3.1830 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ \frac{e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac{e^3 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac{e^2 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac{e \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-((10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(40*b^2*(b*d - a*e)*(a + b*x)^4)
- (e*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(240*b^2*(b*d - a*e)^2*(a +
b*x)^3) + (e^2*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(192*b^2*(b*d - a*e
)^3*(a + b*x)^2) - (e^3*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(128*b^2*(
b*d - a*e)^4*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(5*b*(b*d - a*e)*(a + b*
x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.684671, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac{e^3 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac{e^2 \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac{e \sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac{\sqrt{d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-((10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(40*b^2*(b*d - a*e)*(a + b*x)^4)
- (e*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(240*b^2*(b*d - a*e)^2*(a +
b*x)^3) + (e^2*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(192*b^2*(b*d - a*e
)^3*(a + b*x)^2) - (e^3*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(128*b^2*(
b*d - a*e)^4*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(5*b*(b*d - a*e)*(a + b*
x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt
[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

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Rubi in Sympy [A]  time = 138.788, size = 299, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} + \frac{e^{3} \sqrt{d + e x} \left (7 A b e + 3 B a e - 10 B b d\right )}{128 b^{2} \left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{e^{2} \sqrt{d + e x} \left (7 A b e + 3 B a e - 10 B b d\right )}{192 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} + \frac{e \sqrt{d + e x} \left (7 A b e + 3 B a e - 10 B b d\right )}{240 b^{2} \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} - \frac{\sqrt{d + e x} \left (7 A b e + 3 B a e - 10 B b d\right )}{40 b^{2} \left (a + b x\right )^{4} \left (a e - b d\right )} + \frac{e^{4} \left (7 A b e + 3 B a e - 10 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{5}{2}} \left (a e - b d\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(d + e*x)**(3/2)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) + e**3*sqrt(d + e*x)
*(7*A*b*e + 3*B*a*e - 10*B*b*d)/(128*b**2*(a + b*x)*(a*e - b*d)**4) + e**2*sqrt(
d + e*x)*(7*A*b*e + 3*B*a*e - 10*B*b*d)/(192*b**2*(a + b*x)**2*(a*e - b*d)**3) +
e*sqrt(d + e*x)*(7*A*b*e + 3*B*a*e - 10*B*b*d)/(240*b**2*(a + b*x)**3*(a*e - b*
d)**2) - sqrt(d + e*x)*(7*A*b*e + 3*B*a*e - 10*B*b*d)/(40*b**2*(a + b*x)**4*(a*e
- b*d)) + e**4*(7*A*b*e + 3*B*a*e - 10*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a
*e - b*d))/(128*b**(5/2)*(a*e - b*d)**(9/2))

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Mathematica [A]  time = 0.874788, size = 255, normalized size = 0.81 \[ \frac{e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (-15 e^3 (a+b x)^4 (3 a B e+7 A b e-10 b B d)-10 e^2 (a+b x)^3 (a e-b d) (3 a B e+7 A b e-10 b B d)+48 (a+b x) (b d-a e)^3 (-11 a B e+A b e+10 b B d)-8 e (a+b x)^2 (b d-a e)^2 (3 a B e+7 A b e-10 b B d)+384 (A b-a B) (b d-a e)^4\right )}{1920 b^2 (a+b x)^5 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[d + e*x])/(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)^4 + 48*(b*d - a*e)^3*(10*b*B*d + A*
b*e - 11*a*B*e)*(a + b*x) - 8*e*(b*d - a*e)^2*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*(a
+ b*x)^2 - 10*e^2*(-(b*d) + a*e)*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*(a + b*x)^3 -
15*e^3*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*(a + b*x)^4))/(1920*b^2*(b*d - a*e)^4*(a
+ b*x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/
Sqrt[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

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Maple [B]  time = 0.037, size = 1037, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

7/128*e^5/(b*e*x+a*e)^5*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3
*e+b^4*d^4)*(e*x+d)^(9/2)*A+3/128*e^5/(b*e*x+a*e)^5*b^2/(a^4*e^4-4*a^3*b*d*e^3+6
*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/2)*a*B-5/64*e^4/(b*e*x+a*e)^5
*b^3/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(e*x+d)^(9/
2)*B*d+49/192*e^5/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3
)*(e*x+d)^(7/2)*A+7/64*e^5/(b*e*x+a*e)^5*b/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-
b^3*d^3)*(e*x+d)^(7/2)*a*B-35/96*e^4/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*b*d*e^2+3*
a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)*B*d+7/15*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e
+b^2*d^2)*(e*x+d)^(5/2)*A*b+1/5*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e
*x+d)^(5/2)*a*B-2/3*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)*
B*b*d+79/192*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(3/2)*A-7/64*e^5/(b*e*x+a*e)^5/
b/(a*e-b*d)*(e*x+d)^(3/2)*a*B-29/96*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(3/2)*B*
d-7/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A-3/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^
(1/2)*a*B+5/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d+7/128*e^5/b/(a^4*e^4-4*a^3*
b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/(b*(a*e-b*d))^(1/2)*arctan((e*x
+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A+3/128*e^5/b^2/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^
2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(
a*e-b*d))^(1/2))*a*B-5/64*e^4/b/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3
*d^3*e+b^4*d^4)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*
B*d

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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)*sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.324939, 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)*sqrt(e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(2*(96*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(22*B*a^2*b^3 + 93*A*a*b^4)*d^3*e +
4*(109*B*a^3*b^2 + 526*A*a^2*b^3)*d^2*e^2 - 10*(12*B*a^4*b + 121*A*a^3*b^2)*d*e
^3 + 15*(3*B*a^5 + 7*A*a^4*b)*e^4 + 15*(10*B*b^5*d*e^3 - (3*B*a*b^4 + 7*A*b^5)*e
^4)*x^4 - 10*(10*B*b^5*d^2*e^2 - (73*B*a*b^4 + 7*A*b^5)*d*e^3 + 7*(3*B*a^2*b^3 +
7*A*a*b^4)*e^4)*x^3 + 2*(40*B*b^5*d^3*e - 2*(121*B*a*b^4 + 14*A*b^5)*d^2*e^2 +
(709*B*a^2*b^3 + 161*A*a*b^4)*d*e^3 - 64*(3*B*a^3*b^2 + 7*A*a^2*b^3)*e^4)*x^2 +
2*(240*B*b^5*d^4 - 8*(113*B*a*b^4 - 3*A*b^5)*d^3*e + 2*(589*B*a^2*b^3 - 64*A*a*b
^4)*d^2*e^2 - (409*B*a^3*b^2 - 289*A*a^2*b^3)*d*e^3 + 5*(21*B*a^4*b - 79*A*a^3*b
^2)*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) + 15*(10*B*a^5*b*d*e^4 - (3*B*a^6
+ 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (3*B*a*b^5 + 7*A*b^6)*e^5)*x^5 + 5*(10*B*a*
b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (3*B*a
^3*b^3 + 7*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 + 7*A*a^3
*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (3*B*a^5*b + 7*A*a^4*b^2)*e^5)*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^5*b^6*d^4 - 4*a^6*b^5*d^3*e + 6*a^7*b^4*d^2*e^2 - 4*a^8*b^3*d*e^3 +
a^9*b^2*e^4 + (b^11*d^4 - 4*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 - 4*a^3*b^8*d*e^3
+ a^4*b^7*e^4)*x^5 + 5*(a*b^10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4
*b^7*d*e^3 + a^5*b^6*e^4)*x^4 + 10*(a^2*b^9*d^4 - 4*a^3*b^8*d^3*e + 6*a^4*b^7*d^
2*e^2 - 4*a^5*b^6*d*e^3 + a^6*b^5*e^4)*x^3 + 10*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e +
6*a^5*b^6*d^2*e^2 - 4*a^6*b^5*d*e^3 + a^7*b^4*e^4)*x^2 + 5*(a^4*b^7*d^4 - 4*a^5
*b^6*d^3*e + 6*a^6*b^5*d^2*e^2 - 4*a^7*b^4*d*e^3 + a^8*b^3*e^4)*x)*sqrt(b^2*d -
a*b*e)), -1/1920*((96*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(22*B*a^2*b^3 + 93*A*a*b^4)*d
^3*e + 4*(109*B*a^3*b^2 + 526*A*a^2*b^3)*d^2*e^2 - 10*(12*B*a^4*b + 121*A*a^3*b^
2)*d*e^3 + 15*(3*B*a^5 + 7*A*a^4*b)*e^4 + 15*(10*B*b^5*d*e^3 - (3*B*a*b^4 + 7*A*
b^5)*e^4)*x^4 - 10*(10*B*b^5*d^2*e^2 - (73*B*a*b^4 + 7*A*b^5)*d*e^3 + 7*(3*B*a^2
*b^3 + 7*A*a*b^4)*e^4)*x^3 + 2*(40*B*b^5*d^3*e - 2*(121*B*a*b^4 + 14*A*b^5)*d^2*
e^2 + (709*B*a^2*b^3 + 161*A*a*b^4)*d*e^3 - 64*(3*B*a^3*b^2 + 7*A*a^2*b^3)*e^4)*
x^2 + 2*(240*B*b^5*d^4 - 8*(113*B*a*b^4 - 3*A*b^5)*d^3*e + 2*(589*B*a^2*b^3 - 64
*A*a*b^4)*d^2*e^2 - (409*B*a^3*b^2 - 289*A*a^2*b^3)*d*e^3 + 5*(21*B*a^4*b - 79*A
*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) - 15*(10*B*a^5*b*d*e^4 - (3
*B*a^6 + 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (3*B*a*b^5 + 7*A*b^6)*e^5)*x^5 + 5*(
10*B*a*b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 -
(3*B*a^3*b^3 + 7*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 +
7*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (3*B*a^5*b + 7*A*a^4*b^2)*e^5)*x
)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^5*b^6*d^4 - 4*a
^6*b^5*d^3*e + 6*a^7*b^4*d^2*e^2 - 4*a^8*b^3*d*e^3 + a^9*b^2*e^4 + (b^11*d^4 - 4
*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 - 4*a^3*b^8*d*e^3 + a^4*b^7*e^4)*x^5 + 5*(a*b^
10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4*b^7*d*e^3 + a^5*b^6*e^4)*x^
4 + 10*(a^2*b^9*d^4 - 4*a^3*b^8*d^3*e + 6*a^4*b^7*d^2*e^2 - 4*a^5*b^6*d*e^3 + a^
6*b^5*e^4)*x^3 + 10*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e + 6*a^5*b^6*d^2*e^2 - 4*a^6*b
^5*d*e^3 + a^7*b^4*e^4)*x^2 + 5*(a^4*b^7*d^4 - 4*a^5*b^6*d^3*e + 6*a^6*b^5*d^2*e
^2 - 4*a^7*b^4*d*e^3 + a^8*b^3*e^4)*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)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.311823, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done