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

Optimal. Leaf size=45 \[ -\frac {2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {636} \begin {gather*} -\frac {2 (-2 a B-x (b B-2 A c)+A b)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x))/((b^2 - 4*a*c)*Sqrt[a + 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 (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (A b-2 a B-(b B-2 A c) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 44, normalized size = 0.98 \begin {gather*} \frac {2 B (2 a+b x)-2 A (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*(2*a + b*x) - 2*A*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])

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IntegrateAlgebraic [A]  time = 0.39, size = 44, normalized size = 0.98 \begin {gather*} -\frac {2 (-2 a B+A b+2 A c x-b B x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b - 2*a*B - b*B*x + 2*A*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

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fricas [A]  time = 0.53, size = 74, normalized size = 1.64 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x + a} {\left (2 \, B a - A b + {\left (B b - 2 \, A c\right )} x\right )}}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(c*x^2 + b*x + a)*(2*B*a - A*b + (B*b - 2*A*c)*x)/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*
b*c)*x)

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giac [A]  time = 0.24, size = 55, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (\frac {{\left (B b - 2 \, A c\right )} x}{b^{2} - 4 \, a c} + \frac {2 \, B a - A b}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {4 A c x -2 B b x +2 A b -4 B a}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(c*x^2+b*x+a)^(1/2)*(2*A*c*x-B*b*x+A*b-2*B*a)/(4*a*c-b^2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 1.39, size = 44, normalized size = 0.98 \begin {gather*} \frac {2\,A\,b-4\,B\,a+4\,A\,c\,x-2\,B\,b\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*A*b - 4*B*a + 4*A*c*x - 2*B*b*x)/((4*a*c - b^2)*(a + 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 (a + b x + c x^{2}\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+a)**(3/2),x)

[Out]

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

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