3.1825 \(\int \frac{(A+B x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)
Optimal. Leaf size=393 \[ -\frac{231 e^4 \sqrt{b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}+\frac{231 e^4 \sqrt{d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}+\frac{77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac{231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}-\frac{33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
[Out]
(231*e^4*(10*b*B*d + 3*A*b*e - 13*a*B*e)*Sqrt[d + e*x])/(128*b^7) + (77*e^4*(10*
b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(3/2))/(128*b^6*(b*d - a*e)) - (231*e^3*(1
0*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(5/2))/(640*b^5*(b*d - a*e)*(a + b*x)) -
(33*e^2*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(7/2))/(320*b^4*(b*d - a*e)*(
a + b*x)^2) - (11*e*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(9/2))/(240*b^3*(b
*d - a*e)*(a + b*x)^3) - ((10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(11/2))/(40*
b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(13/2))/(5*b*(b*d - a*e)*(
a + b*x)^5) - (231*e^4*Sqrt[b*d - a*e]*(10*b*B*d + 3*A*b*e - 13*a*B*e)*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))
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Rubi [A] time = 0.797334, antiderivative size = 393, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{231 e^4 \sqrt{b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}+\frac{231 e^4 \sqrt{d+e x} (-13 a B e+3 A b e+10 b B d)}{128 b^7}+\frac{77 e^4 (d+e x)^{3/2} (-13 a B e+3 A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac{231 e^3 (d+e x)^{5/2} (-13 a B e+3 A b e+10 b B d)}{640 b^5 (a+b x) (b d-a e)}-\frac{33 e^2 (d+e x)^{7/2} (-13 a B e+3 A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac{11 e (d+e x)^{9/2} (-13 a B e+3 A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{11/2} (-13 a B e+3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{13/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(11/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(231*e^4*(10*b*B*d + 3*A*b*e - 13*a*B*e)*Sqrt[d + e*x])/(128*b^7) + (77*e^4*(10*
b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(3/2))/(128*b^6*(b*d - a*e)) - (231*e^3*(1
0*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(5/2))/(640*b^5*(b*d - a*e)*(a + b*x)) -
(33*e^2*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(7/2))/(320*b^4*(b*d - a*e)*(
a + b*x)^2) - (11*e*(10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(9/2))/(240*b^3*(b
*d - a*e)*(a + b*x)^3) - ((10*b*B*d + 3*A*b*e - 13*a*B*e)*(d + e*x)^(11/2))/(40*
b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(13/2))/(5*b*(b*d - a*e)*(
a + b*x)^5) - (231*e^4*Sqrt[b*d - a*e]*(10*b*B*d + 3*A*b*e - 13*a*B*e)*ArcTanh[(
Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))
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Rubi in Sympy [A] time = 154.078, size = 386, normalized size = 0.98 \[ \frac{\left (d + e x\right )^{\frac{13}{2}} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 A b e - 13 B a e + 10 B b d\right )}{40 b^{2} \left (a + b x\right )^{4} \left (a e - b d\right )} + \frac{11 e \left (d + e x\right )^{\frac{9}{2}} \left (3 A b e - 13 B a e + 10 B b d\right )}{240 b^{3} \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{33 e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (3 A b e - 13 B a e + 10 B b d\right )}{320 b^{4} \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{231 e^{3} \left (d + e x\right )^{\frac{5}{2}} \left (3 A b e - 13 B a e + 10 B b d\right )}{640 b^{5} \left (a + b x\right ) \left (a e - b d\right )} - \frac{77 e^{4} \left (d + e x\right )^{\frac{3}{2}} \left (3 A b e - 13 B a e + 10 B b d\right )}{128 b^{6} \left (a e - b d\right )} + \frac{231 e^{4} \sqrt{d + e x} \left (3 A b e - 13 B a e + 10 B b d\right )}{128 b^{7}} - \frac{231 e^{4} \sqrt{a e - b d} \left (3 A b e - 13 B a e + 10 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
(d + e*x)**(13/2)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) + (d + e*x)**(11/2)
*(3*A*b*e - 13*B*a*e + 10*B*b*d)/(40*b**2*(a + b*x)**4*(a*e - b*d)) + 11*e*(d +
e*x)**(9/2)*(3*A*b*e - 13*B*a*e + 10*B*b*d)/(240*b**3*(a + b*x)**3*(a*e - b*d))
+ 33*e**2*(d + e*x)**(7/2)*(3*A*b*e - 13*B*a*e + 10*B*b*d)/(320*b**4*(a + b*x)**
2*(a*e - b*d)) + 231*e**3*(d + e*x)**(5/2)*(3*A*b*e - 13*B*a*e + 10*B*b*d)/(640*
b**5*(a + b*x)*(a*e - b*d)) - 77*e**4*(d + e*x)**(3/2)*(3*A*b*e - 13*B*a*e + 10*
B*b*d)/(128*b**6*(a*e - b*d)) + 231*e**4*sqrt(d + e*x)*(3*A*b*e - 13*B*a*e + 10*
B*b*d)/(128*b**7) - 231*e**4*sqrt(a*e - b*d)*(3*A*b*e - 13*B*a*e + 10*B*b*d)*ata
n(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(15/2))
_______________________________________________________________________________________
Mathematica [A] time = 1.10335, size = 299, normalized size = 0.76 \[ -\frac{231 e^4 \sqrt{b d-a e} (-13 a B e+3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{15/2}}-\frac{\sqrt{d+e x} \left (-1280 e^4 (a+b x)^5 (-18 a B e+3 A b e+16 b B d)+45 e^3 (a+b x)^4 (b d-a e) (-791 a B e+281 A b e+510 b B d)+10 e^2 (a+b x)^3 (b d-a e)^2 (-2107 a B e+1077 A b e+1030 b B d)+48 (a+b x) (b d-a e)^4 (-61 a B e+51 A b e+10 b B d)+8 e (a+b x)^2 (b d-a e)^3 (-1253 a B e+843 A b e+410 b B d)+384 (A b-a B) (b d-a e)^5-1280 b B e^5 x (a+b x)^5\right )}{1920 b^7 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(11/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
-(Sqrt[d + e*x]*(384*(A*b - a*B)*(b*d - a*e)^5 + 48*(b*d - a*e)^4*(10*b*B*d + 51
*A*b*e - 61*a*B*e)*(a + b*x) + 8*e*(b*d - a*e)^3*(410*b*B*d + 843*A*b*e - 1253*a
*B*e)*(a + b*x)^2 + 10*e^2*(b*d - a*e)^2*(1030*b*B*d + 1077*A*b*e - 2107*a*B*e)*
(a + b*x)^3 + 45*e^3*(b*d - a*e)*(510*b*B*d + 281*A*b*e - 791*a*B*e)*(a + b*x)^4
- 1280*e^4*(16*b*B*d + 3*A*b*e - 18*a*B*e)*(a + b*x)^5 - 1280*b*B*e^5*x*(a + b*
x)^5))/(1920*b^7*(a + b*x)^5) - (231*e^4*Sqrt[b*d - a*e]*(10*b*B*d + 3*A*b*e - 1
3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))
_______________________________________________________________________________________
Maple [B] time = 0.067, size = 1633, normalized size = 4.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
-12*e^5/b^7*a*B*(e*x+d)^(1/2)+2*e^5/b^6*A*(e*x+d)^(1/2)+2/3*e^4/b^6*B*(e*x+d)^(3
/2)+10*e^4/b^6*B*d*(e*x+d)^(1/2)+3349/96*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^5
-515/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*d^6+4075/96*e^4/b/(b*e*x+a*e)^5*B*(e
*x+d)^(7/2)*d^3+693/128*e^5/b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*A*d-693/128*e^6/b^6/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*A*a+3003/128*e^6/b^7/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*a^2*B-1467/128*e^10/b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a
^6+843/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A*a-843/128*e^5/b/(b*e*x+a*e)^5*(
e*x+d)^(9/2)*A*d-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^3-4955/32*e
^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d^2+12485/64*e^7/b^4/(b*e*x+a*e)^5*(e*x
+d)^(1/2)*B*a^3*d^3-17635/128*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^4+4919
/12*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^2*d^3+2931/32*e^7/b^3/(b*e*x+a*e)^5*
(e*x+d)^(3/2)*A*a^2*d^2-977/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^3+22607
/96*e^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4*d-36421/192*e^5/b^2/(b*e*x+a*e)^5*
(e*x+d)^(3/2)*B*a*d^4+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^4-1327/
32*e^6/b^2/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a*d+2701/16*e^6/b^3/(b*e*x+a*e)^5*B*(e*
x+d)^(7/2)*a^2*d-9477/64*e^5/b^2/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d^2-42283/96*e^
7/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d^2-393/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^
(5/2)*A*a^2*d+393/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d^2+4619/15*e^7/b^4/
(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3*d-2113/5*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B
*a^2*d^2+3833/15*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^3-977/16*e^8/b^4/(b*e
*x+a*e)^5*(e*x+d)^(3/2)*A*a^3*d+3903/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a
*d-5313/128*e^5/b^6/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*B*d*a+6617/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^5+8365/128*e^9/b^6/
(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5*d-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)
*A*a^4*d+2185/64*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d^2-2373/128*e^6/b^3/
(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a^2+131/5*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^
3-131/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^3-1253/15*e^8/b^5/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*B*a^4+977/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^4+977/64*e^5/b/
(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^4-9629/192*e^9/b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B
*a^5+437/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^5-437/128*e^5/b/(b*e*x+a*e
)^5*(e*x+d)^(1/2)*A*d^5+1327/64*e^7/b^3/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a^2+1327/6
4*e^5/b/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d^2-12131/192*e^7/b^4/(b*e*x+a*e)^5*B*(e*x
+d)^(7/2)*a^3+1155/64*e^4/b^5/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e
-b*d))^(1/2))*B*d^2-765/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d^2-172/3*e^4/b/(
b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^4
_______________________________________________________________________________________
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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.320598, 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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
[-1/3840*(3465*(10*B*a^5*b*d*e^4 - (13*B*a^6 - 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4
- (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (13*B*a^2*b^4 - 3*A*a*
b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (13*B*a^3*b^3 - 3*A*a^2*b^4)*e^5)*x^3 +
10*(10*B*a^3*b^3*d*e^4 - (13*B*a^4*b^2 - 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^
2*d*e^4 - (13*B*a^5*b - 3*A*a^4*b^2)*e^5)*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*(1280*B*b^6*e^
5*x^6 - 96*(B*a*b^5 + 4*A*b^6)*d^5 - 176*(2*B*a^2*b^4 + 3*A*a*b^5)*d^4*e - 396*(
3*B*a^3*b^3 + 2*A*a^2*b^4)*d^3*e^2 - 1386*(4*B*a^4*b^2 + A*a^3*b^3)*d^2*e^3 + 11
55*(43*B*a^5*b - 3*A*a^4*b^2)*d*e^4 - 3465*(13*B*a^6 - 3*A*a^5*b)*e^5 + 1280*(16
*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 - 5*(4590*B*b^6*d^2*e^3 - (32189*
B*a*b^5 - 2529*A*b^6)*d*e^4 + 2123*(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 - 10*(103
0*B*b^6*d^3*e^2 + 3*(1671*B*a*b^5 + 359*A*b^6)*d^2*e^3 - 22*(1757*B*a^2*b^4 - 13
2*A*a*b^5)*d*e^4 + 2607*(13*B*a^3*b^3 - 3*A*a^2*b^4)*e^5)*x^3 - 2*(1640*B*b^6*d^
4*e + 2*(2759*B*a*b^5 + 1686*A*b^6)*d^3*e^2 + 33*(797*B*a^2*b^4 + 183*A*a*b^5)*d
^2*e^3 - 33*(6547*B*a^3*b^3 - 477*A*a^2*b^4)*d*e^4 + 14784*(13*B*a^4*b^2 - 3*A*a
^3*b^3)*e^5)*x^2 - 2*(240*B*b^6*d^5 + 8*(107*B*a*b^5 + 153*A*b^6)*d^4*e + 22*(13
1*B*a^2*b^4 + 84*A*a*b^5)*d^3*e^2 + 99*(137*B*a^3*b^3 + 33*A*a^2*b^4)*d^2*e^3 -
462*(253*B*a^4*b^2 - 18*A*a^3*b^3)*d*e^4 + 8085*(13*B*a^5*b - 3*A*a^4*b^2)*e^5)*
x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 +
5*a^4*b^8*x + a^5*b^7), -1/1920*(3465*(10*B*a^5*b*d*e^4 - (13*B*a^6 - 3*A*a^5*b)
*e^5 + (10*B*b^6*d*e^4 - (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 -
(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (13*B*a^3*b^3 -
3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (13*B*a^4*b^2 - 3*A*a^3*b^3)*e^
5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt(-(b*d -
a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (1280*B*b^6*e^5*x^6 - 96*(
B*a*b^5 + 4*A*b^6)*d^5 - 176*(2*B*a^2*b^4 + 3*A*a*b^5)*d^4*e - 396*(3*B*a^3*b^3
+ 2*A*a^2*b^4)*d^3*e^2 - 1386*(4*B*a^4*b^2 + A*a^3*b^3)*d^2*e^3 + 1155*(43*B*a^5
*b - 3*A*a^4*b^2)*d*e^4 - 3465*(13*B*a^6 - 3*A*a^5*b)*e^5 + 1280*(16*B*b^6*d*e^4
- (13*B*a*b^5 - 3*A*b^6)*e^5)*x^5 - 5*(4590*B*b^6*d^2*e^3 - (32189*B*a*b^5 - 25
29*A*b^6)*d*e^4 + 2123*(13*B*a^2*b^4 - 3*A*a*b^5)*e^5)*x^4 - 10*(1030*B*b^6*d^3*
e^2 + 3*(1671*B*a*b^5 + 359*A*b^6)*d^2*e^3 - 22*(1757*B*a^2*b^4 - 132*A*a*b^5)*d
*e^4 + 2607*(13*B*a^3*b^3 - 3*A*a^2*b^4)*e^5)*x^3 - 2*(1640*B*b^6*d^4*e + 2*(275
9*B*a*b^5 + 1686*A*b^6)*d^3*e^2 + 33*(797*B*a^2*b^4 + 183*A*a*b^5)*d^2*e^3 - 33*
(6547*B*a^3*b^3 - 477*A*a^2*b^4)*d*e^4 + 14784*(13*B*a^4*b^2 - 3*A*a^3*b^3)*e^5)
*x^2 - 2*(240*B*b^6*d^5 + 8*(107*B*a*b^5 + 153*A*b^6)*d^4*e + 22*(131*B*a^2*b^4
+ 84*A*a*b^5)*d^3*e^2 + 99*(137*B*a^3*b^3 + 33*A*a^2*b^4)*d^2*e^3 - 462*(253*B*a
^4*b^2 - 18*A*a^3*b^3)*d*e^4 + 8085*(13*B*a^5*b - 3*A*a^4*b^2)*e^5)*x)*sqrt(e*x
+ d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x
+ a^5*b^7)]
_______________________________________________________________________________________
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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.328542, 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)^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]
Done