Optimal. Leaf size=77 \[ -\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} \frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac {c^2 d^2}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^5} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^5}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^4}+\frac {c^2 d^2}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}+\frac {2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac {c^2 d^2}{2 e^3 (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 61, normalized size = 0.79 \begin {gather*} -\frac {3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^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 = 0.75, size = 83, normalized size = 1.08
method | result | size |
gosper | \(-\frac {6 c^{2} d^{2} x^{2} e^{2}+8 a c d \,e^{3} x +4 c^{2} d^{3} e x +3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(72\) |
risch | \(\frac {-\frac {c^{2} d^{2} x^{2}}{2 e}-\frac {d c \left (2 e^{2} a +c \,d^{2}\right ) x}{3 e^{2}}-\frac {3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(75\) |
default | \(-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {c^{2} d^{2}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {2 c d \left (e^{2} a -c \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{3}}\) | \(83\) |
norman | \(\frac {-\frac {d^{2} \left (3 a^{2} e^{7}+2 a c \,d^{2} e^{5}+c^{2} d^{4} e^{3}\right )}{12 e^{6}}-\frac {\left (a^{2} e^{7}+6 a c \,d^{2} e^{5}+5 c^{2} d^{4} e^{3}\right ) x^{2}}{4 e^{4}}-\frac {2 d \left (a c \,e^{5}+2 c^{2} d^{2} e^{3}\right ) x^{3}}{3 e^{3}}-\frac {d \left (a^{2} e^{7}+2 a c \,d^{2} e^{5}+c^{2} d^{4} e^{3}\right ) x}{2 e^{5}}-\frac {e \,c^{2} d^{2} x^{4}}{2}}{\left (e x +d \right )^{6}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 100, normalized size = 1.30 \begin {gather*} -\frac {6 \, c^{2} d^{2} x^{2} e^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\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 = 3.25, size = 99, normalized size = 1.29 \begin {gather*} -\frac {4 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + 3 \, a^{2} e^{4} + 2 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}}{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 = 1.02, size = 114, normalized size = 1.48 \begin {gather*} \frac {- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} - 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 8 a c d e^{3} - 4 c^{2} d^{3} 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 = 1.12, size = 68, normalized size = 0.88 \begin {gather*} -\frac {{\left (6 \, c^{2} d^{2} x^{2} e^{2} + 4 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-3\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 85, normalized size = 1.10 \begin {gather*} -\frac {\frac {a^2\,e}{4}-d\,\left (\frac {c^2\,x^3}{3}-\frac {2\,a\,c\,x}{3}\right )-\frac {c^2\,e\,x^4}{12}+\frac {a\,c\,d^2}{6\,e}}{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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