3.945 \(\int \frac{(A+B x) (a+b x+c x^2)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=346 \[ \frac{\sqrt{a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{128 a^2 x}+\frac{\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{192 a^2 x^3}+\frac{1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5} \]

[Out]

((10*a*b*B*(b^2 - 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(b^2 + 12*a*c) - A*(3*b^3 - 28*
a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(128*a^2*x) + ((4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c
) - A*(3*b^3 - 20*a*b*c))*x)*(a + b*x + c*x^2)^(3/2))/(192*a^2*x^3) - ((8*a*A + 5*(A*b + 2*a*B)*x)*(a + b*x +
c*x^2)^(5/2))/(40*a*x^5) + ((10*a*B*(b^4 - 24*a*b^2*c - 48*a^2*c^2) - A*(3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2))*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(5/2)) + (c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/2

________________________________________________________________________________________

Rubi [A]  time = 0.491384, antiderivative size = 346, 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 = {810, 812, 843, 621, 206, 724} \[ \frac{\sqrt{a+b x+c x^2} \left (-A \left (128 a^2 c^2-28 a b^2 c+3 b^4\right )+2 c x \left (10 a B \left (12 a c+b^2\right )-A \left (3 b^3-28 a b c\right )\right )+10 a b B \left (b^2-20 a c\right )\right )}{128 a^2 x}+\frac{\left (10 a B \left (-48 a^2 c^2-24 a b^2 c+b^4\right )-A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (4 a \left (-16 a A c-10 a b B+3 A b^2\right )-3 x \left (10 a B \left (4 a c+b^2\right )-A \left (3 b^3-20 a b c\right )\right )\right )}{192 a^2 x^3}+\frac{1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{\left (a+b x+c x^2\right )^{5/2} (5 x (2 a B+A b)+8 a A)}{40 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

((10*a*b*B*(b^2 - 20*a*c) - A*(3*b^4 - 28*a*b^2*c + 128*a^2*c^2) + 2*c*(10*a*B*(b^2 + 12*a*c) - A*(3*b^3 - 28*
a*b*c))*x)*Sqrt[a + b*x + c*x^2])/(128*a^2*x) + ((4*a*(3*A*b^2 - 10*a*b*B - 16*a*A*c) - 3*(10*a*B*(b^2 + 4*a*c
) - A*(3*b^3 - 20*a*b*c))*x)*(a + b*x + c*x^2)^(3/2))/(192*a^2*x^3) - ((8*a*A + 5*(A*b + 2*a*B)*x)*(a + b*x +
c*x^2)^(5/2))/(40*a*x^5) + ((10*a*B*(b^4 - 24*a*b^2*c - 48*a^2*c^2) - A*(3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2))*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(5/2)) + (c^(3/2)*(5*b*B + 2*A*c)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/2

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 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 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^6} \, dx &=-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}-\frac{\int \frac{\left (\frac{1}{2} \left (3 A b^2-10 a b B-16 a A c\right )-(A b+10 a B) c x\right ) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx}{8 a}\\ &=\frac{\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac{\int \frac{\left (\frac{1}{4} \left (-10 a b B \left (b^2-20 a c\right )+4 A \left (\frac{3 b^4}{4}-7 a b^2 c+32 a^2 c^2\right )\right )+\frac{1}{2} c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{x^2} \, dx}{32 a^2}\\ &=\frac{\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 a^2 x}+\frac{\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}-\frac{\int \frac{\frac{1}{4} \left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-4 A \left (\frac{3 b^5}{4}-10 a b^3 c+60 a^2 b c^2\right )\right )-32 a^2 c^2 (5 b B+2 A c) x}{x \sqrt{a+b x+c x^2}} \, dx}{64 a^2}\\ &=\frac{\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 a^2 x}+\frac{\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac{1}{2} \left (c^2 (5 b B+2 A c)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx-\frac{\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{256 a^2}\\ &=\frac{\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 a^2 x}+\frac{\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\left (c^2 (5 b B+2 A c)\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 (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\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 )}{128 a^2}\\ &=\frac{\left (10 a b B \left (b^2-20 a c\right )-A \left (3 b^4-28 a b^2 c+128 a^2 c^2\right )+2 c \left (10 a B \left (b^2+12 a c\right )-A \left (3 b^3-28 a b c\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 a^2 x}+\frac{\left (4 a \left (3 A b^2-10 a b B-16 a A c\right )-3 \left (10 a B \left (b^2+4 a c\right )-A \left (3 b^3-20 a b c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 a^2 x^3}-\frac{(8 a A+5 (A b+2 a B) x) \left (a+b x+c x^2\right )^{5/2}}{40 a x^5}+\frac{\left (10 a B \left (b^4-24 a b^2 c-48 a^2 c^2\right )-A \left (3 b^5-40 a b^3 c+240 a^2 b c^2\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{256 a^{5/2}}+\frac{1}{2} c^{3/2} (5 b B+2 A c) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.889059, size = 289, normalized size = 0.84 \[ -\frac{\sqrt{a+x (b+c x)} \left (4 a^2 x^2 \left (2 A \left (93 b^2+311 b c x+368 c^2 x^2\right )+5 B x \left (59 b^2+278 b c x-96 c^2 x^2\right )\right )+16 a^3 x (A (63 b+88 c x)+5 B x (17 b+27 c x))+96 a^4 (4 A+5 B x)+30 a b^2 x^3 (A (b+18 c x)+5 b B x)-45 A b^4 x^4\right )}{1920 a^2 x^5}-\frac{\left (A \left (240 a^2 b c^2-40 a b^3 c+3 b^5\right )+10 a B \left (48 a^2 c^2+24 a b^2 c-b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{256 a^{5/2}}+\frac{1}{2} c^{3/2} (2 A c+5 b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^6,x]

[Out]

-(Sqrt[a + x*(b + c*x)]*(-45*A*b^4*x^4 + 96*a^4*(4*A + 5*B*x) + 30*a*b^2*x^3*(5*b*B*x + A*(b + 18*c*x)) + 16*a
^3*x*(5*B*x*(17*b + 27*c*x) + A*(63*b + 88*c*x)) + 4*a^2*x^2*(5*B*x*(59*b^2 + 278*b*c*x - 96*c^2*x^2) + 2*A*(9
3*b^2 + 311*b*c*x + 368*c^2*x^2))))/(1920*a^2*x^5) - ((10*a*B*(-b^4 + 24*a*b^2*c + 48*a^2*c^2) + A*(3*b^5 - 40
*a*b^3*c + 240*a^2*b*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])])/(256*a^(5/2)) + (c^(3/2)*(5
*b*B + 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/2

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Maple [B]  time = 0.02, size = 1371, normalized size = 4. \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^6,x)

[Out]

-5/192*B/a^3*b^3*c*(c*x^2+b*x+a)^(3/2)*x-19/48*B/a^3*b*c/x*(c*x^2+b*x+a)^(7/2)+35/48*B/a^2*b*c^2*(c*x^2+b*x+a)
^(3/2)*x+25/16*B/a*b*c^2*(c*x^2+b*x+a)^(1/2)*x+19/48*B/a^3*b*c^2*(c*x^2+b*x+a)^(5/2)*x-11/160*A/a^4*b^2*c^2*(c
*x^2+b*x+a)^(5/2)*x+11/160*A/a^4*b^2*c/x*(c*x^2+b*x+a)^(7/2)-13/96*A/a^3*b^2*c^2*(c*x^2+b*x+a)^(3/2)*x+1/128*A
/a^4*b^4*c*(c*x^2+b*x+a)^(3/2)*x+3/640*A/a^5*b^4*c*(c*x^2+b*x+a)^(5/2)*x+3/128*A/a^3*b^4*(c*x^2+b*x+a)^(1/2)*x
*c-7/32*A/a^2*b^2*c^2*(c*x^2+b*x+a)^(1/2)*x+19/240*A/a^3*b*c/x^2*(c*x^2+b*x+a)^(7/2)-5/64*B/a^2*b^3*(c*x^2+b*x
+a)^(1/2)*x*c-1/64*B/a^4*b^3*c*(c*x^2+b*x+a)^(5/2)*x-5/64*B/a^2*b^4*(c*x^2+b*x+a)^(1/2)-5/192*B/a^3*b^4*(c*x^2
+b*x+a)^(3/2)-1/64*B/a^4*b^4*(c*x^2+b*x+a)^(5/2)-1/5*A/a/x^5*(c*x^2+b*x+a)^(7/2)-3/256*A/a^(5/2)*b^5*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+3/128*A/a^3*b^5*(c*x^2+b*x+a)^(1/2)+1/128*A/a^4*b^5*(c*x^2+b*x+a)^(3/2)+3
/640*A/a^5*b^5*(c*x^2+b*x+a)^(5/2)+5/2*B*b*c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+5/8*B/a*c^2*(c*
x^2+b*x+a)^(3/2)+3/8*B/a^2*c^2*(c*x^2+b*x+a)^(5/2)-1/4*B/a/x^4*(c*x^2+b*x+a)^(7/2)-15/8*B*a^(1/2)*c^2*ln((2*a+
b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/128*B/a^(3/2)*b^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+15/8*B
*c^2*(c*x^2+b*x+a)^(1/2)+A*c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+37/96*B/a^3*b^2*c*(c*x^2+b*x+a)
^(5/2)+55/32*B/a*b^2*c*(c*x^2+b*x+a)^(1/2)-15/16*B/a^(1/2)*b^2*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
-3/8*B/a^2*c/x^2*(c*x^2+b*x+a)^(7/2)+1/24*B/a^2*b/x^3*(c*x^2+b*x+a)^(7/2)+65/96*B/a^2*b^2*c*(c*x^2+b*x+a)^(3/2
)+3/40*A/a^2*b/x^4*(c*x^2+b*x+a)^(7/2)-21/320*A/a^4*b^3*c*(c*x^2+b*x+a)^(5/2)-17/64*A/a^2*b^3*c*(c*x^2+b*x+a)^
(1/2)+31/48*A/a^2*b*c^2*(c*x^2+b*x+a)^(3/2)+109/240*A/a^3*b*c^2*(c*x^2+b*x+a)^(5/2)+23/16*A/a*b*c^2*(c*x^2+b*x
+a)^(1/2)+5/32*A/a^(3/2)*b^3*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/16*A/a^(1/2)*b*c^2*ln((2*a+b*x
+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-2/15*A/a^2*c/x^3*(c*x^2+b*x+a)^(7/2)+8/15*A/a^3*c^3*(c*x^2+b*x+a)^(5/2)*x-8
/15*A/a^3*c^2/x*(c*x^2+b*x+a)^(7/2)+2/3*A/a^2*c^3*(c*x^2+b*x+a)^(3/2)*x+A/a*c^3*(c*x^2+b*x+a)^(1/2)*x-1/80*A/a
^3*b^2/x^3*(c*x^2+b*x+a)^(7/2)-23/192*A/a^3*b^3*c*(c*x^2+b*x+a)^(3/2)-1/320*A/a^4*b^3/x^2*(c*x^2+b*x+a)^(7/2)-
3/640*A/a^5*b^4/x*(c*x^2+b*x+a)^(7/2)+1/96*B/a^3*b^2/x^2*(c*x^2+b*x+a)^(7/2)+1/64*B/a^4*b^3/x*(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 45.411, size = 3448, normalized size = 9.97 \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^6,x, algorithm="fricas")

[Out]

[1/7680*(1920*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(c)*x^5*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*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a
*b^3)*c)*sqrt(a)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27
*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 9
3*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/7680*(3840
*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(-c)*x^5*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*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*sqrt(a)*x^
5*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*(1920*B*a^
3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4
- 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A
*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/3840*(15*(10*B*a*b^4 - 3*A*b^
5 - 240*(2*B*a^3 + A*a^2*b)*c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*sqrt(-a)*x^5*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)) - 960*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(c)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150
*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^
2*b^3 + 4*(270*B*a^4 + 311*A*a^3*b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 +
 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x + a))/(a^3*x^5), -1/3840*(15*(10*B*a*b^4 - 3*A*b^5 - 240*(2*B*a^3 + A*a^2*b)*
c^2 - 40*(6*B*a^2*b^2 - A*a*b^3)*c)*sqrt(-a)*x^5*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)) + 1920*(5*B*a^3*b*c + 2*A*a^3*c^2)*sqrt(-c)*x^5*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*(1920*B*a^3*c^2*x^5 - 384*A*a^5 - (150*B*a^2*b^3 - 45*A*a*b^4 + 2944*A*
a^3*c^2 + 20*(278*B*a^3*b + 27*A*a^2*b^2)*c)*x^4 - 2*(590*B*a^3*b^2 + 15*A*a^2*b^3 + 4*(270*B*a^4 + 311*A*a^3*
b)*c)*x^3 - 8*(170*B*a^4*b + 93*A*a^3*b^2 + 176*A*a^4*c)*x^2 - 48*(10*B*a^5 + 21*A*a^4*b)*x)*sqrt(c*x^2 + b*x
+ a))/(a^3*x^5)]

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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^{6}}\, 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**6,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**6, x)

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Giac [B]  time = 2.28287, size = 2120, normalized size = 6.13 \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^6,x, algorithm="giac")

[Out]

sqrt(c*x^2 + b*x + a)*B*c^2 - 1/2*(5*B*b*c^2 + 2*A*c^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)
+ b))/sqrt(c) - 1/128*(10*B*a*b^4 - 3*A*b^5 - 240*B*a^2*b^2*c + 40*A*a*b^3*c - 480*B*a^3*c^2 - 240*A*a^2*b*c^2
)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/1920*(150*(sqrt(c)*x - sqrt(c*x^2 +
 b*x + a))^9*B*a*b^4*sqrt(c) - 45*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*b^5*sqrt(c) + 7920*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^9*B*a^2*b^2*c^(3/2) + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a*b^3*c^(3/2) + 4320*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^3*c^(5/2) + 7920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^2*b*c^(5/2) +
 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^2*b^3*c + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^3*b*
c^2 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^2*b^2*c^2 + 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*
A*a^3*c^3 + 580*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^2*b^4*sqrt(c) + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*A*a*b^5*sqrt(c) - 13920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^3*b^2*c^(3/2) + 6160*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^7*A*a^2*b^3*c^(3/2) - 4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^4*c^(5/2) - 2400*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^3*b*c^(5/2) - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^3*b^3*c + 38
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^2*b^4*c - 57600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^4*b*c^2
 - 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^4*c^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b^
4*sqrt(c) + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^5*sqrt(c) + 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*B*a^4*b^2*c^(3/2) + 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^3*b^3*c^(3/2) + 19200*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*A*a^4*b*c^(5/2) + 70400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^5*b*c^2 + 19200*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^4*b^2*c^2 + 35840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^5*c^3 + 700
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^4*b^4*sqrt(c) - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^5
*sqrt(c) - 16800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^5*b^2*c^(3/2) + 2800*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^3*A*a^4*b^3*c^(3/2) + 4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^6*c^(5/2) + 2400*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*A*a^5*b*c^(5/2) - 44800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^6*b*c^2 - 17920*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*A*a^6*c^3 - 150*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^5*b^4*sqrt(c) + 45*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b^5*sqrt(c) + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^6*b^2*c^(3/2)
- 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*b^3*c^(3/2) - 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^7*c
^(5/2) + 3600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^6*b*c^(5/2) + 8960*B*a^7*b*c^2 + 5888*A*a^7*c^3)/(((sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^5*a^2*sqrt(c))