e x)^3}-\frac {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{e^5 (d+e x)} \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)^6} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^6}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^5}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^4}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac {c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 116, normalized size = 0.88 \begin {gather*} -\frac {b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 b c e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 143, normalized size = 1.08
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {c \left (b e +2 c d \right ) x^{3}}{e^{2}}-\frac {\left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) x^{2}}{3 e^{3}}-\frac {d \left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) x}{6 e^{4}}-\frac {d^{2} \left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right )}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(127\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {\left (b c e +2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) x^{2}}{3 e^{3}}-\frac {d \left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right ) x}{6 e^{4}}-\frac {d^{2} \left (b^{2} e^{2}+3 b c d e +6 d^{2} c^{2}\right )}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(129\) |
gosper | \(-\frac {30 c^{2} x^{4} e^{4}+30 b c \,e^{4} x^{3}+60 c^{2} d \,e^{3} x^{3}+10 b^{2} e^{4} x^{2}+30 b c d \,e^{3} x^{2}+60 c^{2} d^{2} e^{2} x^{2}+5 b^{2} d \,e^{3} x +15 b c \,d^{2} e^{2} x +30 c^{2} d^{3} e x +d^{2} e^{2} b^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5} \left (e x +d \right )^{5}}\) | \(140\) |
default | \(\frac {d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )^{2}}-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} e^{2}-6 b c d e +6 d^{2} c^{2}}{3 e^{5} \left (e x +d \right )^{3}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 168, normalized size = 1.27 \begin {gather*} -\frac {30 \, c^{2} x^{4} e^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 167, normalized size = 1.27 \begin {gather*} -\frac {6 \, c^{2} d^{4} + 10 \, {\left (3 \, c^{2} x^{4} + 3 \, b c x^{3} + b^{2} x^{2}\right )} e^{4} + 5 \, {\left (12 \, c^{2} d x^{3} + 6 \, b c d x^{2} + b^{2} d x\right )} e^{3} + {\left (60 \, c^{2} d^{2} x^{2} + 15 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 3 \, {\left (10 \, c^{2} d^{3} x + b c d^{3}\right )} e}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.90, size = 196, normalized size = 1.48 \begin {gather*} \frac {- b^{2} d^{2} e^{2} - 3 b c d^{3} e - 6 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 30 b c e^{4} - 60 c^{2} d e^{3}\right ) + x^{2} \left (- 10 b^{2} e^{4} - 30 b c d e^{3} - 60 c^{2} d^{2} e^{2}\right ) + x \left (- 5 b^{2} d e^{3} - 15 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 132, normalized size = 1.00 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 5 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 169, normalized size = 1.28 \begin {gather*} -\frac {\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{3\,e^3}+\frac {c^2\,x^4}{e}+\frac {d^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{30\,e^5}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}+\frac {d\,x\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{6\,e^4}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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