3.10.77 \(\int \frac {A+B x}{x^3 (a+b x+c x^2)^{5/2}} \, dx\) [977]

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

[Out]

2/3*(A*b^2-a*b*B-2*a*A*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/x^2/(c*x^2+b*x+a)^(3/2)-5/8*(-4*A*a*c+7*A*b^2-4*B*a*b
)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(9/2)-2/3*(4*a*b*B*(-6*a*c+b^2)-A*(40*a^2*c^2-42*a*b^2*
c+7*b^4)-c*(-36*A*a*b*c+7*A*b^3+32*B*a^2*c-4*B*a*b^2)*x)/a^2/(-4*a*c+b^2)^2/x^2/(c*x^2+b*x+a)^(1/2)+1/6*(4*a*b
*B*(-28*a*c+5*b^2)-A*(240*a^2*c^2-216*a*b^2*c+35*b^4))*(c*x^2+b*x+a)^(1/2)/a^3/(-4*a*c+b^2)^2/x^2-1/12*(4*a*B*
(128*a^2*c^2-100*a*b^2*c+15*b^4)-A*(1296*a^2*b*c^2-760*a*b^3*c+105*b^5))*(c*x^2+b*x+a)^(1/2)/a^4/(-4*a*c+b^2)^
2/x

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Rubi [A]
time = 0.31, antiderivative size = 381, 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 = {836, 848, 820, 738, 212} \begin {gather*} -\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 (-A \left (40 a^2 c^2-42 a b^2 c+7 b^4\right )-c x \left (32 a^2 B c-36 a A b c-4 a b^2 B+7 A b^3\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 {\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 {\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 {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}} \end {gather*}

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 212

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

Rule 738

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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 836

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 848

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^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 ) \text {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*}

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Mathematica [A]
time = 2.76, size = 424, normalized size = 1.11 \begin {gather*} \frac {\frac {\sqrt {a} \left (-96 a^5 c^2 (A+2 B x)+105 A b^5 x^3 (b+c x)^2+16 a^4 c \left (A \left (3 b^2+21 b c x-40 c^2 x^2\right )-2 B x \left (-3 b^2+32 b c x+24 c^2 x^2\right )\right )-10 a b^3 x^2 (b+c x) \left (6 b B x (b+c x)+A \left (-14 b^2+83 b c x+76 c^2 x^2\right )\right )-2 a^3 \left (3 A \left (b^4+28 b^3 c x-392 b^2 c^2 x^2-224 b c^3 x^3+80 c^4 x^4\right )+2 B x \left (3 b^4-148 b^3 c x+48 b^2 c^2 x^2+312 b c^3 x^3+128 c^4 x^4\right )\right )+a^2 b x \left (40 b B x \left (-2 b^3+9 b^2 c x+21 b c^2 x^2+10 c^3 x^3\right )+3 A \left (7 b^4-372 b^3 c x+232 b^2 c^2 x^2+1008 b c^3 x^3+432 c^4 x^4\right )\right )\right )}{\left (b^2-4 a c\right )^2 x^2 (a+x (b+c x))^{3/2}}+105 A b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+60 a (b B+A c) \tanh ^{-1}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{12 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a]*(-96*a^5*c^2*(A + 2*B*x) + 105*A*b^5*x^3*(b + c*x)^2 + 16*a^4*c*(A*(3*b^2 + 21*b*c*x - 40*c^2*x^2) -
2*B*x*(-3*b^2 + 32*b*c*x + 24*c^2*x^2)) - 10*a*b^3*x^2*(b + c*x)*(6*b*B*x*(b + c*x) + A*(-14*b^2 + 83*b*c*x +
76*c^2*x^2)) - 2*a^3*(3*A*(b^4 + 28*b^3*c*x - 392*b^2*c^2*x^2 - 224*b*c^3*x^3 + 80*c^4*x^4) + 2*B*x*(3*b^4 -
148*b^3*c*x + 48*b^2*c^2*x^2 + 312*b*c^3*x^3 + 128*c^4*x^4)) + a^2*b*x*(40*b*B*x*(-2*b^3 + 9*b^2*c*x + 21*b*c^
2*x^2 + 10*c^3*x^3) + 3*A*(7*b^4 - 372*b^3*c*x + 232*b^2*c^2*x^2 + 1008*b*c^3*x^3 + 432*c^4*x^4))))/((b^2 - 4*
a*c)^2*x^2*(a + x*(b + c*x))^(3/2)) + 105*A*b^2*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] + 60*a*(b
*B + A*c)*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(12*a^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(779\) vs. \(2(355)=710\).
time = 0.83, size = 780, normalized size = 2.05

method	result	size
default	\(A \left (-\frac {1}{2 a \,x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {7 b \left (-\frac {1}{a x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {5 b \left (\frac {1}{3 a \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 a}+\frac {\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}-\frac {4 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{a}\right )}{4 a}-\frac {5 c \left (\frac {1}{3 a \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 a}+\frac {\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}\right )+B \left (-\frac {1}{a x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {5 b \left (\frac {1}{3 a \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 a}+\frac {\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}-\frac {4 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{a}\right )\)	\(780\)
risch	\(\text {Expression too large to display}\)	\(8765\)

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,method=_RETURNVERBOSE)

[Out]

A*(-1/2/a/x^2/(c*x^2+b*x+a)^(3/2)-7/4*b/a*(-1/a/x/(c*x^2+b*x+a)^(3/2)-5/2*b/a*(1/3/a/(c*x^2+b*x+a)^(3/2)-1/2*b
/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))+1/a*(1/a
/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+
a)^(1/2))/x)))-4*c/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+
a)^(1/2)))-5/2*c/a*(1/3/a/(c*x^2+b*x+a)^(3/2)-1/2*b/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4
*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))+1/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x))))+B*(-1/a/x/(c*x^2+b*x+a)^(3/2)-5/2*b/a*(1/3/
a/(c*x^2+b*x+a)^(3/2)-1/2*b/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c
*x^2+b*x+a)^(1/2))+1/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*
a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)))-4*c/a*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b
^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))

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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^3/(c*x^2+b*x+a)^(5/2),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 deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1024 vs. \(2 (355) = 710\).
time = 9.98, size = 2057, normalized size = 5.40 \begin {gather*} \text {Too large to display} \end {gather*}

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (355) = 710\).
time = 1.06, size = 765, normalized size = 2.01 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {{\left (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}\right )} x}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}} + \frac {3 \, {\left (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}\right )}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )} x + \frac {3 \, {\left (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}\right )}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )} x + \frac {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}}{a^{15} b^{4} - 8 \, a^{16} b^{2} c + 16 \, a^{17} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (4 \, B a b - 7 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + 13 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 24 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{4}} \end {gather*}

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]

-2/3*((((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/(a^15*b^4 - 8*a^16*b^2*c + 16*a^17*c^2) + 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)/(a^15*b^4 - 8*a^16*b^2*c + 16*a^17*c
^2))*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)/(a^15*b^4 - 8*a^16*b^2*c + 16*a^17*c^2))*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)/(a^15*b^4 - 8
*a^16*b^2*c + 16*a^17*c^2))/(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 - s
qrt(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 - sqr
t(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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