3.1181 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^7} \, dx\)
Optimal. Leaf size=402 \[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{512 d^4 (d+e x)^2 (c d-b e)^4}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{192 d^3 (d+e x)^4 (c d-b e)^3}+\frac{b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{60 d^2 (d+e x)^5 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)} \]
[Out]
-(b^2*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2
])/(512*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d
+ (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(192*d^3*(c*d - b*e)^3*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5/2
))/(6*d*(c*d - b*e)*(d + e*x)^6) - ((7*A*e*(2*c*d - b*e) - B*d*(2*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(60*d^2*(
c*d - b*e)^2*(d + e*x)^5) + (b^4*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*ArcTanh[(b*d
+ (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(1024*d^(9/2)*(c*d - b*e)^(9/2))
________________________________________________________________________________________
Rubi [A] time = 0.577745, antiderivative size = 402, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used =
{834, 806, 720, 724, 206} \[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{512 d^4 (d+e x)^2 (c d-b e)^4}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{192 d^3 (d+e x)^4 (c d-b e)^3}+\frac{b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{60 d^2 (d+e x)^5 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]
[Out]
-(b^2*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2
])/(512*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d
+ (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(192*d^3*(c*d - b*e)^3*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5/2
))/(6*d*(c*d - b*e)*(d + e*x)^6) - ((7*A*e*(2*c*d - b*e) - B*d*(2*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(60*d^2*(
c*d - b*e)^2*(d + e*x)^5) + (b^4*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*ArcTanh[(b*d
+ (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(1024*d^(9/2)*(c*d - b*e)^(9/2))
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 (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{\int \frac{\left (\frac{1}{2} (-12 A c d+b (5 B d+7 A e))-c (B d-A e) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 d (c d-b e)}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac{\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{24 d^2 (c d-b e)^2}\\ &=\frac{\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}-\frac{\left (b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{128 d^3 (c d-b e)^3}\\ &=-\frac{b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac{\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac{\left (b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{1024 d^4 (c d-b e)^4}\\ &=-\frac{b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac{\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}-\frac{\left (b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{512 d^4 (c d-b e)^4}\\ &=-\frac{b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac{\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac{(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac{b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}\\ \end{align*}
Mathematica [A] time = 1.4991, size = 352, normalized size = 0.88 \[ \frac{(x (b+c x))^{3/2} \left (\frac{\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \left (\frac{b^2 \sqrt{x} \sqrt{b+c x} (5 b d+3 b e x+2 c d x)}{8 d^2 (d+e x)^2 (b e-c d)}+\frac{3 b^4 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{8 d^{5/2} (b e-c d)^{3/2}}-\frac{2 x^{3/2} (b+c x)^{5/2}}{(d+e x)^4}+\frac{b \sqrt{x} (b+c x)^{5/2}}{(d+e x)^3 (c d-b e)}\right )}{32 d (b+c x)^{3/2} (c d-b e)^2}-\frac{x^{5/2} (b+c x) (7 A e (b e-2 c d)+B d (5 b e+2 c d))}{10 d (d+e x)^5 (c d-b e)}+\frac{x^{5/2} (b+c x) (A e-B d)}{(d+e x)^6}\right )}{6 d x^{3/2} (b e-c d)} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]
[Out]
((x*(b + c*x))^(3/2)*(((-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^6 - ((7*A*e*(-2*c*d + b*e) + B*d*(2*c*d + 5
*b*e))*x^(5/2)*(b + c*x))/(10*d*(c*d - b*e)*(d + e*x)^5) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*
B*d + 7*A*e))*((-2*x^(3/2)*(b + c*x)^(5/2))/(d + e*x)^4 + (b*Sqrt[x]*(b + c*x)^(5/2))/((c*d - b*e)*(d + e*x)^3
) + (b^2*Sqrt[x]*Sqrt[b + c*x]*(5*b*d + 2*c*d*x + 3*b*e*x))/(8*d^2*(-(c*d) + b*e)*(d + e*x)^2) + (3*b^4*ArcTan
[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(8*d^(5/2)*(-(c*d) + b*e)^(3/2))))/(32*d*(c*d - b*e)^2
*(b + c*x)^(3/2))))/(6*d*(-(c*d) + b*e)*x^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.036, size = 29243, normalized size = 72.7 \begin{align*} \text{output 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)^(3/2)/(e*x+d)^7,x)
[Out]
result too large to display
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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)^(3/2)/(e*x+d)^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.23007, size = 7985, normalized size = 19.86 \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)^(3/2)/(e*x+d)^7,x, algorithm="fricas")
[Out]
[1/15360*(15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 -
12*(B*b^5*c - 2*A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c - 2*A
*b^4*c^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4
*e^4 + (5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b
^6 - 24*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^2 + (5*B*b^6 - 24*A*b^5
*c)*d^5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*x)*
sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(105*
A*b^6*d^6*e^3 + 180*(B*b^4*c^2 - 2*A*b^3*c^3)*d^9 - 15*(17*B*b^5*c - 48*A*b^4*c^2)*d^8*e + 15*(5*B*b^6 - 31*A*
b^5*c)*d^7*e^2 + (256*B*c^6*d^8*e - 105*A*b^6*d*e^8 - 64*(17*B*b*c^5 - 2*A*c^6)*d^7*e^2 + 16*(103*B*b^2*c^4 -
28*A*b*c^5)*d^6*e^3 - 32*(28*B*b^3*c^3 - 13*A*b^2*c^4)*d^5*e^4 - 10*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^4*e^5 + (205
*B*b^5*c - 466*A*b^4*c^2)*d^3*e^6 - 5*(15*B*b^6 - 79*A*b^5*c)*d^2*e^7)*x^5 + (1536*B*c^6*d^9 - 595*A*b^6*d^2*e
^7 - 256*(26*B*b*c^5 - 3*A*c^6)*d^8*e + 64*(163*B*b^2*c^4 - 43*A*b*c^5)*d^7*e^2 - 40*(155*B*b^3*c^3 - 68*A*b^2
*c^4)*d^6*e^3 + 4*(37*B*b^4*c^2 + 68*A*b^3*c^3)*d^5*e^4 + (1165*B*b^5*c - 2656*A*b^4*c^2)*d^4*e^5 - (425*B*b^6
- 2243*A*b^5*c)*d^3*e^6)*x^4 - 6*(231*A*b^6*d^3*e^6 - 32*(11*B*b*c^5 + 10*A*c^6)*d^9 + 8*(215*B*b^2*c^4 + 148
*A*b*c^5)*d^8*e - 4*(817*B*b^3*c^3 + 318*A*b^2*c^4)*d^7*e^2 + 2*(1387*B*b^4*c^2 + 18*A*b^3*c^3)*d^6*e^3 - (103
9*B*b^5*c - 1014*A*b^4*c^2)*d^5*e^4 + 3*(55*B*b^6 - 291*A*b^5*c)*d^4*e^5)*x^3 - 2*(843*A*b^6*d^4*e^5 - 48*(B*b
^2*c^4 + 30*A*b*c^5)*d^9 + 4*(101*B*b^3*c^3 + 1704*A*b^2*c^4)*d^8*e - 4*(421*B*b^4*c^2 + 2879*A*b^3*c^3)*d^7*e
^2 + (1823*B*b^5*c + 9712*A*b^4*c^2)*d^6*e^3 - 5*(99*B*b^6 + 883*A*b^5*c)*d^5*e^4)*x^2 + 5*(119*A*b^6*d^5*e^4
- 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^9 + 14*(17*B*b^4*c^2 - 36*A*b^3*c^3)*d^8*e - (299*B*b^5*c - 878*A*b^4*c^2)*d^
7*e^2 + (85*B*b^6 - 541*A*b^5*c)*d^6*e^3)*x)*sqrt(c*x^2 + b*x))/(c^5*d^16 - 5*b*c^4*d^15*e + 10*b^2*c^3*d^14*e
^2 - 10*b^3*c^2*d^13*e^3 + 5*b^4*c*d^12*e^4 - b^5*d^11*e^5 + (c^5*d^10*e^6 - 5*b*c^4*d^9*e^7 + 10*b^2*c^3*d^8*
e^8 - 10*b^3*c^2*d^7*e^9 + 5*b^4*c*d^6*e^10 - b^5*d^5*e^11)*x^6 + 6*(c^5*d^11*e^5 - 5*b*c^4*d^10*e^6 + 10*b^2*
c^3*d^9*e^7 - 10*b^3*c^2*d^8*e^8 + 5*b^4*c*d^7*e^9 - b^5*d^6*e^10)*x^5 + 15*(c^5*d^12*e^4 - 5*b*c^4*d^11*e^5 +
10*b^2*c^3*d^10*e^6 - 10*b^3*c^2*d^9*e^7 + 5*b^4*c*d^8*e^8 - b^5*d^7*e^9)*x^4 + 20*(c^5*d^13*e^3 - 5*b*c^4*d^
12*e^4 + 10*b^2*c^3*d^11*e^5 - 10*b^3*c^2*d^10*e^6 + 5*b^4*c*d^9*e^7 - b^5*d^8*e^8)*x^3 + 15*(c^5*d^14*e^2 - 5
*b*c^4*d^13*e^3 + 10*b^2*c^3*d^12*e^4 - 10*b^3*c^2*d^11*e^5 + 5*b^4*c*d^10*e^6 - b^5*d^9*e^7)*x^2 + 6*(c^5*d^1
5*e - 5*b*c^4*d^14*e^2 + 10*b^2*c^3*d^13*e^3 - 10*b^3*c^2*d^12*e^4 + 5*b^4*c*d^11*e^5 - b^5*d^10*e^6)*x), 1/76
80*(15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 - 12*(B
*b^5*c - 2*A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c - 2*A*b^4*c
^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4*e^4 +
(5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b^6 - 2
4*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^2 + (5*B*b^6 - 24*A*b^5*c)*d^
5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*x)*sqrt(-
c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (105*A*b^6*d^6*e^3 + 180*(B*b
^4*c^2 - 2*A*b^3*c^3)*d^9 - 15*(17*B*b^5*c - 48*A*b^4*c^2)*d^8*e + 15*(5*B*b^6 - 31*A*b^5*c)*d^7*e^2 + (256*B*
c^6*d^8*e - 105*A*b^6*d*e^8 - 64*(17*B*b*c^5 - 2*A*c^6)*d^7*e^2 + 16*(103*B*b^2*c^4 - 28*A*b*c^5)*d^6*e^3 - 32
*(28*B*b^3*c^3 - 13*A*b^2*c^4)*d^5*e^4 - 10*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^4*e^5 + (205*B*b^5*c - 466*A*b^4*c^2
)*d^3*e^6 - 5*(15*B*b^6 - 79*A*b^5*c)*d^2*e^7)*x^5 + (1536*B*c^6*d^9 - 595*A*b^6*d^2*e^7 - 256*(26*B*b*c^5 - 3
*A*c^6)*d^8*e + 64*(163*B*b^2*c^4 - 43*A*b*c^5)*d^7*e^2 - 40*(155*B*b^3*c^3 - 68*A*b^2*c^4)*d^6*e^3 + 4*(37*B*
b^4*c^2 + 68*A*b^3*c^3)*d^5*e^4 + (1165*B*b^5*c - 2656*A*b^4*c^2)*d^4*e^5 - (425*B*b^6 - 2243*A*b^5*c)*d^3*e^6
)*x^4 - 6*(231*A*b^6*d^3*e^6 - 32*(11*B*b*c^5 + 10*A*c^6)*d^9 + 8*(215*B*b^2*c^4 + 148*A*b*c^5)*d^8*e - 4*(817
*B*b^3*c^3 + 318*A*b^2*c^4)*d^7*e^2 + 2*(1387*B*b^4*c^2 + 18*A*b^3*c^3)*d^6*e^3 - (1039*B*b^5*c - 1014*A*b^4*c
^2)*d^5*e^4 + 3*(55*B*b^6 - 291*A*b^5*c)*d^4*e^5)*x^3 - 2*(843*A*b^6*d^4*e^5 - 48*(B*b^2*c^4 + 30*A*b*c^5)*d^9
+ 4*(101*B*b^3*c^3 + 1704*A*b^2*c^4)*d^8*e - 4*(421*B*b^4*c^2 + 2879*A*b^3*c^3)*d^7*e^2 + (1823*B*b^5*c + 971
2*A*b^4*c^2)*d^6*e^3 - 5*(99*B*b^6 + 883*A*b^5*c)*d^5*e^4)*x^2 + 5*(119*A*b^6*d^5*e^4 - 24*(B*b^3*c^3 - 2*A*b^
2*c^4)*d^9 + 14*(17*B*b^4*c^2 - 36*A*b^3*c^3)*d^8*e - (299*B*b^5*c - 878*A*b^4*c^2)*d^7*e^2 + (85*B*b^6 - 541*
A*b^5*c)*d^6*e^3)*x)*sqrt(c*x^2 + b*x))/(c^5*d^16 - 5*b*c^4*d^15*e + 10*b^2*c^3*d^14*e^2 - 10*b^3*c^2*d^13*e^3
+ 5*b^4*c*d^12*e^4 - b^5*d^11*e^5 + (c^5*d^10*e^6 - 5*b*c^4*d^9*e^7 + 10*b^2*c^3*d^8*e^8 - 10*b^3*c^2*d^7*e^9
+ 5*b^4*c*d^6*e^10 - b^5*d^5*e^11)*x^6 + 6*(c^5*d^11*e^5 - 5*b*c^4*d^10*e^6 + 10*b^2*c^3*d^9*e^7 - 10*b^3*c^2
*d^8*e^8 + 5*b^4*c*d^7*e^9 - b^5*d^6*e^10)*x^5 + 15*(c^5*d^12*e^4 - 5*b*c^4*d^11*e^5 + 10*b^2*c^3*d^10*e^6 - 1
0*b^3*c^2*d^9*e^7 + 5*b^4*c*d^8*e^8 - b^5*d^7*e^9)*x^4 + 20*(c^5*d^13*e^3 - 5*b*c^4*d^12*e^4 + 10*b^2*c^3*d^11
*e^5 - 10*b^3*c^2*d^10*e^6 + 5*b^4*c*d^9*e^7 - b^5*d^8*e^8)*x^3 + 15*(c^5*d^14*e^2 - 5*b*c^4*d^13*e^3 + 10*b^2
*c^3*d^12*e^4 - 10*b^3*c^2*d^11*e^5 + 5*b^4*c*d^10*e^6 - b^5*d^9*e^7)*x^2 + 6*(c^5*d^15*e - 5*b*c^4*d^14*e^2 +
10*b^2*c^3*d^13*e^3 - 10*b^3*c^2*d^12*e^4 + 5*b^4*c*d^11*e^5 - b^5*d^10*e^6)*x)]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**7,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.91292, size = 7760, normalized size = 19.3 \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)^(3/2)/(e*x+d)^7,x, algorithm="giac")
[Out]
-1/512*(12*B*b^5*c*d^2 - 24*A*b^4*c^2*d^2 - 5*B*b^6*d*e + 24*A*b^5*c*d*e - 7*A*b^6*e^2)*arctan(-((sqrt(c)*x -
sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*
c*d^5*e^3 + b^4*d^4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/7680*(49152*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^8*d^11*e
+ 16384*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^(17/2)*d^12 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^(15/2)
*d^10*e^2 + 69632*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(15/2)*d^11*e + 8192*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^6*A*c^(17/2)*d^11*e + 49152*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^8*d^12 + 40960*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^9*B*c^7*d^9*e^3 - 36864*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b*c^7*d^10*e^2 + 24576*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^7*A*c^8*d^10*e^2 - 36864*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^7*d^11*e + 24576*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^5*A*b*c^8*d^11*e + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^(15/2)*d^12 + 15360*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^10*B*c^(13/2)*d^8*e^4 - 138240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^(13/2)*d^9*e^
3 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^(15/2)*d^9*e^3 - 254976*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^
2*c^(13/2)*d^10*e^2 + 40960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^(15/2)*d^10*e^2 - 138240*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^4*B*b^3*c^(13/2)*d^11*e + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(15/2)*d^11*e + 40960
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^7*d^12 - 117760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^6*d^8*e^4 +
20480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*c^7*d^8*e^4 - 211968*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^6*d^
9*e^3 - 211968*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^6*d^10*e^2 - 117760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3
*B*b^4*c^6*d^11*e + 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^7*d^11*e + 15360*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*B*b^4*c^(13/2)*d^12 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^(11/2)*d^7*e^5 - 46080*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^8*A*b*c^(13/2)*d^8*e^4 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3*c^(11/2)*d^9*e^3
- 101376*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(13/2)*d^9*e^3 - 46080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A
*b^3*c^(13/2)*d^10*e^2 - 48384*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(11/2)*d^11*e + 7680*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^2*A*b^4*c^(13/2)*d^11*e + 3072*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*c^6*d^12 + 61440*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^5*d^7*e^5 - 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b*c^6*d^7*e^5 + 276480
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^3*c^5*d^8*e^4 - 119808*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^6*d^8*
e^4 + 276480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^5*d^9*e^3 - 119808*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*
b^3*c^6*d^9*e^3 + 80640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*c^5*d^10*e^2 - 43520*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*A*b^4*c^6*d^10*e^2 - 9984*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^5*d^11*e + 1536*(sqrt(c)*x - sqrt(c
*x^2 + b*x))*A*b^5*c^6*d^11*e + 256*B*b^6*c^(11/2)*d^12 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^(9/
2)*d^6*e^6 + 153600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(9/2)*d^7*e^5 - 122880*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^8*A*b^2*c^(11/2)*d^7*e^5 + 230400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^(9/2)*d^8*e^4 - 55296*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^(11/2)*d^8*e^4 + 164160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c^(9/2)*d
^9*e^3 - 51840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^(11/2)*d^9*e^3 + 43968*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^2*B*b^6*c^(9/2)*d^10*e^2 - 18432*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^5*c^(11/2)*d^10*e^2 - 832*B*b^7*c^(9/
2)*d^11*e + 128*A*b^6*c^(11/2)*d^11*e + 112640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^4*d^6*e^6 + 122880*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2*c^5*d^6*e^6 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^4*d^7*e^5
- 12288*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^5*d^7*e^5 + 34560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^
4*d^8*e^4 + 41472*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^4*c^5*d^8*e^4 + 36160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
3*B*b^6*c^4*d^9*e^3 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^5*d^9*e^3 + 9792*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*B*b^7*c^4*d^10*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^5*d^10*e^2 - 61440*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^10*B*b^3*c^(7/2)*d^5*e^7 + 337920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(9/2)*d^6*e^6 - 841
92*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*c^(7/2)*d^7*e^5 + 100800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^
(9/2)*d^7*e^5 - 38160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^6*c^(7/2)*d^8*e^4 + 60480*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^4*A*b^5*c^(9/2)*d^8*e^4 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(7/2)*d^9*e^3 + 3840*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*A*b^6*c^(9/2)*d^9*e^3 + 816*B*b^8*c^(7/2)*d^10*e^2 - 320*A*b^7*c^(9/2)*d^10*e^2 - 1433
60*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^4*c^3*d^5*e^7 - 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^3*c^4*d^5
*e^7 - 85536*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c^3*d^6*e^6 + 336960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*
b^4*c^4*d^6*e^6 - 82656*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^6*c^3*d^7*e^5 + 87360*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^5*A*b^5*c^4*d^7*e^5 - 25600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^7*c^3*d^8*e^4 + 33920*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^3*A*b^6*c^4*d^8*e^4 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^8*c^3*d^9*e^3 + 1152*(sqrt(c)*x
- sqrt(c*x^2 + b*x))*A*b^7*c^4*d^9*e^3 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^4*c^(5/2)*d^4*e^8 - 1177
20*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^5*c^(5/2)*d^5*e^7 - 317520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^4*c^
(7/2)*d^5*e^7 - 58936*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^6*c^(5/2)*d^6*e^6 + 95424*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^6*A*b^5*c^(7/2)*d^6*e^6 - 34920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7*c^(5/2)*d^7*e^5 + 15120*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^(7/2)*d^7*e^5 - 8280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8*c^(5/2)*d^8*e
^4 + 11136*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^7*c^(7/2)*d^8*e^4 - 80*B*b^9*c^(5/2)*d^9*e^3 + 96*A*b^8*c^(7/
2)*d^9*e^3 + 54960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^5*c^2*d^4*e^8 + 2720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
9*A*b^4*c^3*d^4*e^8 - 18408*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^6*c^2*d^5*e^7 - 419328*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^7*A*b^5*c^3*d^5*e^7 - 1128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^7*c^2*d^6*e^6 - 71808*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^5*A*b^6*c^3*d^6*e^6 - 4680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^8*c^2*d^7*e^5 - 18080*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c^3*d^7*e^5 - 1560*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^9*c^2*d^8*e^4 + 21
12*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*c^3*d^8*e^4 + 1980*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^5*c^(3/2)*d
^3*e^9 - 3960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^4*c^(5/2)*d^3*e^9 + 62070*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^8*B*b^6*c^(3/2)*d^4*e^8 + 99480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^5*c^(5/2)*d^4*e^8 + 29608*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^6*B*b^7*c^(3/2)*d^5*e^7 - 242840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^6*c^(5/2)*d^5*e^7 +
15420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^8*c^(3/2)*d^6*e^6 - 64440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^7*
c^(5/2)*d^6*e^6 + 1740*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^9*c^(3/2)*d^7*e^5 - 12384*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*A*b^8*c^(5/2)*d^7*e^5 - 130*B*b^10*c^(3/2)*d^8*e^4 + 176*A*b^9*c^(5/2)*d^8*e^4 + 180*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^11*B*b^5*c*d^2*e^10 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^4*c^2*d^2*e^10 - 2680*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^9*B*b^6*c*d^3*e^9 + 15720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^5*c^2*d^3*e^9 + 320
64*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^7*c*d^4*e^8 + 170520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^6*c^2*d^4*
e^8 + 19944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^8*c*d^5*e^7 - 47400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^7*
c^2*d^5*e^7 + 7180*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^9*c*d^6*e^6 - 16320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3
*A*b^8*c^2*d^6*e^6 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^10*c*d^7*e^5 - 3120*(sqrt(c)*x - sqrt(c*x^2 + b*x
))*A*b^9*c^2*d^7*e^5 - 825*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^6*sqrt(c)*d^2*e^10 + 3960*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^10*A*b^5*c^(3/2)*d^2*e^10 - 3825*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^7*sqrt(c)*d^3*e^9 + 6390*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^6*c^(3/2)*d^3*e^9 + 8430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^8*sqrt(c)
*d^4*e^8 + 115328*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^7*c^(3/2)*d^4*e^8 + 4950*(sqrt(c)*x - sqrt(c*x^2 + b*x
))^4*B*b^9*sqrt(c)*d^5*e^7 + 14460*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^8*c^(3/2)*d^5*e^7 + 1275*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*B*b^10*sqrt(c)*d^6*e^6 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9*c^(3/2)*d^6*e^6 + 7
5*B*b^11*sqrt(c)*d^7*e^5 - 290*A*b^10*c^(3/2)*d^7*e^5 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^6*d*e^11 + 3
60*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^5*c*d*e^11 - 425*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^7*d^2*e^10 -
3140*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^6*c*d^2*e^10 - 990*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^8*d^3*e^9
- 13896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^7*c*d^3*e^9 + 990*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^9*d^4*e^
8 + 38784*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^8*c*d^4*e^8 + 425*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^10*d^5
*e^7 + 9440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^9*c*d^5*e^7 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^11*d^6*
e^6 + 900*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^10*c*d^6*e^6 - 1155*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^6*sqr
t(c)*d*e^11 - 5355*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^7*sqrt(c)*d^2*e^10 - 9702*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^6*A*b^8*sqrt(c)*d^3*e^9 + 6930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^9*sqrt(c)*d^4*e^8 + 1785*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*A*b^10*sqrt(c)*d^5*e^7 + 105*A*b^11*sqrt(c)*d^6*e^6 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x
))^11*A*b^6*e^12 - 595*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^7*d*e^11 - 1386*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7
*A*b^8*d^2*e^10 - 1686*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^9*d^3*e^9 + 595*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3
*A*b^10*d^4*e^8 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^11*d^5*e^7)/((c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*
c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4*e^9)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*sqrt(c)*d + b*d)^6)