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

Optimal. Leaf size=158 \[ -\frac{\sqrt{a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]) - ((3*A*b^2 - 2*a*
b*B - 8*a*A*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 - 4*a*c)*x) + ((3*A*b - 2*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*
Sqrt[a + b*x + c*x^2])])/(2*a^(5/2))

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Rubi [A]  time = 0.119215, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {822, 806, 724, 206} \[ -\frac{\sqrt{a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(a*(b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]) - ((3*A*b^2 - 2*a*
b*B - 8*a*A*c)*Sqrt[a + b*x + c*x^2])/(a^2*(b^2 - 4*a*c)*x) + ((3*A*b - 2*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*
Sqrt[a + b*x + c*x^2])])/(2*a^(5/2))

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} \left (-3 A b^2+2 a b B+8 a A c\right )-(A b-2 a B) c x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}}-\frac{\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}-\frac{(3 A b-2 a B) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{2 a^2}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}}-\frac{\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}+\frac{(3 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{a^2}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt{a+b x+c x^2}}-\frac{\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.191771, size = 150, normalized size = 0.95 \[ \frac{\frac{2 \sqrt{a} \left (-4 a^2 c (A-B x)+a A \left (b^2-10 b c x-8 c^2 x^2\right )-2 a b B x (b+c x)+3 A b^2 x (b+c x)\right )}{x \sqrt{a+x (b+c x)}}-\left (b^2-4 a c\right ) (3 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{2 a^{5/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[a]*(-4*a^2*c*(A - B*x) + 3*A*b^2*x*(b + c*x) - 2*a*b*B*x*(b + c*x) + a*A*(b^2 - 10*b*c*x - 8*c^2*x^2)
))/(x*Sqrt[a + x*(b + c*x)]) - (3*A*b - 2*a*B)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*
x)])])/(2*a^(5/2)*(-b^2 + 4*a*c))

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Maple [B]  time = 0.012, size = 330, normalized size = 2.1 \begin{align*}{\frac{B}{a}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{Bcbx}{a \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}B}{a \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{ax}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{A{b}^{2}cx}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,A{b}^{3}}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-8\,{\frac{A{c}^{2}x}{a \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{Abc}{a \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B/a/(c*x^2+b*x+a)^(1/2)-2*B*b/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c-B*b^2/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-B/
a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-A/a/x/(c*x^2+b*x+a)^(1/2)-3/2*A/a^2*b/(c*x^2+b*x+a)^(1/2
)+3*A/a^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c+3/2*A/a^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3/2*A/a^(5/2)*
b*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-8*A*c^2/a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-4*A*c/a/(4*a*c-b^2
)/(c*x^2+b*x+a)^(1/2)*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.71862, size = 1400, normalized size = 8.86 \begin{align*} \left [\frac{{\left ({\left (4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} -{\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \,{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt{a} \log \left (-\frac{8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \,{\left (A a^{2} b^{2} - 4 \, A a^{3} c -{\left (8 \, A a^{2} c^{2} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{4 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}, -\frac{{\left ({\left (4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} -{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} -{\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \,{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \,{\left (A a^{2} b^{2} - 4 \, A a^{3} c -{\left (8 \, A a^{2} c^{2} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} -{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*x^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a
*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*x)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*
x^2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(A*a^2*b^2 - 4*A*a^3*c - (8*A*a^2*c^2 + (2
*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c*x^2 + b*x + a))
/((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3 - 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x), -1/2*(((4*(2*B*a^2 - 3*A*a
*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*x^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*b^
2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*x)*sqrt(-a)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(
a*c*x^2 + a*b*x + a^2)) + 2*(A*a^2*b^2 - 4*A*a^3*c - (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*x^2 - (2*B*a^2*
b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^3*b^2*c - 4*a^4*c^2)*x^3 + (a^3*b^3
- 4*a^4*b*c)*x^2 + (a^4*b^2 - 4*a^5*c)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.54441, size = 297, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (\frac{{\left (B a^{3} b c - A a^{2} b^{2} c + 2 \, A a^{3} c^{2}\right )} x}{a^{4} b^{2} - 4 \, a^{5} c} + \frac{B a^{3} b^{2} - A a^{2} b^{3} - 2 \, B a^{4} c + 3 \, A a^{3} b c}{a^{4} b^{2} - 4 \, a^{5} c}\right )}}{\sqrt{c x^{2} + b x + a}} + \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((B*a^3*b*c - A*a^2*b^2*c + 2*A*a^3*c^2)*x/(a^4*b^2 - 4*a^5*c) + (B*a^3*b^2 - A*a^2*b^3 - 2*B*a^4*c + 3*A*a^
3*b*c)/(a^4*b^2 - 4*a^5*c))/sqrt(c*x^2 + b*x + a) + (2*B*a - 3*A*b)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)/sqrt(-a))/(sqrt(-a)*a^2) + ((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b + 2*A*a*sqrt(c))/(((sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^2 - a)*a^2)