3.1184 \(\int (A+B x) (d+e x) (b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=264 \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}-\frac{\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \]

[Out]

(5*b^4*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*A
*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2*d + 9*b^2*B*
e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B*e - 16*c*(B*d + A*e) - 14*B*c*e*x
)*(b*x + c*x^2)^(7/2))/(112*c^2) - (5*b^6*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(16384*c^(11/2))

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Rubi [A]  time = 0.249938, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {779, 612, 620, 206} \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}-\frac{\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^4*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*A
*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2*d + 9*b^2*B*
e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B*e - 16*c*(B*d + A*e) - 14*B*c*e*x
)*(b*x + c*x^2)^(7/2))/(112*c^2) - (5*b^6*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(16384*c^(11/2))

Rule 779

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

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

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 (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx &=-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{\left (\frac{9}{2} b^2 B e+8 c (2 A c d-b (B d+A e))\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac{\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac{5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{\left (5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \sqrt{b x+c x^2} \, dx}{4096 c^4}\\ &=\frac{5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac{5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{16384 c^5}\\ &=\frac{5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac{5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.821753, size = 315, normalized size = 1.19 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (128 b^3 c^4 x^2 (2 A (7 d+3 e x)+3 B x (2 d+e x))+256 b^2 c^5 x^3 (A (378 d+296 e x)+B x (296 d+243 e x))+56 b^5 c^2 (20 A (3 d+e x)+B x (20 d+9 e x))-16 b^4 c^3 x (28 A (5 d+2 e x)+B x (56 d+27 e x))-210 b^6 c (8 A e+8 B d+3 B e x)+1024 b c^6 x^4 (4 A (35 d+29 e x)+B x (116 d+99 e x))+2048 c^7 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+945 b^7 B e\right )-\frac{105 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{344064 c^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^7*B*e - 210*b^6*c*(8*B*d + 8*A*e + 3*B*e*x) + 128*b^3*c^4*x^2*(3*B*x*(2*d +
e*x) + 2*A*(7*d + 3*e*x)) + 2048*c^7*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x)) + 56*b^5*c^2*(20*A*(3*d +
e*x) + B*x*(20*d + 9*e*x)) - 16*b^4*c^3*x*(28*A*(5*d + 2*e*x) + B*x*(56*d + 27*e*x)) + 1024*b*c^6*x^4*(4*A*(35
*d + 29*e*x) + B*x*(116*d + 99*e*x)) + 256*b^2*c^5*x^3*(B*x*(296*d + 243*e*x) + A*(378*d + 296*e*x))) - (105*b
^(11/2)*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c
*x)/b])))/(344064*c^(11/2))

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Maple [B]  time = 0.007, size = 716, normalized size = 2.7 \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)*(e*x+d)*(c*x^2+b*x)^(5/2),x)

[Out]

-15/2048*B*e*b^5/c^4*(c*x^2+b*x)^(3/2)+3/128*B*e*b^3/c^3*(c*x^2+b*x)^(5/2)-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*A*
e-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*B*d-1/24*b^2/c^2*(c*x^2+b*x)^(5/2)*A*e-1/24*b^2/c^2*(c*x^2+b*x)^(5/2)*B*d+4
5/16384*B*e*b^7/c^5*(c*x^2+b*x)^(1/2)-45/32768*B*e*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*
B*e*x*(c*x^2+b*x)^(7/2)/c-9/112*B*e*b/c^2*(c*x^2+b*x)^(7/2)+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*B*d+5/2048*b^7/c^(
9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*A*e-5/192*d*A*b^3/c^2*(c*x^
2+b*x)^(3/2)+1/12*d*A/c*(c*x^2+b*x)^(5/2)*b+5/2048*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e+5
/512*d*A*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*d*A*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+5/192*b^3/
c^2*(c*x^2+b*x)^(3/2)*x*B*d+5/192*b^3/c^2*(c*x^2+b*x)^(3/2)*x*A*e+3/64*B*e*b^2/c^2*(c*x^2+b*x)^(5/2)*x-15/1024
*B*e*b^4/c^3*(c*x^2+b*x)^(3/2)*x+45/8192*B*e*b^6/c^4*(c*x^2+b*x)^(1/2)*x-1/12*b/c*(c*x^2+b*x)^(5/2)*x*A*e-1/12
*b/c*(c*x^2+b*x)^(5/2)*x*B*d-5/96*d*A*b^2/c*(c*x^2+b*x)^(3/2)*x+5/256*d*A*b^4/c^2*(c*x^2+b*x)^(1/2)*x-5/512*b^
5/c^3*(c*x^2+b*x)^(1/2)*x*A*e-5/512*b^5/c^3*(c*x^2+b*x)^(1/2)*x*B*d+1/6*d*A*(c*x^2+b*x)^(5/2)*x+1/7*(c*x^2+b*x
)^(7/2)/c*A*e+1/7*(c*x^2+b*x)^(7/2)/c*B*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67133, size = 1820, normalized size = 6.89 \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)*(e*x+d)*(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/688128*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c)) + 2*(43008*B*c^8*e*x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16*(29*B*b*c
^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B*b^2*c^6 + 70*A*b*c^7)*d + (3*B*b^3*c^5
+ 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B*b^3*c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^3 - 56*(16
*(B*b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680*(B*b^6*c^2 - 2*A*b^5*c^3)*d + 105*(9
*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B*b^5*c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)*sqrt(c*x^
2 + b*x))/c^6, -1/344064*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*sqrt(-c)*arctan(sqrt(c
*x^2 + b*x)*sqrt(-c)/(c*x)) - (43008*B*c^8*e*x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16
*(29*B*b*c^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B*b^2*c^6 + 70*A*b*c^7)*d + (3
*B*b^3*c^5 + 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B*b^3*c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^
3 - 56*(16*(B*b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680*(B*b^6*c^2 - 2*A*b^5*c^3)*
d + 105*(9*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B*b^5*c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)
*sqrt(c*x^2 + b*x))/c^6]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.33103, size = 574, normalized size = 2.17 \begin{align*} \frac{1}{344064} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, B c^{2} x e + \frac{16 \, B c^{9} d + 33 \, B b c^{8} e + 16 \, A c^{9} e}{c^{7}}\right )} x + \frac{464 \, B b c^{8} d + 224 \, A c^{9} d + 243 \, B b^{2} c^{7} e + 464 \, A b c^{8} e}{c^{7}}\right )} x + \frac{592 \, B b^{2} c^{7} d + 1120 \, A b c^{8} d + 3 \, B b^{3} c^{6} e + 592 \, A b^{2} c^{7} e}{c^{7}}\right )} x + \frac{3 \,{\left (16 \, B b^{3} c^{6} d + 2016 \, A b^{2} c^{7} d - 9 \, B b^{4} c^{5} e + 16 \, A b^{3} c^{6} e\right )}}{c^{7}}\right )} x - \frac{7 \,{\left (16 \, B b^{4} c^{5} d - 32 \, A b^{3} c^{6} d - 9 \, B b^{5} c^{4} e + 16 \, A b^{4} c^{5} e\right )}}{c^{7}}\right )} x + \frac{35 \,{\left (16 \, B b^{5} c^{4} d - 32 \, A b^{4} c^{5} d - 9 \, B b^{6} c^{3} e + 16 \, A b^{5} c^{4} e\right )}}{c^{7}}\right )} x - \frac{105 \,{\left (16 \, B b^{6} c^{3} d - 32 \, A b^{5} c^{4} d - 9 \, B b^{7} c^{2} e + 16 \, A b^{6} c^{3} e\right )}}{c^{7}}\right )} - \frac{5 \,{\left (16 \, B b^{7} c d - 32 \, A b^{6} c^{2} d - 9 \, B b^{8} e + 16 \, A b^{7} c e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x*e + (16*B*c^9*d + 33*B*b*c^8*e + 16*A*c^9*e)/c^7)*x
+ (464*B*b*c^8*d + 224*A*c^9*d + 243*B*b^2*c^7*e + 464*A*b*c^8*e)/c^7)*x + (592*B*b^2*c^7*d + 1120*A*b*c^8*d +
3*B*b^3*c^6*e + 592*A*b^2*c^7*e)/c^7)*x + 3*(16*B*b^3*c^6*d + 2016*A*b^2*c^7*d - 9*B*b^4*c^5*e + 16*A*b^3*c^6
*e)/c^7)*x - 7*(16*B*b^4*c^5*d - 32*A*b^3*c^6*d - 9*B*b^5*c^4*e + 16*A*b^4*c^5*e)/c^7)*x + 35*(16*B*b^5*c^4*d
- 32*A*b^4*c^5*d - 9*B*b^6*c^3*e + 16*A*b^5*c^4*e)/c^7)*x - 105*(16*B*b^6*c^3*d - 32*A*b^5*c^4*d - 9*B*b^7*c^2
*e + 16*A*b^6*c^3*e)/c^7) - 5/32768*(16*B*b^7*c*d - 32*A*b^6*c^2*d - 9*B*b^8*e + 16*A*b^7*c*e)*log(abs(-2*(sqr
t(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)