Optimal. Leaf size=45 \[ -\frac {(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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 = {644, 32} \[ -\frac {(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 644
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=\frac {(d+e x)^3 \int (d+e x)^{-3+m} \, dx}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ &=-\frac {(d+e x)^{1+m}}{e (2-m) \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 = 31, normalized size = 0.69 \[ \frac {(d+e x)^{m+1}}{e (m-2) \left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.38, size = 122, normalized size = 2.71 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 2 \, c^{2} d^{3} e + {\left (c^{2} e^{4} m - 2 \, c^{2} e^{4}\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} m - 2 \, c^{2} d e^{3}\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e^{2} m - 2 \, c^{2} d^{2} e^{2}\right )} x} \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.91 \[ \frac {\left (e x +d \right )^{m +1}}{\left (m -2\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 51, normalized size = 1.13 \[ \frac {{\left (e x + d\right )}^{m} \sqrt {c}}{c^{2} e^{3} {\left (m - 2\right )} x^{2} + 2 \, c^{2} d e^{2} {\left (m - 2\right )} x + c^{2} d^{2} e {\left (m - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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