3.324 \(\int \frac{d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0445442, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.63012, size = 32, normalized size = 0.97 \[ - \frac{2 b d - x \left (2 b e - 4 c d\right )}{b^{2} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**2+b*x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0430985, size = 30, normalized size = 0.91 \[ \frac{2 (-b d+b e x-2 c d x)}{b^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 37, normalized size = 1.1 \[ -2\,{\frac{x \left ( cx+b \right ) \left ( -bex+2\,cdx+bd \right ) }{{b}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x)^(3/2),x)

[Out]

-2*x*(c*x+b)*(-b*e*x+2*c*d*x+b*d)/b^2/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.700106, size = 74, normalized size = 2.24 \[ -\frac{4 \, c d x}{\sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, e x}{\sqrt{c x^{2} + b x} b} - \frac{2 \, d}{\sqrt{c x^{2} + b x} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219937, size = 42, normalized size = 1.27 \[ -\frac{2 \,{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{\sqrt{c x^{2} + b x} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((d + e*x)/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.223758, size = 46, normalized size = 1.39 \[ -\frac{2 \,{\left (\frac{d}{b} + \frac{{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

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