3.976 \(\int \frac{A+B x}{x^2 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=288 \[ \frac{\sqrt{a+b x+c x^2} \left (2 a b B \left (3 b^2-20 a c\right )-A \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )\right )}{3 a^3 x \left (b^2-4 a c\right )^2}-\frac{2 \left (-c x \left (24 a^2 B c-28 a A b c-2 a b^2 B+5 A b^3\right )-A \left (32 a^2 c^2-32 a b^2 c+5 b^4\right )+2 a b B \left (b^2-8 a c\right )\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{7/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^(3/2)) - (2*(2*a*b*B*
(b^2 - 8*a*c) - A*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2) - c*(5*A*b^3 - 2*a*b^2*B - 28*a*A*b*c + 24*a^2*B*c)*x))/(3
*a^2*(b^2 - 4*a*c)^2*x*Sqrt[a + b*x + c*x^2]) + ((2*a*b*B*(3*b^2 - 20*a*c) - A*(15*b^4 - 100*a*b^2*c + 128*a^2
*c^2))*Sqrt[a + b*x + c*x^2])/(3*a^3*(b^2 - 4*a*c)^2*x) + ((5*A*b - 2*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt
[a + b*x + c*x^2])])/(2*a^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.312436, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {822, 806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (2 a b B \left (3 b^2-20 a c\right )-A \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )\right )}{3 a^3 x \left (b^2-4 a c\right )^2}-\frac{2 \left (-c x \left (24 a^2 B c-28 a A b c-2 a b^2 B+5 A b^3\right )-A \left (32 a^2 c^2-32 a b^2 c+5 b^4\right )+2 a b B \left (b^2-8 a c\right )\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{7/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^(3/2)) - (2*(2*a*b*B*
(b^2 - 8*a*c) - A*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2) - c*(5*A*b^3 - 2*a*b^2*B - 28*a*A*b*c + 24*a^2*B*c)*x))/(3
*a^2*(b^2 - 4*a*c)^2*x*Sqrt[a + b*x + c*x^2]) + ((2*a*b*B*(3*b^2 - 20*a*c) - A*(15*b^4 - 100*a*b^2*c + 128*a^2
*c^2))*Sqrt[a + b*x + c*x^2])/(3*a^3*(b^2 - 4*a*c)^2*x) + ((5*A*b - 2*a*B)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt
[a + b*x + c*x^2])])/(2*a^(7/2))

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 )^{5/2}} \, dx &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (-5 A b^2+2 a b B+16 a A c\right )-3 (A b-2 a B) c x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 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 )}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 a b B \left (b^2-8 a c\right )-A \left (5 b^4-32 a b^2 c+32 a^2 c^2\right )-c \left (5 A b^3-2 a b^2 B-28 a A b c+24 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{\frac{1}{4} \left (-2 a b B \left (3 b^2-20 a c\right )+4 A \left (\frac{15 b^4}{4}-25 a b^2 c+32 a^2 c^2\right )\right )-\frac{1}{2} c \left (2 a B \left (b^2-12 a c\right )-A \left (5 b^3-28 a b c\right )\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{3 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 a b B \left (b^2-8 a c\right )-A \left (5 b^4-32 a b^2 c+32 a^2 c^2\right )-c \left (5 A b^3-2 a b^2 B-28 a A b c+24 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}}+\frac{\left (2 a b B \left (3 b^2-20 a c\right )-A \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{3 a^3 \left (b^2-4 a c\right )^2 x}-\frac{(5 A b-2 a B) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{2 a^3}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 a b B \left (b^2-8 a c\right )-A \left (5 b^4-32 a b^2 c+32 a^2 c^2\right )-c \left (5 A b^3-2 a b^2 B-28 a A b c+24 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}}+\frac{\left (2 a b B \left (3 b^2-20 a c\right )-A \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{3 a^3 \left (b^2-4 a c\right )^2 x}+\frac{(5 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^3}\\ &=\frac{2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (2 a b B \left (b^2-8 a c\right )-A \left (5 b^4-32 a b^2 c+32 a^2 c^2\right )-c \left (5 A b^3-2 a b^2 B-28 a A b c+24 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x \sqrt{a+b x+c x^2}}+\frac{\left (2 a b B \left (3 b^2-20 a c\right )-A \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{3 a^3 \left (b^2-4 a c\right )^2 x}+\frac{(5 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^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.493636, size = 285, normalized size = 0.99 \[ \frac{2 \left (-\frac{\sqrt{a+x (b+c x)} \left (A \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )+2 a b B \left (20 a c-3 b^2\right )\right )}{2 a^2 x \left (b^2-4 a c\right )}+\frac{A \left (-32 a^2 c^2+32 a b^2 c+28 a b c^2 x-5 b^3 c x-5 b^4\right )+2 a B \left (-8 a b c-12 a c^2 x+b^2 c x+b^3\right )}{a x \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{3 \left (b^2-4 a c\right ) (5 A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{4 a^{5/2}}+\frac{A \left (-2 a c+b^2+b c x\right )-a B (b+2 c x)}{x (a+x (b+c x))^{3/2}}\right )}{3 a \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(-((2*a*b*B*(-3*b^2 + 20*a*c) + A*(15*b^4 - 100*a*b^2*c + 128*a^2*c^2))*Sqrt[a + x*(b + c*x)])/(2*a^2*(b^2
- 4*a*c)*x) + (-(a*B*(b + 2*c*x)) + A*(b^2 - 2*a*c + b*c*x))/(x*(a + x*(b + c*x))^(3/2)) + (2*a*B*(b^3 - 8*a*b
*c + b^2*c*x - 12*a*c^2*x) + A*(-5*b^4 + 32*a*b^2*c - 32*a^2*c^2 - 5*b^3*c*x + 28*a*b*c^2*x))/(a*(-b^2 + 4*a*c
)*x*Sqrt[a + x*(b + c*x)]) + (3*(5*A*b - 2*a*B)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c
*x)])])/(4*a^(5/2))))/(3*a*(b^2 - 4*a*c))

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Maple [B]  time = 0.01, size = 709, normalized size = 2.5 \begin{align*} \text{result too large to display} \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)^(5/2),x)

[Out]

1/3*B/a/(c*x^2+b*x+a)^(3/2)-2/3*B/a*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c-1/3*B/a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a
)^(3/2)-16/3*B/a*b*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-8/3*B/a*b^2*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+B/a
^2/(c*x^2+b*x+a)^(1/2)-2*B/a^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c-B/a^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)
-B/a^(5/2)*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)^(3/2)-5/6*A/a^2*b/(c*x^2+b*x+a)^(
3/2)+5/3*A/a^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c+5/6*A/a^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+40/3*A/a^
2*b^2*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+20/3*A/a^2*b^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/2*A/a^3*b/(
c*x^2+b*x+a)^(1/2)+5*A/a^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c+5/2*A/a^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)+5/2*A/a^(7/2)*b*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-16/3*A/a*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*
x-8/3*A/a*c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-128/3*A/a*c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-64/3*A/a*c^2/(
4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b

________________________________________________________________________________________

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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 21.3932, size = 3505, normalized size = 12.17 \begin{align*} \text{result too large to display} \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)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*((16*(2*B*a^3 - 5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5*A*a*b^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*x^5 +
2*(16*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 8*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2 + (2*B*a*b^5 - 5*A*b^6)*c)*x^4 + (2*B*a*
b^6 - 5*A*b^7 + 32*(2*B*a^4 - 5*A*a^3*b)*c^3 - 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*x^3 + 2*(2*B*a^2*b^5 - 5*A*a*b^6
+ 16*(2*B*a^4*b - 5*A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*x^2 + (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(
2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*c)*x)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*s
qrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(3*A*a^3*b^4 - 24*A*a^4*b^2*c + 48*A*a^5*c^2 + (128
*A*a^3*c^4 + 20*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 3*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2)*x^4 - 6*(4*(2*B*a^4 - 13*A*a^
3*b)*c^3 - 7*(2*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 + (2*B*a^2*b^4 - 5*A*a*b^5)*c)*x^3 - 3*(2*B*a^2*b^5 - 5*A*a*b^6 -
16*A*a^3*b^2*c^2 - 64*A*a^4*c^3 - 6*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*x^2 - 4*(2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(B
*a^5 - 4*A*a^4*b)*c^2 - (14*B*a^4*b^2 - 37*A*a^3*b^3)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^4*b^4*c^2 - 8*a^5*b^2*c
^3 + 16*a^6*c^4)*x^5 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^4 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)
*x^3 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^2 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x), 1/6*(3*((16*(2*
B*a^3 - 5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5*A*a*b^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*x^5 + 2*(16*(2*B*a^3*b -
5*A*a^2*b^2)*c^3 - 8*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2 + (2*B*a*b^5 - 5*A*b^6)*c)*x^4 + (2*B*a*b^6 - 5*A*b^7 + 32
*(2*B*a^4 - 5*A*a^3*b)*c^3 - 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*x^3 + 2*(2*B*a^2*b^5 - 5*A*a*b^6 + 16*(2*B*a^4*b -
5*A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*x^2 + (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*
b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*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*(3*A*a^3*b^4 - 24*A*a^4*b^2*c + 48*A*a^5*c^2 + (128*A*a^3*c^4 + 20*(2*B*a^3*b - 5*A
*a^2*b^2)*c^3 - 3*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2)*x^4 - 6*(4*(2*B*a^4 - 13*A*a^3*b)*c^3 - 7*(2*B*a^3*b^2 - 5*A*
a^2*b^3)*c^2 + (2*B*a^2*b^4 - 5*A*a*b^5)*c)*x^3 - 3*(2*B*a^2*b^5 - 5*A*a*b^6 - 16*A*a^3*b^2*c^2 - 64*A*a^4*c^3
- 6*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*x^2 - 4*(2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(B*a^5 - 4*A*a^4*b)*c^2 - (14*B*a^
4*b^2 - 37*A*a^3*b^3)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^5 + 2*(a^4*b^
5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^4 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^3 + 2*(a^5*b^5 - 8*a^6*b^3*c
+ 16*a^7*b*c^2)*x^2 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.19371, size = 652, normalized size = 2.26 \begin{align*} \frac{{\left ({\left (\frac{{\left (3 \, B a^{9} b^{3} c^{2} - 6 \, A a^{8} b^{4} c^{2} - 20 \, B a^{10} b c^{3} + 38 \, A a^{9} b^{2} c^{3} - 40 \, A a^{10} c^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (2 \, B a^{9} b^{4} c - 4 \, A a^{8} b^{5} c - 14 \, B a^{10} b^{2} c^{2} + 27 \, A a^{9} b^{3} c^{2} + 8 \, B a^{11} c^{3} - 36 \, A a^{10} b c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (B a^{9} b^{5} - 2 \, A a^{8} b^{6} - 6 \, B a^{10} b^{3} c + 12 \, A a^{9} b^{4} c - 8 \, A a^{10} b^{2} c^{2} - 16 \, A a^{11} c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{4 \, B a^{10} b^{4} - 7 \, A a^{9} b^{5} - 28 \, B a^{11} b^{2} c + 50 \, A a^{10} b^{3} c + 32 \, B a^{12} c^{2} - 80 \, A a^{11} b c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} + \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \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^{3}} \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)^(5/2),x, algorithm="giac")

[Out]

1/3*((((3*B*a^9*b^3*c^2 - 6*A*a^8*b^4*c^2 - 20*B*a^10*b*c^3 + 38*A*a^9*b^2*c^3 - 40*A*a^10*c^4)*x/(b^4*c^2 - 8
*a*b^2*c^3 + 16*a^2*c^4) + 3*(2*B*a^9*b^4*c - 4*A*a^8*b^5*c - 14*B*a^10*b^2*c^2 + 27*A*a^9*b^3*c^2 + 8*B*a^11*
c^3 - 36*A*a^10*b*c^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 3*(B*a^9*b^5 - 2*A*a^8*b^6 - 6*B*a^10*b^3*c +
12*A*a^9*b^4*c - 8*A*a^10*b^2*c^2 - 16*A*a^11*c^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (4*B*a^10*b^4 -
7*A*a^9*b^5 - 28*B*a^11*b^2*c + 50*A*a^10*b^3*c + 32*B*a^12*c^2 - 80*A*a^11*b*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16
*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) + (2*B*a - 5*A*b)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sq
rt(-a)*a^3) + ((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^3)