3.1875 \(\int \frac{(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=489 \[ -\frac{(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{105 e^3 (a+b x) \sqrt{b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 e^3 (a+b x) \sqrt{d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(105*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*(d +
e*x)^(3/2))/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2*(8*b*B*
d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (3*e*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(32*b^3*(b*d
- a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d + 3*A*b*e - 11*a*B*e
)*(d + e*x)^(9/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - ((A*b - a*B)*(d + e*x)^(11/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) - (105*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])

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Rubi [A]  time = 0.994259, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{105 e^3 (a+b x) \sqrt{b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 e^3 (a+b x) \sqrt{d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(105*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*(d +
e*x)^(3/2))/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (21*e^2*(8*b*B*
d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - (3*e*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(32*b^3*(b*d
- a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d + 3*A*b*e - 11*a*B*e
)*(d + e*x)^(9/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - ((A*b - a*B)*(d + e*x)^(11/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b
*x + b^2*x^2]) - (105*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(13/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.08498, size = 279, normalized size = 0.57 \[ \frac{(a+b x) \left (-\frac{105 e^3 \sqrt{b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{13/2}}-\frac{\sqrt{d+e x} \left (-128 e^3 (a+b x)^4 (-15 a B e+3 A b e+13 b B d)+15 e^2 (a+b x)^3 (b d-a e) (-153 a B e+65 A b e+88 b B d)+8 (a+b x) (b d-a e)^3 (-41 a B e+33 A b e+8 b B d)+10 e (a+b x)^2 (b d-a e)^2 (-103 a B e+63 A b e+40 b B d)+48 (A b-a B) (b d-a e)^4-128 b B e^4 x (a+b x)^4\right )}{3 b^6 (a+b x)^4}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*(-(Sqrt[d + e*x]*(48*(A*b - a*B)*(b*d - a*e)^4 + 8*(b*d - a*e)^3*(8*b
*B*d + 33*A*b*e - 41*a*B*e)*(a + b*x) + 10*e*(b*d - a*e)^2*(40*b*B*d + 63*A*b*e
- 103*a*B*e)*(a + b*x)^2 + 15*e^2*(b*d - a*e)*(88*b*B*d + 65*A*b*e - 153*a*B*e)*
(a + b*x)^3 - 128*e^3*(13*b*B*d + 3*A*b*e - 15*a*B*e)*(a + b*x)^4 - 128*b*B*e^4*
x*(a + b*x)^4))/(3*b^6*(a + b*x)^4) - (105*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*
e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(13/2)))/(64*S
qrt[(a + b*x)^2])

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Maple [B]  time = 0.051, size = 2430, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(-23940*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^2*b^4*d*e^5+10
080*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a*b^5*d^2*e^4+1536*A*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a*b^4*e^5+5670*A*arctan((e*x+d)^(1/2)*b/(b*(a*e
-b*d))^(1/2))*x^2*a^2*b^4*d*e^5+3615*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^4*d
*e+768*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*a^2*b^3*e^4-7680*B*(b*(a*e-b*d))^
(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^3*e^5-35910*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*x^2*a^3*b^3*d*e^5+15120*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^
2*a^2*b^4*d^2*e^4-4590*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^4*d*e^2-7680*B*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^4*b*e^5-2244*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(
1/2)*a^3*b^2*d*e^4+3366*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^2*e^3-2244
*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^4*d^3*e^2+8700*B*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*a^4*b*d*e^4-13206*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b^2*d^2*e^
3+12084*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^3*e^2-5481*B*(b*(a*e-b*d))
^(1/2)*(e*x+d)^(1/2)*a*b^4*d^4*e+2304*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^
2*b^3*e^5+3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^3*d*e^5+152
70*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^3*d*e^2-12975*B*(b*(a*e-b*d))^(1/2)
*(e*x+d)^(5/2)*a*b^4*d^2*e+512*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a^3*b^2*e^4
-11520*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^2*e^5-23940*B*arctan((e*x+d
)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b^2*d*e^5+10080*B*arctan((e*x+d)^(1/2)*b/(b
*(a*e-b*d))^(1/2))*x*a^3*b^3*d^2*e^4-5787*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^
2*b^3*d*e^3+5787*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^4*d^2*e^2+1536*A*(b*(a*
e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^3*b^2*e^5+18683*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3
/2)*a^3*b^2*d*e^3-25131*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^3*d^2*e^2+1482
5*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^4*d^3*e-5985*B*arctan((e*x+d)^(1/2)*b/
(b*(a*e-b*d))^(1/2))*x^4*a*b^5*d*e^5+3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*x^3*a*b^5*d*e^5+512*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^3*a*b^4*e^4-19
20*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^4*a*b^4*e^5+1536*B*(b*(a*e-b*d))^(1/2)*
(e*x+d)^(1/2)*x^4*b^5*d*e^4-1320*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^5*d^2+356
0*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^5*d^3-945*A*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*a^5*b*e^6-3224*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^5*d^4-3465*
B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^5*e^5+984*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*b^5*d^5+13860*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^5*b*e^6+1929
*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b^2*e^4-1929*A*(b*(a*e-b*d))^(1/2)*(e*x
+d)^(3/2)*b^5*d^3*e+945*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b^2*d*
e^5-5025*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^4*b*e^4-5985*B*arctan((e*x+d)^(1/
2)*b/(b*(a*e-b*d))^(1/2))*a^5*b*d*e^5+2520*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*a^4*b^2*d^2*e^4+945*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*b*e^5+561*A
*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^5*d^4*e-945*A*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*x^4*a*b^5*e^6+945*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x
^4*b^6*d*e^5+128*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^4*b^5*e^4+3465*B*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a^2*b^4*e^6+2520*B*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*x^4*b^6*d^2*e^4+384*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x
^4*b^5*e^5-3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^3*a^2*b^4*e^6+92
16*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^3*d*e^4+6144*B*(b*(a*e-b*d))^(1
/2)*(e*x+d)^(1/2)*x*a^3*b^2*d*e^4+6144*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^3*a
*b^4*d*e^4+3465*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^6*e^6-5855*B*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^3*b^2*e^3+13860*B*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*x^3*a^3*b^3*e^6+975*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^4*e^2
-975*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^5*d*e-5670*A*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*x^2*a^3*b^3*e^6-2295*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a^2
*b^3*e^2+20790*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^4*b^2*e^6+229
5*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a^2*b^3*e^3+2295*A*(b*(a*e-b*d))^(1/2)*(e*
x+d)^(5/2)*b^5*d^2*e-3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b^
2*e^6)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/b^6/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304386, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (1
1*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*
e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 4*(8*B
*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt((b*d - a*e)/b)*log((b*e
*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(128*B*
b^5*e^4*x^5 - 16*(B*a*b^4 + 3*A*b^5)*d^4 - 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*
(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b - 3*A*a^3*b^2)*d*e^3 - 315*(
11*B*a^5 - 3*A*a^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^
4 - (1320*B*b^5*d^2*e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3
- 3*A*a*b^4)*e^4)*x^3 - (400*B*b^5*d^3*e + 30*(71*B*a*b^4 + 21*A*b^5)*d^2*e^2 -
9*(2041*B*a^2*b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*
x^2 - (64*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*
a*b^4)*d^2*e^2 - 21*(649*B*a^3*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*
A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^
3*b^7*x + a^4*b^6), -1/192*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 +
(8*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (11*B*a^
2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3
)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(-(b*
d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (128*B*b^5*e^4*x^5 - 16
*(B*a*b^4 + 3*A*b^5)*d^4 - 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A
*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b - 3*A*a^3*b^2)*d*e^3 - 315*(11*B*a^5 - 3*A*a
^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 - (1320*B*b^5*
d^2*e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4
)*x^3 - (400*B*b^5*d^3*e + 30*(71*B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2*b^
3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 - (64*B*b^5*
d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a*b^4)*d^2*e^2 -
21*(649*B*a^3*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*
x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^
6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.360669, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

Done