3.11.57 \(\int \frac {A+B x}{(b x+c x^2)^{3/2}} \, dx\)

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

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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} \begin {gather*} -\frac {2 (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(A*b - (b*B - 2*A*c)*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 {A+B x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (A b-(b B-2 A c) 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 \begin {gather*} \frac {2 b B x-2 A (b+2 c x)}{b^2 \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*b*B*x - 2*A*(b + 2*c*x))/(b^2*Sqrt[x*(b + c*x)])

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IntegrateAlgebraic [A]  time = 0.00, size = 42, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (-A b-2 A c x+b B x)}{b^2 x (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*(-(A*b) + b*B*x - 2*A*c*x)*Sqrt[b*x + c*x^2])/(b^2*x*(b + c*x))

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fricas [A]  time = 0.41, size = 44, normalized size = 1.33 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (A b - {\left (B b - 2 \, A c\right )} x\right )}}{b^{2} c x^{2} + b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(c*x^2 + b*x)*(A*b - (B*b - 2*A*c)*x)/(b^2*c*x^2 + b^3*x)

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giac [A]  time = 0.28, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {A}{b} - \frac {{\left (B b - 2 \, A c\right )} x}{b^{2}}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

-2*(A/b - (B*b - 2*A*c)*x/b^2)/sqrt(c*x^2 + b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x)^(3/2),x)

[Out]

-2*(c*x+b)*x*(2*A*c*x-B*b*x+A*b)/b^2/(c*x^2+b*x)^(3/2)

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maxima [A]  time = 0.60, size = 55, normalized size = 1.67 \begin {gather*} \frac {2 \, B x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, A}{\sqrt {c x^{2} + b x} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

2*B*x/(sqrt(c*x^2 + b*x)*b) - 4*A*c*x/(sqrt(c*x^2 + b*x)*b^2) - 2*A/(sqrt(c*x^2 + b*x)*b)

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mupad [B]  time = 1.65, size = 31, normalized size = 0.94 \begin {gather*} -\frac {2\,A\,b+4\,A\,c\,x-2\,B\,b\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(b*x + c*x^2)^(3/2),x)

[Out]

-(2*A*b + 4*A*c*x - 2*B*b*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 \begin {gather*} \int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((A + B*x)/(x*(b + c*x))**(3/2), x)

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