Optimal. Leaf size=45 \[ -\frac{(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0668953, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 19.0261, size = 39, normalized size = 0.87 \[ - \frac{\left (d + e x\right )^{m + 1}}{e \left (- m + 2\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0344759, 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.
[In] Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.002, size = 41, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( -2+m \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.696291, size = 69, normalized size = 1.53 \[ \frac{{\left (e x + d\right )}^{m} \sqrt{c}}{c^{2} e^{3}{\left (m - 2\right )} x^{2} + 2 \, c^{2} d e^{2}{\left (m - 2\right )} x + c^{2} d^{2} e{\left (m - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235631, size = 165, normalized size = 3.67 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 2 \, c^{2} d^{3} e +{\left (c^{2} e^{4} m - 2 \, c^{2} e^{4}\right )} x^{3} + 3 \,{\left (c^{2} d e^{3} m - 2 \, c^{2} d e^{3}\right )} x^{2} + 3 \,{\left (c^{2} d^{2} e^{2} m - 2 \, c^{2} d^{2} e^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="giac")
[Out]