3.1881 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=424 \[ -\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac{35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e)/(24*b*(b*d - a*e)^
2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A
*b*e + a*B*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e + a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*S
qrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b*e + a*B*e
)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d -
a*e)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 1.05375, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac{35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]
*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e)/(24*b*(b*d - a*e)^
2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A
*b*e + a*B*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e + a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*S
qrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b*e + a*B*e
)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d -
a*e)^(11/2)*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 2.13316, size = 254, normalized size = 0.6 \[ \frac{(a+b x) \left (\frac{35 e^3 (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{11/2}}+\frac{\sqrt{d+e x} \left (\frac{3 e^2 (-35 a B e+187 A b e-152 b B d)}{a+b x}+\frac{2 e (a e-b d) (-35 a B e+123 A b e-88 b B d)}{(a+b x)^2}-\frac{8 (b d-a e)^2 (7 a B e-15 A b e+8 b B d)}{(a+b x)^3}+\frac{48 (a B-A b) (b d-a e)^3}{(a+b x)^4}+\frac{384 e^3 (A e-B d)}{d+e x}\right )}{3 (b d-a e)^5}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(3/2)*(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)/(a + b*x)^4 - (8*(
b*d - a*e)^2*(8*b*B*d - 15*A*b*e + 7*a*B*e))/(a + b*x)^3 + (2*e*(-(b*d) + a*e)*(
-88*b*B*d + 123*A*b*e - 35*a*B*e))/(a + b*x)^2 + (3*e^2*(-152*b*B*d + 187*A*b*e
- 35*a*B*e))/(a + b*x) + (384*e^3*(-(B*d) + A*e))/(d + e*x)))/(3*(b*d - a*e)^5)
+ (35*e^3*(8*b*B*d - 9*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(Sqrt[b]*(b*d - a*e)^(11/2))))/(64*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.05, size = 1493, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/192*(-5040*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x^2*a^
2*b^3*d*e^3-3360*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*x*a
^3*b^2*d*e^3+945*A*(b*(a*e-b*d))^(1/2)*x^4*b^4*e^4-279*B*(b*(a*e-b*d))^(1/2)*x*a
^4*e^4-64*B*(b*(a*e-b*d))^(1/2)*x*b^4*d^4+2511*A*(b*(a*e-b*d))^(1/2)*x*a^3*b*e^4
+72*A*(b*(a*e-b*d))^(1/2)*x*b^4*d^3*e+384*A*(b*(a*e-b*d))^(1/2)*a^4*e^4-48*A*(b*
(a*e-b*d))^(1/2)*b^4*d^4-105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+
d)^(1/2)*x^4*a*b^4*e^4-840*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)
^(1/2)*x^4*b^5*d*e^3+3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^
(1/2)*x^3*a*b^4*e^4-420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1
/2)*x^3*a^2*b^3*e^4+5670*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(
1/2)*x^2*a^2*b^3*e^4-630*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(
1/2)*x^2*a^3*b^2*e^4+3780*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^
(1/2)*x*a^3*b^2*e^4-420*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1
/2)*x*a^4*b*e^4-840*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*
a^4*b*d*e^3+1197*A*(b*(a*e-b*d))^(1/2)*x^2*a*b^3*d*e^3-4221*B*(b*(a*e-b*d))^(1/2
)*x^2*a^2*b^2*d*e^3-1050*B*(b*(a*e-b*d))^(1/2)*x^2*a*b^3*d^2*e^2+1665*A*(b*(a*e-
b*d))^(1/2)*x*a^2*b^2*d*e^3-468*A*(b*(a*e-b*d))^(1/2)*x*a*b^3*d^2*e^2-2417*B*(b*
(a*e-b*d))^(1/2)*x*a^3*b*d*e^3-1428*B*(b*(a*e-b*d))^(1/2)*x*a^2*b^2*d^2*e^2+408*
B*(b*(a*e-b*d))^(1/2)*x*a*b^3*d^3*e-3115*B*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*d*e^3-1
05*B*(b*(a*e-b*d))^(1/2)*x^4*a*b^3*e^4-840*B*(b*(a*e-b*d))^(1/2)*x^4*b^4*d*e^3+3
465*A*(b*(a*e-b*d))^(1/2)*x^3*a*b^3*e^4+315*A*(b*(a*e-b*d))^(1/2)*x^3*b^4*d*e^3-
385*B*(b*(a*e-b*d))^(1/2)*x^3*a^2*b^2*e^4-280*B*(b*(a*e-b*d))^(1/2)*x^3*b^4*d^2*
e^2+4599*A*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^2*e^4-126*A*(b*(a*e-b*d))^(1/2)*x^2*b^4
*d^2*e^2-511*B*(b*(a*e-b*d))^(1/2)*x^2*a^3*b*e^4+112*B*(b*(a*e-b*d))^(1/2)*x^2*b
^4*d^3*e-663*B*(b*(a*e-b*d))^(1/2)*a^4*d*e^3-16*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^4+
975*A*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3-630*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+
264*A*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e-370*B*(b*(a*e-b*d))^(1/2)*a^3*b*d^2*e^2+10
4*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^3*e+945*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*(e*x+d)^(1/2)*a^4*b*e^4-105*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2)
)*(e*x+d)^(1/2)*a^5*e^4+945*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d
)^(1/2)*x^4*b^5*e^4-3360*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(
1/2)*x^3*a*b^4*d*e^3)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^5/((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)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.334054, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/384*(105*(8*B*a^4*b*d*e^3 + (B*a^5 - 9*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 + (B*a*b
^4 - 9*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 + (B*a^2*b^3 - 9*A*a*b^4)*e^4)*x^3 +
6*(8*B*a^2*b^3*d*e^3 + (B*a^3*b^2 - 9*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^
3 + (B*a^4*b - 9*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d)*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)) + 2*(384*A*a^4*e^
4 - 16*(B*a*b^3 + 3*A*b^4)*d^4 + 8*(13*B*a^2*b^2 + 33*A*a*b^3)*d^3*e - 10*(37*B*
a^3*b + 63*A*a^2*b^2)*d^2*e^2 - 39*(17*B*a^4 - 25*A*a^3*b)*d*e^3 - 105*(8*B*b^4*
d*e^3 + (B*a*b^3 - 9*A*b^4)*e^4)*x^4 - 35*(8*B*b^4*d^2*e^2 + (89*B*a*b^3 - 9*A*b
^4)*d*e^3 + 11*(B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 + 7*(16*B*b^4*d^3*e - 6*(25*B*a*
b^3 + 3*A*b^4)*d^2*e^2 - 9*(67*B*a^2*b^2 - 19*A*a*b^3)*d*e^3 - 73*(B*a^3*b - 9*A
*a^2*b^2)*e^4)*x^2 - (64*B*b^4*d^4 - 24*(17*B*a*b^3 + 3*A*b^4)*d^3*e + 12*(119*B
*a^2*b^2 + 39*A*a*b^3)*d^2*e^2 + (2417*B*a^3*b - 1665*A*a^2*b^2)*d*e^3 + 279*(B*
a^4 - 9*A*a^3*b)*e^4)*x)*sqrt(b^2*d - a*b*e))/((a^4*b^5*d^5 - 5*a^5*b^4*d^4*e +
10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b*d*e^4 - a^9*e^5 + (b^9*d^5 - 5
*a*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a^3*b^6*d^2*e^3 + 5*a^4*b^5*d*e^4 - a^5*b
^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a^2*b^7*d^4*e + 10*a^3*b^6*d^3*e^2 - 10*a^4*b^5*d
^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b^3*e^5)*x^3 + 6*(a^2*b^7*d^5 - 5*a^3*b^6*d^4*e +
10*a^4*b^5*d^3*e^2 - 10*a^5*b^4*d^2*e^3 + 5*a^6*b^3*d*e^4 - a^7*b^2*e^5)*x^2 +
4*(a^3*b^6*d^5 - 5*a^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^2 - 10*a^6*b^3*d^2*e^3 + 5*a
^7*b^2*d*e^4 - a^8*b*e^5)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d)), 1/192*(105*(8*B
*a^4*b*d*e^3 + (B*a^5 - 9*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 + (B*a*b^4 - 9*A*b^5)*e^
4)*x^4 + 4*(8*B*a*b^4*d*e^3 + (B*a^2*b^3 - 9*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*
d*e^3 + (B*a^3*b^2 - 9*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 + (B*a^4*b - 9
*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt
(e*x + d))) + (384*A*a^4*e^4 - 16*(B*a*b^3 + 3*A*b^4)*d^4 + 8*(13*B*a^2*b^2 + 33
*A*a*b^3)*d^3*e - 10*(37*B*a^3*b + 63*A*a^2*b^2)*d^2*e^2 - 39*(17*B*a^4 - 25*A*a
^3*b)*d*e^3 - 105*(8*B*b^4*d*e^3 + (B*a*b^3 - 9*A*b^4)*e^4)*x^4 - 35*(8*B*b^4*d^
2*e^2 + (89*B*a*b^3 - 9*A*b^4)*d*e^3 + 11*(B*a^2*b^2 - 9*A*a*b^3)*e^4)*x^3 + 7*(
16*B*b^4*d^3*e - 6*(25*B*a*b^3 + 3*A*b^4)*d^2*e^2 - 9*(67*B*a^2*b^2 - 19*A*a*b^3
)*d*e^3 - 73*(B*a^3*b - 9*A*a^2*b^2)*e^4)*x^2 - (64*B*b^4*d^4 - 24*(17*B*a*b^3 +
3*A*b^4)*d^3*e + 12*(119*B*a^2*b^2 + 39*A*a*b^3)*d^2*e^2 + (2417*B*a^3*b - 1665
*A*a^2*b^2)*d*e^3 + 279*(B*a^4 - 9*A*a^3*b)*e^4)*x)*sqrt(-b^2*d + a*b*e))/((a^4*
b^5*d^5 - 5*a^5*b^4*d^4*e + 10*a^6*b^3*d^3*e^2 - 10*a^7*b^2*d^2*e^3 + 5*a^8*b*d*
e^4 - a^9*e^5 + (b^9*d^5 - 5*a*b^8*d^4*e + 10*a^2*b^7*d^3*e^2 - 10*a^3*b^6*d^2*e
^3 + 5*a^4*b^5*d*e^4 - a^5*b^4*e^5)*x^4 + 4*(a*b^8*d^5 - 5*a^2*b^7*d^4*e + 10*a^
3*b^6*d^3*e^2 - 10*a^4*b^5*d^2*e^3 + 5*a^5*b^4*d*e^4 - a^6*b^3*e^5)*x^3 + 6*(a^2
*b^7*d^5 - 5*a^3*b^6*d^4*e + 10*a^4*b^5*d^3*e^2 - 10*a^5*b^4*d^2*e^3 + 5*a^6*b^3
*d*e^4 - a^7*b^2*e^5)*x^2 + 4*(a^3*b^6*d^5 - 5*a^4*b^5*d^4*e + 10*a^5*b^4*d^3*e^
2 - 10*a^6*b^3*d^2*e^3 + 5*a^7*b^2*d*e^4 - a^8*b*e^5)*x)*sqrt(-b^2*d + a*b*e)*sq
rt(e*x + d))]

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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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.376907, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Done