3.969 \(\int \frac{A+B x}{x^4 (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=317 \[ \frac{\sqrt{a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{9/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[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^3*Sqrt[a + b*x + c*x^2]) - ((7*A*b^2 - 6*
a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(3*a^2*(b^2 - 4*a*c)*x^3) - ((6*a*B*(5*b^2 - 12*a*c) - A*(35*b^3 - 11
6*a*b*c))*Sqrt[a + b*x + c*x^2])/(12*a^3*(b^2 - 4*a*c)*x^2) + ((6*a*b*B*(15*b^2 - 52*a*c) - A*(105*b^4 - 460*a
*b^2*c + 256*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(24*a^4*(b^2 - 4*a*c)*x) - ((6*a*B*(5*b^2 - 4*a*c) - A*(35*b^3 -
60*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.387877, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {822, 834, 806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}-\frac{\sqrt{a+b x+c x^2} \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{9/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^4*(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^3*Sqrt[a + b*x + c*x^2]) - ((7*A*b^2 - 6*
a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(3*a^2*(b^2 - 4*a*c)*x^3) - ((6*a*B*(5*b^2 - 12*a*c) - A*(35*b^3 - 11
6*a*b*c))*Sqrt[a + b*x + c*x^2])/(12*a^3*(b^2 - 4*a*c)*x^2) + ((6*a*b*B*(15*b^2 - 52*a*c) - A*(105*b^4 - 460*a
*b^2*c + 256*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(24*a^4*(b^2 - 4*a*c)*x) - ((6*a*B*(5*b^2 - 4*a*c) - A*(35*b^3 -
60*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(9/2))

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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -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^4 \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^3 \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} \left (-7 A b^2+6 a b B+16 a A c\right )-3 (A b-2 a B) c x}{x^4 \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^3 \sqrt{a+b x+c x^2}}-\frac{\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac{2 \int \frac{\frac{1}{4} \left (-35 A b^3+30 a b^2 B+116 a A b c-72 a^2 B c\right )-c \left (7 A b^2-6 a b B-16 a A c\right ) x}{x^3 \sqrt{a+b x+c x^2}} \, dx}{3 a^2 \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^3 \sqrt{a+b x+c x^2}}-\frac{\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac{\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt{a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}-\frac{\int \frac{\frac{1}{8} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right )+\frac{1}{4} c \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{3 a^3 \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^3 \sqrt{a+b x+c x^2}}-\frac{\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac{\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt{a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac{\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}-\frac{\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{16 a^4}\\ &=\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^3 \sqrt{a+b x+c x^2}}-\frac{\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac{\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt{a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac{\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}+\frac{\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \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 )}{8 a^4}\\ &=\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^3 \sqrt{a+b x+c x^2}}-\frac{\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt{a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac{\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt{a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac{\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}+\frac{\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.446629, size = 294, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{a} \left (4 a^3 \left (2 A \left (b^2+7 b c x+16 c^2 x^2\right )+3 B x \left (b^2+10 b c x-12 c^2 x^2\right )\right )+2 a^2 x \left (A \left (-86 b^2 c x-7 b^3+244 b c^2 x^2+128 c^3 x^3\right )+3 b B x \left (-5 b^2+62 b c x+52 c^2 x^2\right )\right )-16 a^4 c (2 A+3 B x)-5 a b^2 x^2 \left (A \left (-7 b^2+106 b c x+92 c^2 x^2\right )+18 b B x (b+c x)\right )+105 A b^4 x^3 (b+c x)\right )}{x^3 \sqrt{a+x (b+c x)}}-3 \left (b^2-4 a c\right ) \left (5 A \left (7 b^3-12 a b c\right )+6 a B \left (4 a c-5 b^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{48 a^{9/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^4*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((2*Sqrt[a]*(-16*a^4*c*(2*A + 3*B*x) + 105*A*b^4*x^3*(b + c*x) + 4*a^3*(3*B*x*(b^2 + 10*b*c*x - 12*c^2*x^2) +
2*A*(b^2 + 7*b*c*x + 16*c^2*x^2)) - 5*a*b^2*x^2*(18*b*B*x*(b + c*x) + A*(-7*b^2 + 106*b*c*x + 92*c^2*x^2)) + 2
*a^2*x*(3*b*B*x*(-5*b^2 + 62*b*c*x + 52*c^2*x^2) + A*(-7*b^3 - 86*b^2*c*x + 244*b*c^2*x^2 + 128*c^3*x^3))))/(x
^3*Sqrt[a + x*(b + c*x)]) - 3*(b^2 - 4*a*c)*(6*a*B*(-5*b^2 + 4*a*c) + 5*A*(7*b^3 - 12*a*b*c))*ArcTanh[(2*a + b
*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(48*a^(9/2)*(-b^2 + 4*a*c))

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Maple [B]  time = 0.012, size = 708, normalized size = 2.2 \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^4/(c*x^2+b*x+a)^(3/2),x)

[Out]

-1/3*A/a/x^3/(c*x^2+b*x+a)^(1/2)+7/12*A/a^2*b/x^2/(c*x^2+b*x+a)^(1/2)-35/24*A/a^3*b^2/x/(c*x^2+b*x+a)^(1/2)-35
/16*A/a^4*b^3/(c*x^2+b*x+a)^(1/2)+35/8*A/a^4*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c+35/16*A/a^4*b^5/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)+35/16*A/a^(9/2)*b^3*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-115/6*A/a^3*b^2*c^2/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-115/12*A/a^3*b^3*c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+15/4*A/a^3*b*c/(c*x^2+b*x+
a)^(1/2)-15/4*A/a^(7/2)*b*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+4/3*A/a^2*c/x/(c*x^2+b*x+a)^(1/2)+32
/3*A/a^2*c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16/3*A/a^2*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b-1/2*B/a/x^2/(c
*x^2+b*x+a)^(1/2)+5/4*B/a^2*b/x/(c*x^2+b*x+a)^(1/2)+15/8*B/a^3*b^2/(c*x^2+b*x+a)^(1/2)-15/4*B/a^3*b^3/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)*x*c-15/8*B/a^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-15/8*B/a^(7/2)*b^2*ln((2*a+b*x+2*a^
(1/2)*(c*x^2+b*x+a)^(1/2))/x)+13*B/a^2*b*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+13/2*B/a^2*b^2*c/(4*a*c-b^2)/(c
*x^2+b*x+a)^(1/2)-3/2*B/a^2*c/(c*x^2+b*x+a)^(1/2)+3/2*B/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x
)

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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^4/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.6571, size = 2403, normalized size = 7.58 \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^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a*b^3)*c^2 + 5*(6*B*a*b^4 - 7*A*b^5)*c)*x^5 +
(30*B*a*b^5 - 35*A*b^6 + 48*(2*B*a^3*b - 5*A*a^2*b^2)*c^2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x^4 + (30*B*a^2*
b^4 - 35*A*a*b^5 + 48*(2*B*a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 25*A*a^2*b^3)*c)*x^3)*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*(8*A*a^4*b^2 - 32*A*a^5*
c + (256*A*a^3*c^3 + 4*(78*B*a^3*b - 115*A*a^2*b^2)*c^2 - 15*(6*B*a^2*b^3 - 7*A*a*b^4)*c)*x^4 - (90*B*a^2*b^4
- 105*A*a*b^5 + 8*(18*B*a^4 - 61*A*a^3*b)*c^2 - 2*(186*B*a^3*b^2 - 265*A*a^2*b^3)*c)*x^3 - (30*B*a^3*b^3 - 35*
A*a^2*b^4 - 128*A*a^4*c^2 - 4*(30*B*a^4*b - 43*A*a^3*b^2)*c)*x^2 + 2*(6*B*a^4*b^2 - 7*A*a^3*b^3 - 4*(6*B*a^5 -
7*A*a^4*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^5*b^2*c - 4*a^6*c^2)*x^5 + (a^5*b^3 - 4*a^6*b*c)*x^4 + (a^6*b^2 -
4*a^7*c)*x^3), 1/48*(3*((48*(2*B*a^3 - 5*A*a^2*b)*c^3 - 8*(18*B*a^2*b^2 - 25*A*a*b^3)*c^2 + 5*(6*B*a*b^4 - 7*
A*b^5)*c)*x^5 + (30*B*a*b^5 - 35*A*b^6 + 48*(2*B*a^3*b - 5*A*a^2*b^2)*c^2 - 8*(18*B*a^2*b^3 - 25*A*a*b^4)*c)*x
^4 + (30*B*a^2*b^4 - 35*A*a*b^5 + 48*(2*B*a^4 - 5*A*a^3*b)*c^2 - 8*(18*B*a^3*b^2 - 25*A*a^2*b^3)*c)*x^3)*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*(8*A*a^4*b^2 - 32*A*a^5
*c + (256*A*a^3*c^3 + 4*(78*B*a^3*b - 115*A*a^2*b^2)*c^2 - 15*(6*B*a^2*b^3 - 7*A*a*b^4)*c)*x^4 - (90*B*a^2*b^4
- 105*A*a*b^5 + 8*(18*B*a^4 - 61*A*a^3*b)*c^2 - 2*(186*B*a^3*b^2 - 265*A*a^2*b^3)*c)*x^3 - (30*B*a^3*b^3 - 35
*A*a^2*b^4 - 128*A*a^4*c^2 - 4*(30*B*a^4*b - 43*A*a^3*b^2)*c)*x^2 + 2*(6*B*a^4*b^2 - 7*A*a^3*b^3 - 4*(6*B*a^5
- 7*A*a^4*b)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^5*b^2*c - 4*a^6*c^2)*x^5 + (a^5*b^3 - 4*a^6*b*c)*x^4 + (a^6*b^2
- 4*a^7*c)*x^3)]

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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**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.47213, size = 1077, normalized size = 3.4 \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^4/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

2*((B*a^5*b^3*c - A*a^4*b^4*c - 3*B*a^6*b*c^2 + 4*A*a^5*b^2*c^2 - 2*A*a^6*c^3)*x/(a^8*b^2 - 4*a^9*c) + (B*a^5*
b^4 - A*a^4*b^5 - 4*B*a^6*b^2*c + 5*A*a^5*b^3*c + 2*B*a^7*c^2 - 5*A*a^6*b*c^2)/(a^8*b^2 - 4*a^9*c))/sqrt(c*x^2
+ b*x + a) + 1/8*(30*B*a*b^2 - 35*A*b^3 - 24*B*a^2*c + 60*A*a*b*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)/sqrt(-a))/(sqrt(-a)*a^4) - 1/24*(42*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2 - 57*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^5*A*b^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*c + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*A*a*b*c + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^2*b*sqrt(c) - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*A*a*b^2*sqrt(c) + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^2*c^(3/2) - 96*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^3*B*a^2*b^2 + 136*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*A*a^2*b*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c) + 144*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*A*a^2*b^2*sqrt(c) - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^3*c^(3/2) + 54*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*B*a^3*b^2 - 87*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 + 24*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*B*a^4*c - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b*c + 96*B*a^4*b*sqrt(c) - 144*A*a^3*b^2*sq
rt(c) + 80*A*a^4*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^3*a^4)