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

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

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {636} \[ -\frac {2 (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}} \]

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])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.83, size = 44, normalized size = 1.33 \[ -\frac {2 \, \sqrt {c x^{2} + b x} {\left (b d + {\left (2 \, c d - b e\right )} x\right )}}{b^{2} c x^{2} + b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 34, normalized size = 1.03 \[ -\frac {2 \, {\left (\frac {d}{b} + \frac {{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt {c x^{2} + b x}} \]

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)

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maple [A]  time = 0.05, size = 37, normalized size = 1.12 \[ -\frac {2 \left (c x +b \right ) \left (-b e x +2 c d x +b d \right ) x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}} \]

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*(c*x+b)*x*(-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 = 1.33, size = 55, normalized size = 1.67 \[ -\frac {4 \, c d x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, e x}{\sqrt {c x^{2} + b x} b} - \frac {2 \, d}{\sqrt {c x^{2} + b x} b} \]

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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mupad [B]  time = 0.27, size = 31, normalized size = 0.94 \[ -\frac {2\,b\,d-2\,b\,e\,x+4\,c\,d\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]

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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