Optimal. Leaf size=31 \[ -\frac {1}{5 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {643, 629} \[ -\frac {1}{5 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 629
Rule 643
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=c \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {1}{5 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.65 \[ -\frac {1}{5 e \left (c (d+e x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 111, normalized size = 3.58 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \, {\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.90 \[ -\frac {1}{5 \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.31, size = 75, normalized size = 2.42 \[ -\frac {1}{5 \, {\left (c^{\frac {5}{2}} e^{6} x^{5} + 5 \, c^{\frac {5}{2}} d e^{5} x^{4} + 10 \, c^{\frac {5}{2}} d^{2} e^{4} x^{3} + 10 \, c^{\frac {5}{2}} d^{3} e^{3} x^{2} + 5 \, c^{\frac {5}{2}} d^{4} e^{2} x + c^{\frac {5}{2}} d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 37, normalized size = 1.19 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{5\,c^3\,e\,{\left (d+e\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.22, size = 42, normalized size = 1.35 \[ \begin {cases} - \frac {1}{5 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {5}{2}}} & \text {for}\: e \neq 0 \\\frac {x}{d \left (c d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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