3.1085 \(\int \frac{(d+e x)^m}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

(e*x + d)^m/(sqrt(c)*e*m)

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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]

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

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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]

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

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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]

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