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

Optimal. Leaf size=284 \[ -\frac{5 \left (8 a b B \left (12 a c+b^2\right )-A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 x \left (A \left (4 a c+b^2\right )+8 a b B\right )+4 a (4 a B+A b)\right )}{96 a x^3}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c x \left (A \left (12 a c+b^2\right )+16 a b B\right )-A \left (b^3-20 a b c\right )+8 a B \left (4 a c+b^2\right )\right )}{64 a x}+\frac{5}{8} \sqrt{c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4} \]

[Out]

(-5*(8*a*B*(b^2 + 4*a*c) - A*(b^3 - 20*a*b*c) - 2*c*(16*a*b*B + A*(b^2 + 12*a*c))*x)*Sqrt[a + b*x + c*x^2])/(6
4*a*x) - (5*(4*a*(A*b + 4*a*B) + 3*(8*a*b*B + A*(b^2 + 4*a*c))*x)*(a + b*x + c*x^2)^(3/2))/(96*a*x^3) - ((A -
2*B*x)*(a + b*x + c*x^2)^(5/2))/(4*x^4) - (5*(8*a*b*B*(b^2 + 12*a*c) - A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*ArcT
anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(3/2)) + (5*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*A
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/8

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Rubi [A]  time = 0.38643, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {812, 810, 843, 621, 206, 724} \[ -\frac{5 \left (8 a b B \left (12 a c+b^2\right )-A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 x \left (A \left (4 a c+b^2\right )+8 a b B\right )+4 a (4 a B+A b)\right )}{96 a x^3}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c x \left (A \left (12 a c+b^2\right )+16 a b B\right )-A \left (b^3-20 a b c\right )+8 a B \left (4 a c+b^2\right )\right )}{64 a x}+\frac{5}{8} \sqrt{c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(8*a*B*(b^2 + 4*a*c) - A*(b^3 - 20*a*b*c) - 2*c*(16*a*b*B + A*(b^2 + 12*a*c))*x)*Sqrt[a + b*x + c*x^2])/(6
4*a*x) - (5*(4*a*(A*b + 4*a*B) + 3*(8*a*b*B + A*(b^2 + 4*a*c))*x)*(a + b*x + c*x^2)^(3/2))/(96*a*x^3) - ((A -
2*B*x)*(a + b*x + c*x^2)^(5/2))/(4*x^4) - (5*(8*a*b*B*(b^2 + 12*a*c) - A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*ArcT
anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(3/2)) + (5*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*A
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/8

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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])

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]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^5} \, dx &=-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac{5}{16} \int \frac{(-2 (A b+4 a B)-4 (b B+A c) x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac{5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac{5 \int \frac{\left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )+2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{x^2} \, dx}{64 a}\\ &=-\frac{5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 a x}-\frac{5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac{5 \int \frac{-8 a b B \left (b^2+12 a c\right )+A \left (b^4-24 a b^2 c-48 a^2 c^2\right )-16 a c \left (3 b^2 B+4 A b c+4 a B c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx}{128 a}\\ &=-\frac{5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 a x}-\frac{5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac{1}{8} \left (5 c \left (3 b^2 B+4 A b c+4 a B c\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx+\frac{\left (5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{128 a}\\ &=-\frac{5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 a x}-\frac{5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}+\frac{1}{4} \left (5 c \left (3 b^2 B+4 A b c+4 a B c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )-\frac{\left (5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\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 )}{64 a}\\ &=-\frac{5 \left (8 a B \left (b^2+4 a c\right )-A \left (b^3-20 a b c\right )-2 c \left (16 a b B+A \left (b^2+12 a c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 a x}-\frac{5 \left (4 a (A b+4 a B)+3 \left (8 a b B+A \left (b^2+4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 a x^3}-\frac{(A-2 B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^4}-\frac{5 \left (8 a b B \left (b^2+12 a c\right )-A \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{3/2}}+\frac{5}{8} \sqrt{c} \left (3 b^2 B+4 A b c+4 a B c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.727573, size = 255, normalized size = 0.9 \[ -\frac{\sqrt{a+x (b+c x)} \left (8 a^2 x \left (17 A b+27 A c x+26 b B x+56 B c x^2\right )+16 a^3 (3 A+4 B x)+2 a x^2 \left (A \left (59 b^2+278 b c x-96 c^2 x^2\right )-12 B x \left (-11 b^2+18 b c x+4 c^2 x^2\right )\right )+15 A b^3 x^3\right )}{192 a x^4}+\frac{5 \left (A \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-8 a b B \left (12 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{128 a^{3/2}}+\frac{5}{8} \sqrt{c} \left (4 a B c+4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + x*(b + c*x)]*(15*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 8*a^2*x*(17*A*b + 26*b*B*x + 27*A*c*x + 56*B*c*
x^2) + 2*a*x^2*(A*(59*b^2 + 278*b*c*x - 96*c^2*x^2) - 12*B*x*(-11*b^2 + 18*b*c*x + 4*c^2*x^2))))/(192*a*x^4) +
(5*(-8*a*b*B*(b^2 + 12*a*c) + A*(b^4 - 24*a*b^2*c - 48*a^2*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b
+ c*x)])])/(128*a^(3/2)) + (5*Sqrt[c]*(3*b^2*B + 4*A*b*c + 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x
*(b + c*x)])])/8

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Maple [B]  time = 0.016, size = 1094, normalized size = 3.9 \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)*(c*x^2+b*x+a)^(5/2)/x^5,x)

[Out]

5/24*B/a^2*b^2*c*(c*x^2+b*x+a)^(3/2)*x+1/8*B/a^3*b^2*c*(c*x^2+b*x+a)^(5/2)*x+5/8*B/a*b^2*(c*x^2+b*x+a)^(1/2)*x
*c-5/192*A/a^3*b^3*c*(c*x^2+b*x+a)^(3/2)*x-19/48*A/a^3*b*c/x*(c*x^2+b*x+a)^(7/2)+35/48*A/a^2*b*c^2*(c*x^2+b*x+
a)^(3/2)*x+25/16*A/a*b*c^2*(c*x^2+b*x+a)^(1/2)*x+19/48*A/a^3*b*c^2*(c*x^2+b*x+a)^(5/2)*x-5/64*A/a^2*b^3*(c*x^2
+b*x+a)^(1/2)*x*c-1/64*A/a^4*b^3*c*(c*x^2+b*x+a)^(5/2)*x+5/128*A/a^(3/2)*b^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+
a)^(1/2))/x)-5/192*A/a^3*b^4*(c*x^2+b*x+a)^(3/2)-1/64*A/a^4*b^4*(c*x^2+b*x+a)^(5/2)-5/64*A/a^2*b^4*(c*x^2+b*x+
a)^(1/2)+5/2*A*b*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/8*A/a*c^2*(c*x^2+b*x+a)^(3/2)+3/8*A/a^2
*c^2*(c*x^2+b*x+a)^(5/2)-1/4*A/a/x^4*(c*x^2+b*x+a)^(7/2)-15/8*A*a^(1/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a
)^(1/2))/x)-1/3*B/a/x^3*(c*x^2+b*x+a)^(7/2)+5/2*B*a*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/16*B
/a^(1/2)*b^3*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/2*B*c^2*(c*x^2+b*x+a)^(1/2)*x+15/8*B*b^2*c^(1/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5*B*b*c*(c*x^2+b*x+a)^(1/2)+5/24*B/a^2*b^3*(c*x^2+b*x+a)^(3/2)+1/8
*B/a^3*b^3*(c*x^2+b*x+a)^(5/2)+5/8*B/a*b^3*(c*x^2+b*x+a)^(1/2)+15/8*A*c^2*(c*x^2+b*x+a)^(1/2)+65/96*A/a^2*b^2*
c*(c*x^2+b*x+a)^(3/2)+1/96*A/a^3*b^2/x^2*(c*x^2+b*x+a)^(7/2)+1/64*A/a^4*b^3/x*(c*x^2+b*x+a)^(7/2)+37/96*A/a^3*
b^2*c*(c*x^2+b*x+a)^(5/2)-3/8*A/a^2*c/x^2*(c*x^2+b*x+a)^(7/2)+55/32*A/a*b^2*c*(c*x^2+b*x+a)^(1/2)+25/12*B/a*b*
c*(c*x^2+b*x+a)^(3/2)-1/12*B/a^2*b/x^2*(c*x^2+b*x+a)^(7/2)-1/8*B/a^3*b^2/x*(c*x^2+b*x+a)^(7/2)+17/12*B/a^2*b*c
*(c*x^2+b*x+a)^(5/2)-15/4*B*a^(1/2)*b*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+4/3*B/a^2*c^2*(c*x^2+b*x
+a)^(5/2)*x-4/3*B/a^2*c/x*(c*x^2+b*x+a)^(7/2)+5/3*B/a*c^2*(c*x^2+b*x+a)^(3/2)*x-15/16*A/a^(1/2)*b^2*c*ln((2*a+
b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/24*A/a^2*b/x^3*(c*x^2+b*x+a)^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 29.5851, size = 3058, normalized size = 10.77 \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)*(c*x^2+b*x+a)^(5/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(240*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(c)*x^4*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 +
b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*b^2)*c)*sqr
t(a)*x^4*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*(96
*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 + 15*A*a*b^3 + 4*(112*B*a^3 +
139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a^4 + 17*A*a^3*b)*x)*sqrt(c*x^
2 + b*x + a))/(a^2*x^4), -1/768*(480*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(-c)*x^4*arctan(1/2*sqrt(c*x^2
+ b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*
b + A*a*b^2)*c)*sqrt(a)*x^4*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*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 + 15*A*a*b^
3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a^4 + 17*A*
a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4), 1/384*(15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^2*b + A*a*
b^2)*c)*sqrt(-a)*x^4*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)) + 120*(3*B
*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(c)*x^4*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x
+ b)*sqrt(c) - 4*a*c) + 2*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b^2 +
15*A*a*b^3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8*B*a
^4 + 17*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4), 1/384*(15*(8*B*a*b^3 - A*b^4 + 48*A*a^2*c^2 + 24*(4*B*a^
2*b + A*a*b^2)*c)*sqrt(-a)*x^4*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))
- 240*(3*B*a^2*b^2 + 4*(B*a^3 + A*a^2*b)*c)*sqrt(-c)*x^4*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)
/(c^2*x^2 + b*c*x + a*c)) + 2*(96*B*a^2*c^2*x^5 - 48*A*a^4 + 48*(9*B*a^2*b*c + 4*A*a^2*c^2)*x^4 - (264*B*a^2*b
^2 + 15*A*a*b^3 + 4*(112*B*a^3 + 139*A*a^2*b)*c)*x^3 - 2*(104*B*a^3*b + 59*A*a^2*b^2 + 108*A*a^3*c)*x^2 - 8*(8
*B*a^4 + 17*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**5,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**5, x)

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Giac [B]  time = 1.69552, size = 1570, normalized size = 5.53 \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)*(c*x^2+b*x+a)^(5/2)/x^5,x, algorithm="giac")

[Out]

1/4*(2*B*c^2*x + (9*B*b*c^2 + 4*A*c^3)/c)*sqrt(c*x^2 + b*x + a) - 5/8*(3*B*b^2*c + 4*B*a*c^2 + 4*A*b*c^2)*log(
abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/sqrt(c) + 5/64*(8*B*a*b^3 - A*b^4 + 96*B*a^2*b*c + 24*
A*a*b^2*c + 48*A*a^2*c^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a) + 1/192*(264*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*b^4 + 864*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^7*B*a^2*b*c + 792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c + 432*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^7*A*a^2*c^2 + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^2*b^2*sqrt(c) + 384*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^6*A*a*b^3*sqrt(c) + 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*c^(3/2) + 230
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b*c^(3/2) - 584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*b^3
+ 73*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b^4 - 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b*c - 60
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*c^2 - 230
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^3*b^2*sqrt(c) - 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^4*c
^(3/2) - 3456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^3*b*c^(3/2) + 440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*a^3*b^3 - 55*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b^4 + 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B
*a^4*b*c + 1320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^2*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A
*a^4*c^2 + 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^4*b^2*sqrt(c) + 2432*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*B*a^5*c^(3/2) + 3584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^4*b*c^(3/2) - 120*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*B*a^4*b^3 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*B*a^5*b*c - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^2*c + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*A*a^5*c^2 - 384*B*a^5*b^2*sqrt(c) - 896*B*a^6*c^(3/2) - 896*A*a^5*b*c^(3/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2 - a)^4*a)