3.1816 \(\int \frac{(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)
Optimal. Leaf size=332 \[ -\frac{21 e^2 (b d-a e)^{3/2} (-11 a B e+5 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}}+\frac{21 e^2 \sqrt{d+e x} (b d-a e) (-11 a B e+5 A b e+6 b B d)}{8 b^6}+\frac{7 e^2 (d+e x)^{3/2} (-11 a B e+5 A b e+6 b B d)}{8 b^5}+\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+5 A b e+6 b B d)}{40 b^4 (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+5 A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+5 A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{11/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
[Out]
(21*e^2*(b*d - a*e)*(6*b*B*d + 5*A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(8*b^6) + (7*e
^2*(6*b*B*d + 5*A*b*e - 11*a*B*e)*(d + e*x)^(3/2))/(8*b^5) + (21*e^2*(6*b*B*d +
5*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(40*b^4*(b*d - a*e)) - (3*e*(6*b*B*d + 5*A*
b*e - 11*a*B*e)*(d + e*x)^(7/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - ((6*b*B*d + 5*A
*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B
)*(d + e*x)^(11/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (21*e^2*(b*d - a*e)^(3/2)*(6
*b*B*d + 5*A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
8*b^(13/2))
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Rubi [A] time = 0.780549, antiderivative size = 332, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{21 e^2 (b d-a e)^{3/2} (-11 a B e+5 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}}+\frac{21 e^2 \sqrt{d+e x} (b d-a e) (-11 a B e+5 A b e+6 b B d)}{8 b^6}+\frac{7 e^2 (d+e x)^{3/2} (-11 a B e+5 A b e+6 b B d)}{8 b^5}+\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+5 A b e+6 b B d)}{40 b^4 (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+5 A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+5 A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{11/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(21*e^2*(b*d - a*e)*(6*b*B*d + 5*A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(8*b^6) + (7*e
^2*(6*b*B*d + 5*A*b*e - 11*a*B*e)*(d + e*x)^(3/2))/(8*b^5) + (21*e^2*(6*b*B*d +
5*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(40*b^4*(b*d - a*e)) - (3*e*(6*b*B*d + 5*A*
b*e - 11*a*B*e)*(d + e*x)^(7/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - ((6*b*B*d + 5*A
*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B
)*(d + e*x)^(11/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (21*e^2*(b*d - a*e)^(3/2)*(6
*b*B*d + 5*A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(
8*b^(13/2))
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Rubi in Sympy [A] time = 127.331, size = 326, normalized size = 0.98 \[ \frac{\left (d + e x\right )^{\frac{11}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (5 A b e - 11 B a e + 6 B b d\right )}{12 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{3 e \left (d + e x\right )^{\frac{7}{2}} \left (5 A b e - 11 B a e + 6 B b d\right )}{8 b^{3} \left (a + b x\right ) \left (a e - b d\right )} - \frac{21 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (5 A b e - 11 B a e + 6 B b d\right )}{40 b^{4} \left (a e - b d\right )} + \frac{7 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (5 A b e - 11 B a e + 6 B b d\right )}{8 b^{5}} - \frac{21 e^{2} \sqrt{d + e x} \left (a e - b d\right ) \left (5 A b e - 11 B a e + 6 B b d\right )}{8 b^{6}} + \frac{21 e^{2} \left (a e - b d\right )^{\frac{3}{2}} \left (5 A b e - 11 B a e + 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{13}{2}}} \]
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)**2,x)
[Out]
(d + e*x)**(11/2)*(A*b - B*a)/(3*b*(a + b*x)**3*(a*e - b*d)) + (d + e*x)**(9/2)*
(5*A*b*e - 11*B*a*e + 6*B*b*d)/(12*b**2*(a + b*x)**2*(a*e - b*d)) + 3*e*(d + e*x
)**(7/2)*(5*A*b*e - 11*B*a*e + 6*B*b*d)/(8*b**3*(a + b*x)*(a*e - b*d)) - 21*e**2
*(d + e*x)**(5/2)*(5*A*b*e - 11*B*a*e + 6*B*b*d)/(40*b**4*(a*e - b*d)) + 7*e**2*
(d + e*x)**(3/2)*(5*A*b*e - 11*B*a*e + 6*B*b*d)/(8*b**5) - 21*e**2*sqrt(d + e*x)
*(a*e - b*d)*(5*A*b*e - 11*B*a*e + 6*B*b*d)/(8*b**6) + 21*e**2*(a*e - b*d)**(3/2
)*(5*A*b*e - 11*B*a*e + 6*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*
b**(13/2))
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Mathematica [A] time = 0.896217, size = 262, normalized size = 0.79 \[ \frac{\sqrt{d+e x} \left (16 e^2 \left (150 a^2 B e^2-20 a b e (3 A e+13 B d)+b^2 d (65 A e+108 B d)\right )+16 b e^3 x (-20 a B e+5 A b e+21 b B d)-\frac{15 e (b d-a e)^2 (-89 a B e+55 A b e+34 b B d)}{a+b x}-\frac{10 (b d-a e)^3 (-31 a B e+25 A b e+6 b B d)}{(a+b x)^2}-\frac{40 (A b-a B) (b d-a e)^4}{(a+b x)^3}+48 b^2 B e^4 x^2\right )}{120 b^6}-\frac{21 e^2 (b d-a e)^{3/2} (-11 a B e+5 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(Sqrt[d + e*x]*(16*e^2*(150*a^2*B*e^2 - 20*a*b*e*(13*B*d + 3*A*e) + b^2*d*(108*B
*d + 65*A*e)) + 16*b*e^3*(21*b*B*d + 5*A*b*e - 20*a*B*e)*x + 48*b^2*B*e^4*x^2 -
(40*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x)^3 - (10*(b*d - a*e)^3*(6*b*B*d + 25*A*b
*e - 31*a*B*e))/(a + b*x)^2 - (15*e*(b*d - a*e)^2*(34*b*B*d + 55*A*b*e - 89*a*B*
e))/(a + b*x)))/(120*b^6) - (21*e^2*(b*d - a*e)^(3/2)*(6*b*B*d + 5*A*b*e - 11*a*
B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(13/2))
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Maple [B] time = 0.046, size = 1285, normalized size = 3.9 \[ \text{result too large to display} \]
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)^2,x)
[Out]
157/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a*d^2+55/4*e^4/b^2/(b*e*x+a*e)^3*(e*
x+d)^(5/2)*A*a*d-53/2*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^2*d-123/4*e^5/b^3/
(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2*d^2+2*e^2/b^4*B*(e*x+d)^(3/2)*d+2/3*e^3/b^4*A*
(e*x+d)^(3/2)+8*e^2/b/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^4-231/8*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*a^3+71/8*e^7/b^6/(b*e*x+
a*e)^3*(e*x+d)^(1/2)*B*a^5-55/8*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*a^2-55/8*e
^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*d^2+89/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*
B*a^3-41/8*e^7/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^4-41/8*e^3/b/(b*e*x+a*e)^3*(e
*x+d)^(1/2)*A*d^4-35/3*e^6/b^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a^3+35/3*e^3/b/(b*e
*x+a*e)^3*A*(e*x+d)^(3/2)*d^3+59/3*e^6/b^5/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^4+105
/8*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*a^2
+105/8*e^3/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A
*d^2-32*e^3/b^5*B*a*d*(e*x+d)^(1/2)+63/4*e^2/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x
+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d^3-17/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*
d^3-15/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^5+2/5*e^2/b^4*B*(e*x+d)^(5/2)+12*
e^2/b^4*B*d^2*(e*x+d)^(1/2)-8/3*e^3/b^5*B*(e*x+d)^(3/2)*a-8*e^4/b^5*A*a*(e*x+d)^
(1/2)+8*e^3/b^4*A*d*(e*x+d)^(1/2)+20*e^4/b^6*a^2*B*(e*x+d)^(1/2)+41/2*e^4/b^2/(b
*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d^3-157/4*e^6/b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^
4*d+273/4*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3*d^2-58*e^4/b^3/(b*e*x+a*e)^3
*(e*x+d)^(1/2)*B*a^2*d^3+191/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^4+41/2*
e^6/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^3*d+35*e^5/b^3/(b*e*x+a*e)^3*A*(e*x+d)^(
3/2)*a^2*d-35*e^4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*d^2-67*e^5/b^4/(b*e*x+a*e)
^3*B*(e*x+d)^(3/2)*a^3*d+83*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2*d^2-131/3*
e^3/b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d^3-105/4*e^4/b^4/(b*(a*e-b*d))^(1/2)*ar
ctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*a*d+147/2*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*a^2*d-483/8*e^3/b^4/(b*(a*e-b*d)
)^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*a*d^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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.317145, 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)^2,x, algorithm="fricas")
[Out]
[1/240*(315*(6*B*a^3*b^2*d^2*e^2 - (17*B*a^4*b - 5*A*a^3*b^2)*d*e^3 + (11*B*a^5
- 5*A*a^4*b)*e^4 + (6*B*b^5*d^2*e^2 - (17*B*a*b^4 - 5*A*b^5)*d*e^3 + (11*B*a^2*b
^3 - 5*A*a*b^4)*e^4)*x^3 + 3*(6*B*a*b^4*d^2*e^2 - (17*B*a^2*b^3 - 5*A*a*b^4)*d*e
^3 + (11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 + 3*(6*B*a^2*b^3*d^2*e^2 - (17*B*a^3*
b^2 - 5*A*a^2*b^3)*d*e^3 + (11*B*a^4*b - 5*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*(48*B*b^5*e^4*x^5 - 20*(B*a*b^4 + 2*A*b^5)*d^4 - 90*(2*B*a^2*b^3 + A*a*b^4)*d
^3*e + 63*(51*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^2 - 210*(31*B*a^4*b - 10*A*a^3*b^2)
*d*e^3 + 315*(11*B*a^5 - 5*A*a^4*b)*e^4 + 16*(21*B*b^5*d*e^3 - (11*B*a*b^4 - 5*A
*b^5)*e^4)*x^4 + 16*(108*B*b^5*d^2*e^2 - (197*B*a*b^4 - 65*A*b^5)*d*e^3 + 9*(11*
B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 - 3*(170*B*b^5*d^3*e - (2513*B*a*b^4 - 275*A*b^5
)*d^2*e^2 + 6*(814*B*a^2*b^3 - 265*A*a*b^4)*d*e^3 - 231*(11*B*a^3*b^2 - 5*A*a^2*
b^3)*e^4)*x^2 - 2*(30*B*b^5*d^4 + 5*(53*B*a*b^4 + 25*A*b^5)*d^3*e - 18*(244*B*a^
2*b^3 - 25*A*a*b^4)*d^2*e^2 + 63*(139*B*a^3*b^2 - 45*A*a^2*b^3)*d*e^3 - 420*(11*
B*a^4*b - 5*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7
*x + a^3*b^6), -1/120*(315*(6*B*a^3*b^2*d^2*e^2 - (17*B*a^4*b - 5*A*a^3*b^2)*d*e
^3 + (11*B*a^5 - 5*A*a^4*b)*e^4 + (6*B*b^5*d^2*e^2 - (17*B*a*b^4 - 5*A*b^5)*d*e^
3 + (11*B*a^2*b^3 - 5*A*a*b^4)*e^4)*x^3 + 3*(6*B*a*b^4*d^2*e^2 - (17*B*a^2*b^3 -
5*A*a*b^4)*d*e^3 + (11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 + 3*(6*B*a^2*b^3*d^2*e
^2 - (17*B*a^3*b^2 - 5*A*a^2*b^3)*d*e^3 + (11*B*a^4*b - 5*A*a^3*b^2)*e^4)*x)*sqr
t(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (48*B*b^5*e^4*x^5
- 20*(B*a*b^4 + 2*A*b^5)*d^4 - 90*(2*B*a^2*b^3 + A*a*b^4)*d^3*e + 63*(51*B*a^3*
b^2 - 5*A*a^2*b^3)*d^2*e^2 - 210*(31*B*a^4*b - 10*A*a^3*b^2)*d*e^3 + 315*(11*B*a
^5 - 5*A*a^4*b)*e^4 + 16*(21*B*b^5*d*e^3 - (11*B*a*b^4 - 5*A*b^5)*e^4)*x^4 + 16*
(108*B*b^5*d^2*e^2 - (197*B*a*b^4 - 65*A*b^5)*d*e^3 + 9*(11*B*a^2*b^3 - 5*A*a*b^
4)*e^4)*x^3 - 3*(170*B*b^5*d^3*e - (2513*B*a*b^4 - 275*A*b^5)*d^2*e^2 + 6*(814*B
*a^2*b^3 - 265*A*a*b^4)*d*e^3 - 231*(11*B*a^3*b^2 - 5*A*a^2*b^3)*e^4)*x^2 - 2*(3
0*B*b^5*d^4 + 5*(53*B*a*b^4 + 25*A*b^5)*d^3*e - 18*(244*B*a^2*b^3 - 25*A*a*b^4)*
d^2*e^2 + 63*(139*B*a^3*b^2 - 45*A*a^2*b^3)*d*e^3 - 420*(11*B*a^4*b - 5*A*a^3*b^
2)*e^4)*x)*sqrt(e*x + d))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*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)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.331995, 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)^2,x, algorithm="giac")
[Out]
Done