3.977 \(\int \frac{A+B x}{x^3 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=381 \[ \frac{\sqrt{a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}-\frac{2 \left (-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{9/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[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^2*(a + b*x + c*x^2)^(3/2)) - (2*(4*a*b*
B*(b^2 - 6*a*c) - A*(7*b^4 - 42*a*b^2*c + 40*a^2*c^2) - c*(7*A*b^3 - 4*a*b^2*B - 36*a*A*b*c + 32*a^2*B*c)*x))/
(3*a^2*(b^2 - 4*a*c)^2*x^2*Sqrt[a + b*x + c*x^2]) + ((4*a*b*B*(5*b^2 - 28*a*c) - A*(35*b^4 - 216*a*b^2*c + 240
*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(6*a^3*(b^2 - 4*a*c)^2*x^2) - ((4*a*B*(15*b^4 - 100*a*b^2*c + 128*a^2*c^2) -
A*(105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2))*Sqrt[a + b*x + c*x^2])/(12*a^4*(b^2 - 4*a*c)^2*x) - (5*(7*A*b^2 -
4*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(9/2))

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Rubi [A]  time = 0.461281, antiderivative size = 381, 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} \[ \frac{\sqrt{a+b x+c x^2} \left (4 a b B \left (5 b^2-28 a c\right )-A \left (240 a^2 c^2-216 a b^2 c+35 b^4\right )\right )}{6 a^3 x^2 \left (b^2-4 a c\right )^2}-\frac{\sqrt{a+b x+c x^2} \left (4 a B \left (128 a^2 c^2-100 a b^2 c+15 b^4\right )-A \left (1296 a^2 b c^2-760 a b^3 c+105 b^5\right )\right )}{12 a^4 x \left (b^2-4 a c\right )^2}-\frac{2 \left (-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\right )-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )+4 a b B \left (b^2-6 a c\right )\right )}{3 a^2 x^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{5 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{9/2}}+\frac{2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^3*(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^2*(a + b*x + c*x^2)^(3/2)) - (2*(4*a*b*
B*(b^2 - 6*a*c) - A*(7*b^4 - 42*a*b^2*c + 40*a^2*c^2) - c*(7*A*b^3 - 4*a*b^2*B - 36*a*A*b*c + 32*a^2*B*c)*x))/
(3*a^2*(b^2 - 4*a*c)^2*x^2*Sqrt[a + b*x + c*x^2]) + ((4*a*b*B*(5*b^2 - 28*a*c) - A*(35*b^4 - 216*a*b^2*c + 240
*a^2*c^2))*Sqrt[a + b*x + c*x^2])/(6*a^3*(b^2 - 4*a*c)^2*x^2) - ((4*a*B*(15*b^4 - 100*a*b^2*c + 128*a^2*c^2) -
A*(105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2))*Sqrt[a + b*x + c*x^2])/(12*a^4*(b^2 - 4*a*c)^2*x) - (5*(7*A*b^2 -
4*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*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^3 \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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (-7 A b^2+4 a b B+20 a A c\right )-4 (A b-2 a B) c x}{x^3 \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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{\frac{1}{4} \left (-4 a b B \left (5 b^2-28 a c\right )+4 A \left (\frac{35 b^4}{4}-54 a b^2 c+60 a^2 c^2\right )\right )-c \left (4 a B \left (b^2-8 a c\right )-A \left (7 b^3-36 a b c\right )\right ) x}{x^3 \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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt{a+b x+c x^2}}+\frac{\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac{2 \int \frac{\frac{1}{8} \left (105 A b^5-60 a b^4 B-760 a A b^3 c+400 a^2 b^2 B c+1296 a^2 A b c^2-512 a^3 B c^2\right )-\frac{1}{4} c \left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{3 a^3 \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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt{a+b x+c x^2}}+\frac{\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac{\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt{a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}+\frac{\left (5 \left (7 A b^2-4 a b B-4 a A c\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{8 a^4}\\ &=\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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt{a+b x+c x^2}}+\frac{\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac{\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt{a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}-\frac{\left (5 \left (7 A b^2-4 a b B-4 a A c\right )\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 )}{4 a^4}\\ &=\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^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (4 a b B \left (b^2-6 a c\right )-A \left (7 b^4-42 a b^2 c+40 a^2 c^2\right )-c \left (7 A b^3-4 a b^2 B-36 a A b c+32 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 x^2 \sqrt{a+b x+c x^2}}+\frac{\left (4 a b B \left (5 b^2-28 a c\right )-A \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )\right ) \sqrt{a+b x+c x^2}}{6 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac{\left (4 a B \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )-A \left (105 b^5-760 a b^3 c+1296 a^2 b c^2\right )\right ) \sqrt{a+b x+c x^2}}{12 a^4 \left (b^2-4 a c\right )^2 x}-\frac{5 \left (7 A b^2-4 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.80291, size = 343, normalized size = 0.9 \[ \frac{\frac{2 \sqrt{a} \sqrt{a+x (b+c x)} \left (-8 a^2 b \left (54 A b c+162 A c^2 x+5 b^2 B+50 b B c x\right )+32 a^3 c (15 A c+7 b B+16 B c x)+10 a b^3 (7 A b+76 A c x+6 b B x)-105 A b^5 x\right )+15 x^2 \left (b^2-4 a c\right )^2 \left (-4 a A c-4 a b B+7 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{2 a^{7/2} \left (4 a c-b^2\right )}-\frac{8 \left (A \left (40 a^2 c^2-42 a b^2 c-36 a b c^2 x+7 b^3 c x+7 b^4\right )+4 a B \left (6 a b c+8 a c^2 x-b^2 c x-b^3\right )\right )}{a \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{8 A \left (-2 a c+b^2+b c x\right )-8 a B (b+2 c x)}{(a+x (b+c x))^{3/2}}}{12 a x^2 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

((-8*a*B*(b + 2*c*x) + 8*A*(b^2 - 2*a*c + b*c*x))/(a + x*(b + c*x))^(3/2) - (8*(4*a*B*(-b^3 + 6*a*b*c - b^2*c*
x + 8*a*c^2*x) + A*(7*b^4 - 42*a*b^2*c + 40*a^2*c^2 + 7*b^3*c*x - 36*a*b*c^2*x)))/(a*(-b^2 + 4*a*c)*Sqrt[a + x
*(b + c*x)]) + (2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-105*A*b^5*x + 10*a*b^3*(7*A*b + 6*b*B*x + 76*A*c*x) + 32*a^3
*c*(7*b*B + 15*A*c + 16*B*c*x) - 8*a^2*b*(5*b^2*B + 54*A*b*c + 50*b*B*c*x + 162*A*c^2*x)) + 15*(b^2 - 4*a*c)^2
*(7*A*b^2 - 4*a*b*B - 4*a*A*c)*x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(2*a^(7/2)*(-b^2 +
4*a*c)))/(12*a*(b^2 - 4*a*c)*x^2)

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Maple [B]  time = 0.011, size = 1051, normalized size = 2.8 \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^3/(c*x^2+b*x+a)^(5/2),x)

[Out]

-35/12*A/a^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c-70/3*A/a^3*b^3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-35
/4*A/a^4*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c+11*A/a^2*b*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+88*A/a^2*b*c
^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5*A/a^3*c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+5/3*B/a^2*b^2/(4*a*c-b^
2)/(c*x^2+b*x+a)^(3/2)*x*c+40/3*B/a^2*b^2*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5*B/a^3*b^2/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*x*c-35/3*A/a^3*b^4*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+11/2*A/a^2*b^2*c/(4*a*c-b^2)/(c*x^2+b*x
+a)^(3/2)+44*A/a^2*b^2*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/2*A/a^3*c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+2
0/3*B/a^2*b^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-16/3*B/a*c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-8/3*B/a*c/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-128/3*B/a*c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-64/3*B/a*c^2/(4*a*c-b^2)^2/(c
*x^2+b*x+a)^(1/2)*b+35/24*A/a^3*b^2/(c*x^2+b*x+a)^(3/2)+35/8*A/a^4*b^2/(c*x^2+b*x+a)^(1/2)-35/8*A/a^(9/2)*b^2*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/2*B/a^(7/2)*b*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-5/
6*A/a^2*c/(c*x^2+b*x+a)^(3/2)-5/2*A/a^3*c/(c*x^2+b*x+a)^(1/2)+5/2*A/a^(7/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x
+a)^(1/2))/x)-1/2*A/a/x^2/(c*x^2+b*x+a)^(3/2)-B/a/x/(c*x^2+b*x+a)^(3/2)-5/6*B/a^2*b/(c*x^2+b*x+a)^(3/2)-5/2*B/
a^3*b/(c*x^2+b*x+a)^(1/2)+7/4*A/a^2*b/x/(c*x^2+b*x+a)^(3/2)-35/24*A/a^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-35
/8*A/a^4*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/2*B/a^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/6*B/a^2*b^3/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)

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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^3/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 48.3296, size = 4424, normalized size = 11.61 \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^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*((64*A*a^3*c^5 + 16*(4*B*a^3*b - 9*A*a^2*b^2)*c^4 - 4*(8*B*a^2*b^3 - 15*A*a*b^4)*c^3 + (4*B*a*b^5 -
7*A*b^6)*c^2)*x^6 + 2*(64*A*a^3*b*c^4 + 16*(4*B*a^3*b^2 - 9*A*a^2*b^3)*c^3 - 4*(8*B*a^2*b^4 - 15*A*a*b^5)*c^2
+ (4*B*a*b^6 - 7*A*b^7)*c)*x^5 + (4*B*a*b^7 - 7*A*b^8 - 24*A*a^2*b^4*c^2 + 128*A*a^4*c^4 + 32*(4*B*a^4*b - 7*A
*a^3*b^2)*c^3 - 2*(12*B*a^2*b^5 - 23*A*a*b^6)*c)*x^4 + 2*(4*B*a^2*b^6 - 7*A*a*b^7 + 64*A*a^4*b*c^3 + 16*(4*B*a
^4*b^2 - 9*A*a^3*b^3)*c^2 - 4*(8*B*a^3*b^4 - 15*A*a^2*b^5)*c)*x^3 + (4*B*a^3*b^5 - 7*A*a^2*b^6 + 64*A*a^5*c^3
+ 16*(4*B*a^5*b - 9*A*a^4*b^2)*c^2 - 4*(8*B*a^4*b^3 - 15*A*a^3*b^4)*c)*x^2)*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*(6*A*a^4*b^4 - 48*A*a^5*b^2*c + 96*A*a
^6*c^2 + (16*(32*B*a^4 - 81*A*a^3*b)*c^4 - 40*(10*B*a^3*b^2 - 19*A*a^2*b^3)*c^3 + 15*(4*B*a^2*b^4 - 7*A*a*b^5)
*c^2)*x^5 + 6*(80*A*a^4*c^4 + 8*(26*B*a^4*b - 63*A*a^3*b^2)*c^3 - 5*(28*B*a^3*b^3 - 53*A*a^2*b^4)*c^2 + 5*(4*B
*a^2*b^5 - 7*A*a*b^6)*c)*x^4 + 3*(20*B*a^2*b^6 - 35*A*a*b^7 + 64*(4*B*a^5 - 7*A*a^4*b)*c^3 + 8*(8*B*a^4*b^2 -
29*A*a^3*b^3)*c^2 - 10*(12*B*a^3*b^4 - 23*A*a^2*b^5)*c)*x^3 + 4*(20*B*a^3*b^5 - 35*A*a^2*b^6 + 160*A*a^5*c^3 +
4*(64*B*a^5*b - 147*A*a^4*b^2)*c^2 - (148*B*a^4*b^3 - 279*A*a^3*b^4)*c)*x^2 + 3*(4*B*a^4*b^4 - 7*A*a^3*b^5 +
16*(4*B*a^6 - 7*A*a^5*b)*c^2 - 8*(4*B*a^5*b^2 - 7*A*a^4*b^3)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^5*b^4*c^2 - 8*a^
6*b^2*c^3 + 16*a^7*c^4)*x^6 + 2*(a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*x^5 + (a^5*b^6 - 6*a^6*b^4*c + 32*a
^8*c^3)*x^4 + 2*(a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*x^3 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*x^2), -1/24*
(15*((64*A*a^3*c^5 + 16*(4*B*a^3*b - 9*A*a^2*b^2)*c^4 - 4*(8*B*a^2*b^3 - 15*A*a*b^4)*c^3 + (4*B*a*b^5 - 7*A*b^
6)*c^2)*x^6 + 2*(64*A*a^3*b*c^4 + 16*(4*B*a^3*b^2 - 9*A*a^2*b^3)*c^3 - 4*(8*B*a^2*b^4 - 15*A*a*b^5)*c^2 + (4*B
*a*b^6 - 7*A*b^7)*c)*x^5 + (4*B*a*b^7 - 7*A*b^8 - 24*A*a^2*b^4*c^2 + 128*A*a^4*c^4 + 32*(4*B*a^4*b - 7*A*a^3*b
^2)*c^3 - 2*(12*B*a^2*b^5 - 23*A*a*b^6)*c)*x^4 + 2*(4*B*a^2*b^6 - 7*A*a*b^7 + 64*A*a^4*b*c^3 + 16*(4*B*a^4*b^2
- 9*A*a^3*b^3)*c^2 - 4*(8*B*a^3*b^4 - 15*A*a^2*b^5)*c)*x^3 + (4*B*a^3*b^5 - 7*A*a^2*b^6 + 64*A*a^5*c^3 + 16*(
4*B*a^5*b - 9*A*a^4*b^2)*c^2 - 4*(8*B*a^4*b^3 - 15*A*a^3*b^4)*c)*x^2)*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*(6*A*a^4*b^4 - 48*A*a^5*b^2*c + 96*A*a^6*c^2 + (16*(32*B*a
^4 - 81*A*a^3*b)*c^4 - 40*(10*B*a^3*b^2 - 19*A*a^2*b^3)*c^3 + 15*(4*B*a^2*b^4 - 7*A*a*b^5)*c^2)*x^5 + 6*(80*A*
a^4*c^4 + 8*(26*B*a^4*b - 63*A*a^3*b^2)*c^3 - 5*(28*B*a^3*b^3 - 53*A*a^2*b^4)*c^2 + 5*(4*B*a^2*b^5 - 7*A*a*b^6
)*c)*x^4 + 3*(20*B*a^2*b^6 - 35*A*a*b^7 + 64*(4*B*a^5 - 7*A*a^4*b)*c^3 + 8*(8*B*a^4*b^2 - 29*A*a^3*b^3)*c^2 -
10*(12*B*a^3*b^4 - 23*A*a^2*b^5)*c)*x^3 + 4*(20*B*a^3*b^5 - 35*A*a^2*b^6 + 160*A*a^5*c^3 + 4*(64*B*a^5*b - 147
*A*a^4*b^2)*c^2 - (148*B*a^4*b^3 - 279*A*a^3*b^4)*c)*x^2 + 3*(4*B*a^4*b^4 - 7*A*a^3*b^5 + 16*(4*B*a^6 - 7*A*a^
5*b)*c^2 - 8*(4*B*a^5*b^2 - 7*A*a^4*b^3)*c)*x)*sqrt(c*x^2 + b*x + a))/((a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c
^4)*x^6 + 2*(a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*x^5 + (a^5*b^6 - 6*a^6*b^4*c + 32*a^8*c^3)*x^4 + 2*(a^6
*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*x^3 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*x^2)]

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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**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.21129, size = 1033, normalized size = 2.71 \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^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-1/3*((((6*B*a^12*b^4*c^2 - 9*A*a^11*b^5*c^2 - 38*B*a^13*b^2*c^3 + 62*A*a^12*b^3*c^3 + 40*B*a^14*c^4 - 96*A*a^
13*b*c^4)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(4*B*a^12*b^5*c - 6*A*a^11*b^6*c - 27*B*a^13*b^3*c^2 + 44
*A*a^12*b^4*c^2 + 36*B*a^14*b*c^3 - 80*A*a^13*b^2*c^3 + 16*A*a^14*c^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x
+ 3*(2*B*a^12*b^6 - 3*A*a^11*b^7 - 12*B*a^13*b^4*c + 20*A*a^12*b^5*c + 8*B*a^14*b^2*c^2 - 25*A*a^13*b^3*c^2 +
16*B*a^15*c^3 - 20*A*a^14*b*c^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (7*B*a^13*b^5 - 10*A*a^12*b^6 - 50
*B*a^14*b^3*c + 78*A*a^13*b^4*c + 80*B*a^15*b*c^2 - 162*A*a^14*b^2*c^2 + 56*A*a^15*c^3)/(b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) - 5/4*(4*B*a*b - 7*A*b^2 + 4*A*a*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b - 11*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*A*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*c + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*B*a^2*sqrt(c) - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b*sqrt(c) - 4*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*B*a^2*b + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2
*c - 8*B*a^3*sqrt(c) + 24*A*a^2*b*sqrt(c))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^2*a^4)