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

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

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Rubi [A]  time = 0.39, antiderivative size = 317, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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])

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 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 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])

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*}

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Mathematica [A]  time = 0.43, size = 294, normalized size = 0.93 \begin {gather*} \frac {\frac {2 \sqrt {a} \left (-16 a^4 c (2 A+3 B x)+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 (-7 b^3-86 b^2 c x+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 )-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 )} \end {gather*}

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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IntegrateAlgebraic [A]  time = 2.11, size = 380, normalized size = 1.20 \begin {gather*} -\frac {15 \left (2 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {\left (-24 a^2 B c-35 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {-32 a^4 A c-48 a^4 B c x+8 a^3 A b^2+56 a^3 A b c x+128 a^3 A c^2 x^2+12 a^3 b^2 B x+120 a^3 b B c x^2-144 a^3 B c^2 x^3-14 a^2 A b^3 x-172 a^2 A b^2 c x^2+488 a^2 A b c^2 x^3+256 a^2 A c^3 x^4-30 a^2 b^3 B x^2+372 a^2 b^2 B c x^3+312 a^2 b B c^2 x^4+35 a A b^4 x^2-530 a A b^3 c x^3-460 a A b^2 c^2 x^4-90 a b^4 B x^3-90 a b^3 B c x^4+105 A b^5 x^3+105 A b^4 c x^4}{24 a^4 x^3 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^3*A*b^2 - 32*a^4*A*c - 14*a^2*A*b^3*x + 12*a^3*b^2*B*x + 56*a^3*A*b*c*x - 48*a^4*B*c*x + 35*a*A*b^4*x^2 -
30*a^2*b^3*B*x^2 - 172*a^2*A*b^2*c*x^2 + 120*a^3*b*B*c*x^2 + 128*a^3*A*c^2*x^2 + 105*A*b^5*x^3 - 90*a*b^4*B*x
^3 - 530*a*A*b^3*c*x^3 + 372*a^2*b^2*B*c*x^3 + 488*a^2*A*b*c^2*x^3 - 144*a^3*B*c^2*x^3 + 105*A*b^4*c*x^4 - 90*
a*b^3*B*c*x^4 - 460*a*A*b^2*c^2*x^4 + 312*a^2*b*B*c^2*x^4 + 256*a^2*A*c^3*x^4)/(24*a^4*(-b^2 + 4*a*c)*x^3*Sqrt
[a + b*x + c*x^2]) + ((-35*A*b^3 - 24*a^2*B*c)*ArcTanh[(Sqrt[c]*x - Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(8*a^(9/2
)) - (15*(b^2*B + 2*A*b*c)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2])/Sqrt[a]])/(4*a^(7/2))

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fricas [A]  time = 2.06, size = 1093, normalized size = 3.45

result too large to display

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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giac [B]  time = 0.28, size = 798, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (\frac {{\left (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}\right )} x}{a^{8} b^{2} - 4 \, a^{9} c} + \frac {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}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (30 \, B a b^{2} - 35 \, A b^{3} - 24 \, B a^{2} c + 60 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4}} - \frac {42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} - 57 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt {c} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 136 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b^{2} \sqrt {c} - 192 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 54 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 87 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 96 \, B a^{4} b \sqrt {c} - 144 \, A a^{3} b^{2} \sqrt {c} + 80 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{4}} \end {gather*}

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)

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maple [B]  time = 0.07, size = 708, normalized size = 2.23 \begin {gather*} \frac {32 A \,c^{3} x}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {115 A \,b^{2} c^{2} x}{6 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {35 A \,b^{4} c x}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}+\frac {13 B b \,c^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {15 B \,b^{3} c x}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {16 A b \,c^{2}}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {115 A \,b^{3} c}{12 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {35 A \,b^{5}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}+\frac {13 B \,b^{2} c}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {15 B \,b^{4}}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {15 A b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {7}{2}}}+\frac {35 A \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {9}{2}}}+\frac {3 B c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {15 B \,b^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {7}{2}}}+\frac {15 A b c}{4 \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {35 A \,b^{3}}{16 \sqrt {c \,x^{2}+b x +a}\, a^{4}}-\frac {3 B c}{2 \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {15 B \,b^{2}}{8 \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {4 A c}{3 \sqrt {c \,x^{2}+b x +a}\, a^{2} x}-\frac {35 A \,b^{2}}{24 \sqrt {c \,x^{2}+b x +a}\, a^{3} x}+\frac {5 B b}{4 \sqrt {c \,x^{2}+b x +a}\, a^{2} x}+\frac {7 A b}{12 \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{2}}-\frac {B}{2 \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}}-\frac {A}{3 \sqrt {c \,x^{2}+b x +a}\, a \,x^{3}} \end {gather*}

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)*c*x+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((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*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((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x)+4/3*A*c/a^2/x/(c*x^2+b*x+a)^(1/2)+32
/3*A*c^3/a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16/3*A*c^2/a^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)*c*x-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((b*x+2*a+2*(c
*x^2+b*x+a)^(1/2)*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*c/a^2/(c*x^2+b*x+a)^(1/2)+3/2*B*c/a^(5/2)*ln((b*x+2*a+2*(c*x^2+b*x+a)^(1/2)*a^(1/2))/x
)

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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)/x^4/(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 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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