Optimal. Leaf size=38 \[ -\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 = {642, 607} \[ -\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 607
Rule 642
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=c \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.68 \[ -\frac {c (d+e x)}{4 e \left (c (d+e x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 97, normalized size = 2.55 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.92 \[ -\frac {1}{4 \left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 61, normalized size = 1.61 \[ -\frac {1}{4 \, {\left (c^{\frac {3}{2}} e^{5} x^{4} + 4 \, c^{\frac {3}{2}} d e^{4} x^{3} + 6 \, c^{\frac {3}{2}} d^{2} e^{3} x^{2} + 4 \, c^{\frac {3}{2}} d^{3} e^{2} x + c^{\frac {3}{2}} d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 37, normalized size = 0.97 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,c^2\,e\,{\left (d+e\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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