Optimal. Leaf size=40 \[ \frac{(d+e x)^{m+1}}{e m \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
[Out]
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Rubi [A] time = 0.0628146, antiderivative size = 40, 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 \[ \frac{(d+e x)^{m+1}}{e m \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.891, size = 36, normalized size = 0.9 \[ \frac{\left (d + e x\right )^{m + 1}}{e m \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
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)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0189427, size = 29, normalized size = 0.72 \[ \frac{(d+e x)^{m+1}}{e m \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 39, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{1+m}}{em}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
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)^(1/2),x)
[Out]
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Maxima [A] time = 0.690884, size = 23, normalized size = 0.57 \[ \frac{{\left (e x + d\right )}^{m}}{\sqrt{c} e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231436, size = 61, normalized size = 1.52 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}^{m}}{c e^{2} m x + c d e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{\sqrt{c \left (d + e x\right )^{2}}}\, dx \]
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)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")
[Out]