3.948 \(\int \frac{(A+B x) (a+b x+c x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=288 \[ -\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{384 a^3 x^6}+\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{6144 a^4 x^4}-\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-16 a b B+9 A b^2\right )}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{32768 a^{11/2}}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8} \]

[Out]

(-5*(b^2 - 4*a*c)^2*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(16384*a^5*x^2) + (5*(b^
2 - 4*a*c)*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(6144*a^4*x^4) - ((9*A*b^2 - 16
*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(5/2))/(384*a^3*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(8*a*x^8) +
((9*A*b - 16*a*B)*(a + b*x + c*x^2)^(7/2))/(112*a^2*x^7) + (5*(b^2 - 4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(32768*a^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.247271, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {834, 806, 720, 724, 206} \[ -\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{384 a^3 x^6}+\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{6144 a^4 x^4}-\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-16 a b B+9 A b^2\right )}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{32768 a^{11/2}}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]

[Out]

(-5*(b^2 - 4*a*c)^2*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(16384*a^5*x^2) + (5*(b^
2 - 4*a*c)*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(6144*a^4*x^4) - ((9*A*b^2 - 16
*a*b*B - 4*a*A*c)*(2*a + b*x)*(a + b*x + c*x^2)^(5/2))/(384*a^3*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(8*a*x^8) +
((9*A*b - 16*a*B)*(a + b*x + c*x^2)^(7/2))/(112*a^2*x^7) + (5*(b^2 - 4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(32768*a^(11/2))

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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx &=-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}-\frac{\int \frac{\left (\frac{1}{2} (9 A b-16 a B)+A c x\right ) \left (a+b x+c x^2\right )^{5/2}}{x^8} \, dx}{8 a}\\ &=-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}+\frac{\left (9 A b^2-16 a b B-4 a A c\right ) \int \frac{\left (a+b x+c x^2\right )^{5/2}}{x^7} \, dx}{32 a^2}\\ &=-\frac{\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{\left (5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right )\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx}{768 a^3}\\ &=\frac{5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{6144 a^4 x^4}-\frac{\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}+\frac{\left (5 \left (b^2-4 a c\right )^2 \left (9 A b^2-16 a b B-4 a A c\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx}{4096 a^4}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \sqrt{a+b x+c x^2}}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{6144 a^4 x^4}-\frac{\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (9 A b^2-16 a b B-4 a A c\right )\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{32768 a^5}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \sqrt{a+b x+c x^2}}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{6144 a^4 x^4}-\frac{\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}+\frac{\left (5 \left (b^2-4 a c\right )^3 \left (9 A b^2-16 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 )}{16384 a^5}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \sqrt{a+b x+c x^2}}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right ) \left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{6144 a^4 x^4}-\frac{\left (9 A b^2-16 a b B-4 a A c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{5/2}}{384 a^3 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}+\frac{5 \left (b^2-4 a c\right )^3 \left (9 A b^2-16 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 )}{32768 a^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.843218, size = 242, normalized size = 0.84 \[ -\frac{\frac{\left (-2 a A c-8 a b B+\frac{9 A b^2}{2}\right ) \left (256 a^{5/2} (2 a+b x) (a+x (b+c x))^{5/2}-5 x^2 \left (b^2-4 a c\right ) \left (16 a^{3/2} (2 a+b x) (a+x (b+c x))^{3/2}-3 x^2 \left (b^2-4 a c\right ) \left (2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)}-x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{6144 a^{9/2} x^6}+\frac{(16 a B-9 A b) (a+x (b+c x))^{7/2}}{14 a x^7}+\frac{A (a+x (b+c x))^{7/2}}{x^8}}{8 a} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]

[Out]

-((A*(a + x*(b + c*x))^(7/2))/x^8 + ((-9*A*b + 16*a*B)*(a + x*(b + c*x))^(7/2))/(14*a*x^7) + (((9*A*b^2)/2 - 8
*a*b*B - 2*a*A*c)*(256*a^(5/2)*(2*a + b*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*x^2*(16*a^(3/2)*(2*a + b*
x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*x^2*(2*Sqrt[a]*(2*a + b*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*
x^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + x*(b + c*x)])]))))/(6144*a^(9/2)*x^6))/(8*a)

________________________________________________________________________________________

Maple [B]  time = 0.046, size = 2263, normalized size = 7.9 \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^9,x)

[Out]

5/3072*B/a^6*b^7*(c*x^2+b*x+a)^(3/2)+1/1024*B/a^7*b^7*(c*x^2+b*x+a)^(5/2)+5/1024*B/a^5*b^7*(c*x^2+b*x+a)^(1/2)
-5/2048*B/a^(9/2)*b^7*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/7*B/a/x^7*(c*x^2+b*x+a)^(7/2)+45/32768*A
/a^(11/2)*b^8*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-5/384*A/a^3*c^4*(c*x^2+b*x+a)^(3/2)-1/128*A/a^4*c^
4*(c*x^2+b*x+a)^(5/2)-5/128*A/a^2*c^4*(c*x^2+b*x+a)^(1/2)+5/128*A/a^(3/2)*c^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x
+a)^(1/2))/x)-15/16384*A/a^7*b^8*(c*x^2+b*x+a)^(3/2)-9/16384*A/a^8*b^8*(c*x^2+b*x+a)^(5/2)-45/16384*A/a^6*b^8*
(c*x^2+b*x+a)^(1/2)-1/8*A*(c*x^2+b*x+a)^(7/2)/a/x^8-1/96*A/a^3*c*b/x^5*(c*x^2+b*x+a)^(7/2)+5/256*A/a^4*c^4*b*(
c*x^2+b*x+a)^(3/2)*x-15/16384*A/a^7*b^7*c*(c*x^2+b*x+a)^(3/2)*x-55/6144*A/a^6*b^4*c/x^2*(c*x^2+b*x+a)^(7/2)+18
5/12288*A/a^6*b^5*c^2*(c*x^2+b*x+a)^(3/2)*x+1/48*B/a^3*b*c/x^4*(c*x^2+b*x+a)^(7/2)+1/32*B/a^4*b*c^2/x^2*(c*x^2
+b*x+a)^(7/2)+5/64*B/a^4*b^2*c^3*(c*x^2+b*x+a)^(3/2)*x+5/64*B/a^5*b^2*c^3*(c*x^2+b*x+a)^(5/2)*x-5/64*B/a^5*b^2
*c^2/x*(c*x^2+b*x+a)^(7/2)+5/64*B/a^3*b^2*c^3*(c*x^2+b*x+a)^(1/2)*x-5/384*B/a^6*b^4*c^2*(c*x^2+b*x+a)^(5/2)*x+
5/384*B/a^6*b^4*c/x*(c*x^2+b*x+a)^(7/2)-45/16384*A/a^6*b^7*(c*x^2+b*x+a)^(1/2)*x*c-9/16384*A/a^8*b^7*c*(c*x^2+
b*x+a)^(5/2)*x-55/1024*A/a^4*b^3*c^3*(c*x^2+b*x+a)^(1/2)*x+145/3072*A/a^6*b^3*c^2/x*(c*x^2+b*x+a)^(7/2)-145/30
72*A/a^6*b^3*c^3*(c*x^2+b*x+a)^(5/2)*x-155/3072*A/a^5*b^3*c^3*(c*x^2+b*x+a)^(3/2)*x-7/512*A/a^5*b^2*c^2/x^2*(c
*x^2+b*x+a)^(7/2)-1/128*A/a^4*b^2*c/x^4*(c*x^2+b*x+a)^(7/2)+95/4096*A/a^5*b^5*c^2*(c*x^2+b*x+a)^(1/2)*x+13/768
*A/a^5*b^3*c/x^3*(c*x^2+b*x+a)^(7/2)+31/4096*A/a^7*b^5*c^2*(c*x^2+b*x+a)^(5/2)*x-31/4096*A/a^7*b^5*c/x*(c*x^2+
b*x+a)^(7/2)-5/128*B/a^4*b^4*c^2*(c*x^2+b*x+a)^(1/2)*x+5/1024*B/a^5*b^6*(c*x^2+b*x+a)^(1/2)*x*c+1/1024*B/a^7*b
^6*c*(c*x^2+b*x+a)^(5/2)*x+5/3072*B/a^6*b^6*c*(c*x^2+b*x+a)^(3/2)*x+1/64*B/a^5*b^3*c/x^2*(c*x^2+b*x+a)^(7/2)-5
/192*B/a^5*b^4*c^2*(c*x^2+b*x+a)^(3/2)*x-1/32*B/a^4*b^2*c/x^3*(c*x^2+b*x+a)^(7/2)-1/128*A/a^4*c^2*b/x^3*(c*x^2
+b*x+a)^(7/2)+5/256*A/a^5*c^4*b*(c*x^2+b*x+a)^(5/2)*x-5/256*A/a^5*c^3*b/x*(c*x^2+b*x+a)^(7/2)+5/256*A/a^3*c^4*
b*(c*x^2+b*x+a)^(1/2)*x-5/32*B/a^2*b*c^3*(c*x^2+b*x+a)^(1/2)-35/1536*B/a^5*b^5*c*(c*x^2+b*x+a)^(3/2)-19/1536*B
/a^6*b^5*c*(c*x^2+b*x+a)^(5/2)-25/512*B/a^4*b^5*c*(c*x^2+b*x+a)^(1/2)+5/64*B/a^4*b^3*c^2*(c*x^2+b*x+a)^(3/2)+1
/16*B/a^5*b^3*c^2*(c*x^2+b*x+a)^(5/2)+5/32*B/a^3*b^3*c^2*(c*x^2+b*x+a)^(1/2)+1/64*B/a^4*b^3/x^4*(c*x^2+b*x+a)^
(7/2)-1/24*B/a^3*b^2/x^5*(c*x^2+b*x+a)^(7/2)+1/12*B/a^2*b/x^6*(c*x^2+b*x+a)^(7/2)-5/96*B/a^3*b*c^3*(c*x^2+b*x+
a)^(3/2)-1/32*B/a^4*b*c^3*(c*x^2+b*x+a)^(5/2)-1/384*B/a^5*b^4/x^3*(c*x^2+b*x+a)^(7/2)-1/1536*B/a^6*b^5/x^2*(c*
x^2+b*x+a)^(7/2)-1/1024*B/a^7*b^6/x*(c*x^2+b*x+a)^(7/2)+3/8192*A/a^7*b^6/x^2*(c*x^2+b*x+a)^(7/2)+9/16384*A/a^8
*b^7/x*(c*x^2+b*x+a)^(7/2)+9/112*A/a^2*b/x^7*(c*x^2+b*x+a)^(7/2)+65/512*A/a^3*b^2*c^3*(c*x^2+b*x+a)^(1/2)+325/
24576*A/a^6*b^6*c*(c*x^2+b*x+a)^(3/2)+59/8192*A/a^7*b^6*c*(c*x^2+b*x+a)^(5/2)+235/8192*A/a^5*b^6*c*(c*x^2+b*x+
a)^(1/2)-305/6144*A/a^5*b^4*c^2*(c*x^2+b*x+a)^(3/2)-235/6144*A/a^6*b^4*c^2*(c*x^2+b*x+a)^(5/2)+75/1024*A/a^(7/
2)*b^4*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/128*A/a^(5/2)*b^2*c^3*ln((2*a+b*x+2*a^(1/2)*(c*x^2
+b*x+a)^(1/2))/x)-35/2048*A/a^(9/2)*b^6*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/48*A/a^2*c/x^6*(c*x^
2+b*x+a)^(7/2)+1/192*A/a^3*c^2/x^4*(c*x^2+b*x+a)^(7/2)+1/128*A/a^4*c^3/x^2*(c*x^2+b*x+a)^(7/2)-205/2048*A/a^4*
b^4*c^2*(c*x^2+b*x+a)^(1/2)-9/1024*A/a^5*b^4/x^4*(c*x^2+b*x+a)^(7/2)+3/128*A/a^4*b^3/x^5*(c*x^2+b*x+a)^(7/2)-3
/64*A/a^3*b^2/x^6*(c*x^2+b*x+a)^(7/2)+25/512*A/a^4*b^2*c^3*(c*x^2+b*x+a)^(3/2)+17/512*A/a^5*b^2*c^3*(c*x^2+b*x
+a)^(5/2)+3/2048*A/a^6*b^5/x^3*(c*x^2+b*x+a)^(7/2)+5/32*B/a^(3/2)*b*c^3*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1
/2))/x)+15/512*B/a^(7/2)*b^5*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/128*B/a^(5/2)*b^3*c^2*ln((2*a+
b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)

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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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 59.8696, size = 2603, normalized size = 9.04 \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^9,x, algorithm="fricas")

[Out]

[-1/1376256*(105*(16*B*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^4*b - 3*A*a^3*b^2)*c^3 + 96*(8*B*a^3*b^3 -
5*A*a^2*b^4)*c^2 - 16*(12*B*a^2*b^5 - 7*A*a*b^6)*c)*sqrt(a)*x^8*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*(43008*A*a^8 - (1680*B*a^2*b^6 - 945*A*a*b^7 - 192*(256*B
*a^5 - 221*A*a^4*b)*c^3 + 112*(528*B*a^4*b^2 - 337*A*a^3*b^3)*c^2 - 140*(128*B*a^3*b^4 - 75*A*a^2*b^5)*c)*x^7
+ 2*(560*B*a^3*b^5 - 315*A*a^2*b^6 + 6720*A*a^5*c^3 + 48*(304*B*a^5*b - 199*A*a^4*b^2)*c^2 - 28*(192*B*a^4*b^3
- 113*A*a^3*b^4)*c)*x^6 - 8*(112*B*a^4*b^4 - 63*A*a^3*b^5 - 48*(384*B*a^6 + 29*A*a^5*b)*c^2 - 8*(120*B*a^5*b^
2 - 71*A*a^4*b^3)*c)*x^5 + 16*(48*B*a^5*b^3 - 27*A*a^4*b^4 + 6608*A*a^6*c^2 + 8*(1576*B*a^6*b + 27*A*a^5*b^2)*
c)*x^4 + 128*(592*B*a^6*b^2 + 3*A*a^5*b^3 + 4*(288*B*a^7 + 307*A*a^6*b)*c)*x^3 + 256*(464*B*a^7*b + 243*A*a^6*
b^2 + 476*A*a^7*c)*x^2 + 3072*(16*B*a^8 + 33*A*a^7*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^8), 1/688128*(105*(16*B
*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^4*b - 3*A*a^3*b^2)*c^3 + 96*(8*B*a^3*b^3 - 5*A*a^2*b^4)*c^2 - 16
*(12*B*a^2*b^5 - 7*A*a*b^6)*c)*sqrt(-a)*x^8*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*(43008*A*a^8 - (1680*B*a^2*b^6 - 945*A*a*b^7 - 192*(256*B*a^5 - 221*A*a^4*b)*c^3 + 112*(528*B
*a^4*b^2 - 337*A*a^3*b^3)*c^2 - 140*(128*B*a^3*b^4 - 75*A*a^2*b^5)*c)*x^7 + 2*(560*B*a^3*b^5 - 315*A*a^2*b^6 +
6720*A*a^5*c^3 + 48*(304*B*a^5*b - 199*A*a^4*b^2)*c^2 - 28*(192*B*a^4*b^3 - 113*A*a^3*b^4)*c)*x^6 - 8*(112*B*
a^4*b^4 - 63*A*a^3*b^5 - 48*(384*B*a^6 + 29*A*a^5*b)*c^2 - 8*(120*B*a^5*b^2 - 71*A*a^4*b^3)*c)*x^5 + 16*(48*B*
a^5*b^3 - 27*A*a^4*b^4 + 6608*A*a^6*c^2 + 8*(1576*B*a^6*b + 27*A*a^5*b^2)*c)*x^4 + 128*(592*B*a^6*b^2 + 3*A*a^
5*b^3 + 4*(288*B*a^7 + 307*A*a^6*b)*c)*x^3 + 256*(464*B*a^7*b + 243*A*a^6*b^2 + 476*A*a^7*c)*x^2 + 3072*(16*B*
a^8 + 33*A*a^7*b)*x)*sqrt(c*x^2 + b*x + a))/(a^6*x^8)]

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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^{9}}\, 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**9,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**9, x)

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Giac [B]  time = 1.52339, size = 4864, normalized size = 16.89 \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^9,x, algorithm="giac")

[Out]

5/16384*(16*B*a*b^7 - 9*A*b^8 - 192*B*a^2*b^5*c + 112*A*a*b^6*c + 768*B*a^3*b^3*c^2 - 480*A*a^2*b^4*c^2 - 1024
*B*a^4*b*c^3 + 768*A*a^3*b^2*c^3 - 256*A*a^4*c^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(
-a)*a^5) - 1/344064*(1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a*b^7 - 945*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^15*A*b^8 - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^2*b^5*c + 11760*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^15*A*a*b^6*c + 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^3*b^3*c^2 - 50400*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^15*A*a^2*b^4*c^2 - 107520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*B*a^4*b*c^3 + 80640*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^15*A*a^3*b^2*c^3 - 26880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*A*a^4*c^4 - 688128*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*B*a^5*c^(7/2) - 12880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^2*b^7 +
7245*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a*b^8 + 154560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*B*a^3*b^5
*c - 90160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^2*b^6*c - 618240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*
B*a^4*b^3*c^2 + 386400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^3*b^4*c^2 - 2157568*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^13*B*a^5*b*c^3 - 618240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*A*a^4*b^2*c^3 - 711424*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^13*A*a^5*c^4 - 6193152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*B*a^5*b^2*c^(5/2) + 6881
28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*B*a^6*c^(7/2) - 4816896*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*A*a^5
*b*c^(7/2) + 42896*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^3*b^7 - 24129*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^11*A*a^2*b^8 - 514752*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^4*b^5*c + 300272*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^11*A*a^3*b^6*c - 5510400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^5*b^3*c^2 - 1286880*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^11*A*a^4*b^4*c^2 - 2286592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*B*a^6*b*c^3 - 963916
8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^5*b^2*c^3 - 1603840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*A*a^6*
c^4 - 5734400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^5*b^4*c^(3/2) - 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^10*B*a^6*b^2*c^(5/2) - 16744448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^5*b^3*c^(5/2) - 3440640*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^10*B*a^7*c^(7/2) - 6881280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*A*a^6*b*c^(7/
2) - 80848*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^4*b^7 + 45477*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^3
*b^8 - 1684032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^5*b^5*c - 565936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
9*A*a^4*b^6*c - 4273920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^6*b^3*c^2 - 12811680*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^9*A*a^5*b^4*c^2 - 2329600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*B*a^7*b*c^3 - 21477120*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^9*A*a^6*b^2*c^3 - 3162880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*A*a^7*c^4 - 688128*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^5*b^6*sqrt(c) + 1146880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^6*b
^4*c^(3/2) - 8945664*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^5*b^5*c^(3/2) - 6881280*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^8*B*a^7*b^2*c^(5/2) - 22937600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^6*b^3*c^(5/2) + 3440640*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*B*a^8*c^(7/2) - 17203200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*A*a^7*b*c
^(7/2) + 17456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^5*b^7 - 52827*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A
*a^4*b^8 + 380352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^6*b^5*c - 2630320*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^7*A*a^5*b^6*c + 2607360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^7*b^3*c^2 - 19692960*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^7*A*a^6*b^4*c^2 + 1111040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a^8*b*c^3 - 24917760*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^7*b^2*c^3 - 3162880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^8*c^4 + 6881
28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^6*b^6*sqrt(c) - 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^
5*b^7*sqrt(c) + 1146880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^7*b^4*c^(3/2) - 6881280*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^6*A*a^6*b^5*c^(3/2) + 9633792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a^8*b^2*c^(5/2) - 270663
68*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^7*b^3*c^(5/2) - 2064384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*B*a
^9*c^(7/2) - 8257536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*A*a^8*b*c^(7/2) + 42896*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^5*B*a^6*b^7 - 24129*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^5*b^8 + 1549632*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*B*a^7*b^5*c - 1764112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^6*b^6*c + 4811520*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*B*a^8*b^3*c^2 - 11608800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^7*b^4*c^2 + 390656
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^9*b*c^3 - 15832320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^8*b^2
*c^3 - 1603840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a^9*c^4 + 3440640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4
*B*a^8*b^4*c^(3/2) - 3440640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^7*b^5*c^(3/2) + 2064384*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^4*B*a^9*b^2*c^(5/2) - 8257536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a^8*b^3*c^(5/2) + 2
064384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^10*c^(7/2) - 6193152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*
a^9*b*c^(7/2) - 12880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^7*b^7 + 7245*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*A*a^6*b^8 + 154560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^8*b^5*c - 90160*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*A*a^7*b^6*c + 2822400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^9*b^3*c^2 - 3054240*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*A*a^8*b^4*c^2 + 1283072*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^10*b*c^3 - 4058880*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^9*b^2*c^3 - 711424*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^10*c^4 + 2
064384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^10*b^2*c^(5/2) - 2064384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
2*A*a^9*b^3*c^(5/2) - 98304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^11*c^(7/2) - 589824*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^2*A*a^10*b*c^(7/2) + 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^8*b^7 - 945*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*A*a^7*b^8 - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^9*b^5*c + 11760*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*A*a^8*b^6*c + 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^10*b^3*c^2 - 50400*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*A*a^9*b^4*c^2 + 580608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^11*b*c^3 - 607488*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*A*a^10*b^2*c^3 - 26880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^11*c^4 + 98304*
B*a^12*c^(7/2) - 98304*A*a^11*b*c^(7/2))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^8*a^5)