3.1084 \(\int (d+e x)^m \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+2)} \]

[Out]

((d + e*x)^(1 + m)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(e*(2 + m))

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Rubi [A]  time = 0.0636315, antiderivative size = 42, 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 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2} (d+e x)^{m+1}}{e (m+2)} \]

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]

((d + e*x)^(1 + m)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(e*(2 + m))

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Rubi in Sympy [A]  time = 18.7519, size = 37, normalized size = 0.88 \[ \frac{\left (d + e x\right )^{m + 1} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e \left (m + 2\right )} \]

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]

(d + e*x)**(m + 1)*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(e*(m + 2))

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Mathematica [A]  time = 0.0278571, size = 31, normalized size = 0.74 \[ \frac{\sqrt{c (d+e x)^2} (d+e x)^{m+1}}{e (m+2)} \]

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]

((d + e*x)^(1 + m)*Sqrt[c*(d + e*x)^2])/(e*(2 + m))

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Maple [A]  time = 0.006, size = 41, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( 2+m \right ) }\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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]

(e*x+d)^(1+m)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/e/(2+m)

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Maxima [A]  time = 0.692862, size = 57, normalized size = 1.36 \[ \frac{{\left (\sqrt{c} e^{2} x^{2} + 2 \, \sqrt{c} d e x + \sqrt{c} d^{2}\right )}{\left (e x + d\right )}^{m}}{e{\left (m + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

(sqrt(c)*e^2*x^2 + 2*sqrt(c)*d*e*x + sqrt(c)*d^2)*(e*x + d)^m/(e*(m + 2))

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Fricas [A]  time = 0.247299, size = 59, normalized size = 1.4 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x + d\right )}{\left (e x + d\right )}^{m}}{e m + 2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)*(e*x + d)^m/(e*m + 2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{m}\, dx \]

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]

Integral(sqrt(c*(d + e*x)**2)*(d + e*x)**m, x)

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GIAC/XCAS [A]  time = 0.229752, size = 92, normalized size = 2.19 \[ \frac{\sqrt{c} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 2 \, \sqrt{c} d x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + \sqrt{c} d^{2} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m e + 2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="giac")

[Out]

(sqrt(c)*x^2*e^(m*ln(x*e + d) + 2) + 2*sqrt(c)*d*x*e^(m*ln(x*e + d) + 1) + sqrt(
c)*d^2*e^(m*ln(x*e + d)))/(m*e + 2*e)