Optimal. Leaf size=39 \[ -\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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} \begin {gather*} -\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 607
Rule 642
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c^2 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c}{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 = 0.01, size = 27, normalized size = 0.69 \begin {gather*} -\frac {1}{4 e (d+e x)^3 \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.28, size = 338, normalized size = 8.67 \begin {gather*} \frac {2 c^3 \left (c d^8 e-175 c d^4 e^5 x^4-704 c d^3 e^6 x^5-1056 c d^2 e^7 x^6-704 c d e^8 x^7-176 c e^9 x^8\right )+2 c^3 \sqrt {c e^2} \left (d^7-d^6 e x+d^5 e^2 x^2-d^4 e^3 x^3+176 d^3 e^4 x^4+528 d^2 e^5 x^5+528 d e^6 x^6+176 e^7 x^7\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{d^4 e x^4 \sqrt {c d^2+2 c d e x+c e^2 x^2} \left (-8 c^4 d^3 e^5-24 c^4 d^2 e^6 x-24 c^4 d e^7 x^2-8 c^4 e^8 x^3\right )+d^4 e x^4 \sqrt {c e^2} \left (8 c^4 d^4 e^4+32 c^4 d^3 e^5 x+48 c^4 d^2 e^6 x^2+32 c^4 d e^7 x^3+8 c^4 e^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 85, normalized size = 2.18 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (c e^{6} x^{5} + 5 \, c d e^{5} x^{4} + 10 \, c d^{2} e^{4} x^{3} + 10 \, c d^{3} e^{3} x^{2} + 5 \, c d^{4} e^{2} x + c d^{5} e\right )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.90 \begin {gather*} -\frac {1}{4 \left (e x +d \right )^{3} \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 61, normalized size = 1.56 \begin {gather*} -\frac {1}{4 \, {\left (\sqrt {c} e^{5} x^{4} + 4 \, \sqrt {c} d e^{4} x^{3} + 6 \, \sqrt {c} d^{2} e^{3} x^{2} + 4 \, \sqrt {c} d^{3} e^{2} x + \sqrt {c} d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 37, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,c\,e\,{\left (d+e\,x\right )}^5} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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