3.2024 \(\int \frac{a+b x}{(d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a+b x}{4 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4} \]

[Out]

-(a + b*x)/(4*e*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.112527, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a+b x}{4 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(a + b*x)/(4*e*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 16.7922, size = 36, normalized size = 0.92 \[ - \frac{a + b x}{4 e \left (d + e x\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**5/((b*x+a)**2)**(1/2),x)

[Out]

-(a + b*x)/(4*e*(d + e*x)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0270914, size = 30, normalized size = 0.77 \[ -\frac{a+b x}{4 e \sqrt{(a+b x)^2} (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(a + b*x)/(4*e*Sqrt[(a + b*x)^2]*(d + e*x)^4)

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Maple [A]  time = 0.005, size = 27, normalized size = 0.7 \[ -{\frac{bx+a}{4\,e \left ( ex+d \right ) ^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^5/((b*x+a)^2)^(1/2),x)

[Out]

-1/4/(e*x+d)^4/e*(b*x+a)/((b*x+a)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27392, size = 62, normalized size = 1.59 \[ -\frac{1}{4 \,{\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^5),x, algorithm="fricas")

[Out]

-1/4/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e)

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Sympy [A]  time = 1.7917, size = 49, normalized size = 1.26 \[ - \frac{1}{4 d^{4} e + 16 d^{3} e^{2} x + 24 d^{2} e^{3} x^{2} + 16 d e^{4} x^{3} + 4 e^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(e*x+d)**5/((b*x+a)**2)**(1/2),x)

[Out]

-1/(4*d**4*e + 16*d**3*e**2*x + 24*d**2*e**3*x**2 + 16*d*e**4*x**3 + 4*e**5*x**4
)

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GIAC/XCAS [A]  time = 0.278631, size = 24, normalized size = 0.62 \[ -\frac{e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{4 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^5),x, algorithm="giac")

[Out]

-1/4*e^(-1)*sign(b*x + a)/(x*e + d)^4