3.1877 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e^3 (a+b x) (-7 a B e-A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac{5 e^2 \sqrt{d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e-A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(-5*e^2*(8*b*B*d - A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (5*e*(8*b*B*d - A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(96*b^
3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - A*b*e - 7*a
*B*e)*(d + e*x)^(5/2))/(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)^(7/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (5*e^3*(8*b*B*d - A*b*e - 7*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(9/2)*(b*d - a*e)^(3/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])

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Rubi [A]  time = 0.783216, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e^3 (a+b x) (-7 a B e-A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac{5 e^2 \sqrt{d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e-A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-5*e^2*(8*b*B*d - A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (5*e*(8*b*B*d - A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(96*b^
3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - A*b*e - 7*a
*B*e)*(d + e*x)^(5/2))/(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)^(7/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (5*e^3*(8*b*B*d - A*b*e - 7*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(9/2)*(b*d - a*e)^(3/2)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.02379, size = 235, normalized size = 0.65 \[ \frac{(a+b x) \left (\frac{5 e^3 (7 a B e+A b e-8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (3 e^2 (a+b x)^3 (-93 a B e+5 A b e+88 b B d)+8 (a+b x) (b d-a e)^2 (-25 a B e+17 A b e+8 b B d)+2 e (a+b x)^2 (b d-a e) (-163 a B e+59 A b e+104 b B d)+48 (A b-a B) (b d-a e)^3\right )}{3 b^4 (a+b x)^4 (b d-a e)}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(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)^3 + 8*(b*d - a*e)^2*(8*b
*B*d + 17*A*b*e - 25*a*B*e)*(a + b*x) + 2*e*(b*d - a*e)*(104*b*B*d + 59*A*b*e -
163*a*B*e)*(a + b*x)^2 + 3*e^2*(88*b*B*d + 5*A*b*e - 93*a*B*e)*(a + b*x)^3))/(3*
b^4*(b*d - a*e)*(a + b*x)^4) + (5*e^3*(-8*b*B*d + A*b*e + 7*a*B*e)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(9/2)*(b*d - a*e)^(3/2))))/(64*Sqrt[(a +
b*x)^2])

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Maple [B]  time = 0.033, size = 1273, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(630*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^3*b^2*e^5-55*A*(b
*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3-55*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/
2)*b^4*d^2*e-15*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+15*A*(b*(a*e-b*d))
^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+90*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
x^2*a^2*b^3*e^5-385*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3-120*B*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*b*d*e^4-480*B*arctan((e*x+d)^(1/2)*b/(b*
(a*e-b*d))^(1/2))*x^3*a*b^4*d*e^4+1095*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3
*d*e+105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*a*b^4*e^5-120*B*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*d*e^4+60*A*arctan((e*x+d)^(1/2)*b
/(b*(a*e-b*d))^(1/2))*x^3*a*b^4*e^5+420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(
1/2))*x^3*a^2*b^3*e^5-279*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*a*b^3*e-73*A*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2+73*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^
4*d*e-1265*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e-480*B*arctan((e*x+d)^
(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*d*e^4+110*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(
3/2)*a*b^3*d*e^2+45*A*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-45*A*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2+435*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*a^3*b*d*e^3-675*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2+465*B*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+1210*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3
/2)*a^2*b^2*d*e^2-720*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^3*
d*e^4+15*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^4*b^5*e^5+15*A*(b*(a*e-
b*d))^(1/2)*(e*x+d)^(7/2)*b^4*e+264*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^4*d-58
4*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^4*d^2+15*A*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*a^4*b*e^5+440*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^4*d^3-105*B*(
b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^4*e^4-120*B*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)
*b^4*d^4+105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^5*e^5+60*A*arctan((
e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b^2*e^5-511*B*(b*(a*e-b*d))^(1/2)*(e*x
+d)^(5/2)*a^2*b^2*e^2+420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^4*b*
e^5)/e*(b*x+a)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)/b^4/((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)^(5/2)/(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 = 0.296219, 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)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(2*(16*(B*a*b^3 + 3*A*b^4)*d^3 + 8*(3*B*a^2*b^2 - A*a*b^3)*d^2*e + 10*(5
*B*a^3*b - A*a^2*b^2)*d*e^2 - 15*(7*B*a^4 + A*a^3*b)*e^3 + 3*(88*B*b^4*d*e^2 - (
93*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + (208*B*b^4*d^2*e + 2*(129*B*a*b^3 + 59*A*b^4)*d
*e^2 - 73*(7*B*a^2*b^2 + A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 + 8*(11*B*a*b^3 + 17*
A*b^4)*d^2*e + 4*(47*B*a^2*b^2 - 9*A*a*b^3)*d*e^2 - 55*(7*B*a^3*b + A*a^2*b^2)*e
^3)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) - 15*(8*B*a^4*b*d*e^3 - (7*B*a^5 + A*a^
4*b)*e^4 + (8*B*b^5*d*e^3 - (7*B*a*b^4 + A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 -
(7*B*a^2*b^3 + A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (7*B*a^3*b^2 + A*a^2*b
^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (7*B*a^4*b + A*a^3*b^2)*e^4)*x)*log((sqrt(
b^2*d - a*b*e)*(b*e*x + 2*b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a
)))/((a^4*b^5*d - a^5*b^4*e + (b^9*d - a*b^8*e)*x^4 + 4*(a*b^8*d - a^2*b^7*e)*x^
3 + 6*(a^2*b^7*d - a^3*b^6*e)*x^2 + 4*(a^3*b^6*d - a^4*b^5*e)*x)*sqrt(b^2*d - a*
b*e)), -1/192*((16*(B*a*b^3 + 3*A*b^4)*d^3 + 8*(3*B*a^2*b^2 - A*a*b^3)*d^2*e + 1
0*(5*B*a^3*b - A*a^2*b^2)*d*e^2 - 15*(7*B*a^4 + A*a^3*b)*e^3 + 3*(88*B*b^4*d*e^2
- (93*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + (208*B*b^4*d^2*e + 2*(129*B*a*b^3 + 59*A*b^
4)*d*e^2 - 73*(7*B*a^2*b^2 + A*a*b^3)*e^3)*x^2 + (64*B*b^4*d^3 + 8*(11*B*a*b^3 +
17*A*b^4)*d^2*e + 4*(47*B*a^2*b^2 - 9*A*a*b^3)*d*e^2 - 55*(7*B*a^3*b + A*a^2*b^
2)*e^3)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) + 15*(8*B*a^4*b*d*e^3 - (7*B*a^5 +
A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (7*B*a*b^4 + A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e
^3 - (7*B*a^2*b^3 + A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (7*B*a^3*b^2 + A*
a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (7*B*a^4*b + A*a^3*b^2)*e^4)*x)*arcta
n(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^4*b^5*d - a^5*b^4*e +
(b^9*d - a*b^8*e)*x^4 + 4*(a*b^8*d - a^2*b^7*e)*x^3 + 6*(a^2*b^7*d - a^3*b^6*e)*
x^2 + 4*(a^3*b^6*d - a^4*b^5*e)*x)*sqrt(-b^2*d + a*b*e))]

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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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.358208, size = 879, normalized size = 2.45 \[ -\frac{5 \,{\left (8 \, B b d e^{3} - 7 \, B a e^{4} - A b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \,{\left (b^{5} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{264 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} d e^{3} - 584 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{3} + 440 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{3} - 120 \, \sqrt{x e + d} B b^{4} d^{4} e^{3} - 279 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{3} e^{4} + 15 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{4} e^{4} + 1095 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{4} + 73 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{4} - 1265 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{4} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{4} + 465 \, \sqrt{x e + d} B a b^{3} d^{3} e^{4} + 15 \, \sqrt{x e + d} A b^{4} d^{3} e^{4} - 511 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{5} - 73 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{5} + 1210 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{5} + 110 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{5} - 675 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{5} - 45 \, \sqrt{x e + d} A a b^{3} d^{2} e^{5} - 385 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{6} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{6} + 435 \, \sqrt{x e + d} B a^{3} b d e^{6} + 45 \, \sqrt{x e + d} A a^{2} b^{2} d e^{6} - 105 \, \sqrt{x e + d} B a^{4} e^{7} - 15 \, \sqrt{x e + d} A a^{3} b e^{7}}{192 \,{\left (b^{5} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/64*(8*B*b*d*e^3 - 7*B*a*e^4 - A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a
*b*e))/((b^5*d*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a*b^4*e*sign(-(x*e + d)*b*
e + b*d*e - a*e^2))*sqrt(-b^2*d + a*b*e)) + 1/192*(264*(x*e + d)^(7/2)*B*b^4*d*e
^3 - 584*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 440*(x*e + d)^(3/2)*B*b^4*d^3*e^3 - 120
*sqrt(x*e + d)*B*b^4*d^4*e^3 - 279*(x*e + d)^(7/2)*B*a*b^3*e^4 + 15*(x*e + d)^(7
/2)*A*b^4*e^4 + 1095*(x*e + d)^(5/2)*B*a*b^3*d*e^4 + 73*(x*e + d)^(5/2)*A*b^4*d*
e^4 - 1265*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 55*(x*e + d)^(3/2)*A*b^4*d^2*e^4 +
465*sqrt(x*e + d)*B*a*b^3*d^3*e^4 + 15*sqrt(x*e + d)*A*b^4*d^3*e^4 - 511*(x*e +
d)^(5/2)*B*a^2*b^2*e^5 - 73*(x*e + d)^(5/2)*A*a*b^3*e^5 + 1210*(x*e + d)^(3/2)*B
*a^2*b^2*d*e^5 + 110*(x*e + d)^(3/2)*A*a*b^3*d*e^5 - 675*sqrt(x*e + d)*B*a^2*b^2
*d^2*e^5 - 45*sqrt(x*e + d)*A*a*b^3*d^2*e^5 - 385*(x*e + d)^(3/2)*B*a^3*b*e^6 -
55*(x*e + d)^(3/2)*A*a^2*b^2*e^6 + 435*sqrt(x*e + d)*B*a^3*b*d*e^6 + 45*sqrt(x*e
+ d)*A*a^2*b^2*d*e^6 - 105*sqrt(x*e + d)*B*a^4*e^7 - 15*sqrt(x*e + d)*A*a^3*b*e
^7)/((b^5*d*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - a*b^4*e*sign(-(x*e + d)*b*e +
b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^4)