3.1831 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac{7 e^4 (-a B e-9 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}+\frac{7 e^3 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac{7 e^2 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac{7 e \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-((A*b - a*B)*Sqrt[d + e*x])/(5*b*(b*d - a*e)*(a + b*x)^5) - ((10*b*B*d - 9*A*b*
e - a*B*e)*Sqrt[d + e*x])/(40*b*(b*d - a*e)^2*(a + b*x)^4) + (7*e*(10*b*B*d - 9*
A*b*e - a*B*e)*Sqrt[d + e*x])/(240*b*(b*d - a*e)^3*(a + b*x)^3) - (7*e^2*(10*b*B
*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(192*b*(b*d - a*e)^4*(a + b*x)^2) + (7*e^3*
(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b*(b*d - a*e)^5*(a + b*x)) - (7
*e^4*(10*b*B*d - 9*A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.726932, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{7 e^4 (-a B e-9 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}+\frac{7 e^3 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac{7 e^2 \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac{7 e \sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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]

-((A*b - a*B)*Sqrt[d + e*x])/(5*b*(b*d - a*e)*(a + b*x)^5) - ((10*b*B*d - 9*A*b*
e - a*B*e)*Sqrt[d + e*x])/(40*b*(b*d - a*e)^2*(a + b*x)^4) + (7*e*(10*b*B*d - 9*
A*b*e - a*B*e)*Sqrt[d + e*x])/(240*b*(b*d - a*e)^3*(a + b*x)^3) - (7*e^2*(10*b*B
*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(192*b*(b*d - a*e)^4*(a + b*x)^2) + (7*e^3*
(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b*(b*d - a*e)^5*(a + b*x)) - (7
*e^4*(10*b*B*d - 9*A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 144.477, size = 292, normalized size = 0.93 \[ \frac{7 e^{3} \sqrt{d + e x} \left (9 A b e + B a e - 10 B b d\right )}{128 b \left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{7 e^{2} \sqrt{d + e x} \left (9 A b e + B a e - 10 B b d\right )}{192 b \left (a + b x\right )^{2} \left (a e - b d\right )^{4}} + \frac{7 e \sqrt{d + e x} \left (9 A b e + B a e - 10 B b d\right )}{240 b \left (a + b x\right )^{3} \left (a e - b d\right )^{3}} + \frac{\sqrt{d + e x} \left (9 A b e + B a e - 10 B b d\right )}{40 b \left (a + b x\right )^{4} \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} + \frac{7 e^{4} \left (9 A b e + B a e - 10 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*e**3*sqrt(d + e*x)*(9*A*b*e + B*a*e - 10*B*b*d)/(128*b*(a + b*x)*(a*e - b*d)**
5) + 7*e**2*sqrt(d + e*x)*(9*A*b*e + B*a*e - 10*B*b*d)/(192*b*(a + b*x)**2*(a*e
- b*d)**4) + 7*e*sqrt(d + e*x)*(9*A*b*e + B*a*e - 10*B*b*d)/(240*b*(a + b*x)**3*
(a*e - b*d)**3) + sqrt(d + e*x)*(9*A*b*e + B*a*e - 10*B*b*d)/(40*b*(a + b*x)**4*
(a*e - b*d)**2) + sqrt(d + e*x)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) + 7*e
**4*(9*A*b*e + B*a*e - 10*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(12
8*b**(3/2)*(a*e - b*d)**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.999787, size = 252, normalized size = 0.81 \[ \frac{7 e^4 (a B e+9 A b e-10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}-\frac{\sqrt{d+e x} \left (105 e^3 (a+b x)^4 (a B e+9 A b e-10 b B d)+70 e^2 (a+b x)^3 (a e-b d) (a B e+9 A b e-10 b B d)+48 (a+b x) (b d-a e)^3 (-a B e-9 A b e+10 b B d)+56 e (a+b x)^2 (b d-a e)^2 (a B e+9 A b e-10 b B d)+384 (A b-a B) (b d-a e)^4\right )}{1920 b (a+b x)^5 (b d-a e)^5} \]

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 - 9*
A*b*e - a*B*e)*(a + b*x) + 56*e*(b*d - a*e)^2*(-10*b*B*d + 9*A*b*e + a*B*e)*(a +
b*x)^2 + 70*e^2*(-(b*d) + a*e)*(-10*b*B*d + 9*A*b*e + a*B*e)*(a + b*x)^3 + 105*
e^3*(-10*b*B*d + 9*A*b*e + a*B*e)*(a + b*x)^4))/(1920*b*(b*d - a*e)^5*(a + b*x)^
5) + (7*e^4*(-10*b*B*d + 9*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

_______________________________________________________________________________________

Maple [B]  time = 0.041, size = 1274, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

63/128*e^5/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^
3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*A+7/128*e^5/(b*e*x+a*e)^5*b^3/(a^
5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)
*(e*x+d)^(9/2)*a*B-35/64*e^4/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2
*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*B*d+147/64*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)^(7/2)*A+49/192*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)^(7/2)*a*B-245/96*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)^(7/2)*B*d+
21/5*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)
^(5/2)*A+7/15*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)^(5/2)*a*B-14/3*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)^(5/2)*B*d+237/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2
)*(e*x+d)^(3/2)*A*b+79/192*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)
^(3/2)*a*B-395/96*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(3/2)*B*
b*d+193/128*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*A-7/128*e^5/(b*e*x+a*e)^5/
b/(a*e-b*d)*(e*x+d)^(1/2)*a*B-93/64*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*B*
d+63/128*e^5/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^
4*d^4*e-b^5*d^5)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*A+7/128*e^5/b/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*
b^4*d^4*e-b^5*d^5)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2
))*a*B-35/64*e^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*
a*b^4*d^4*e-b^5*d^5)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1
/2))*B*d

_______________________________________________________________________________________

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)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.31813, 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)^3*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-1/3840*(2*(96*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(32*B*a^2*b^3 + 123*A*a*b^4)*d^3*e
+ 4*(289*B*a^3*b^2 + 1026*A*a^2*b^3)*d^2*e^2 - 10*(158*B*a^4*b + 447*A*a^3*b^2)*
d*e^3 - 15*(7*B*a^5 - 193*A*a^4*b)*e^4 - 105*(10*B*b^5*d*e^3 - (B*a*b^4 + 9*A*b^
5)*e^4)*x^4 + 70*(10*B*b^5*d^2*e^2 - (71*B*a*b^4 + 9*A*b^5)*d*e^3 + 7*(B*a^2*b^3
+ 9*A*a*b^4)*e^4)*x^3 - 14*(40*B*b^5*d^3*e - 18*(13*B*a*b^4 + 2*A*b^5)*d^2*e^2
+ 3*(221*B*a^2*b^3 + 69*A*a*b^4)*d*e^3 - 64*(B*a^3*b^2 + 9*A*a^2*b^3)*e^4)*x^2 +
2*(240*B*b^5*d^4 - 8*(163*B*a*b^4 + 27*A*b^5)*d^3*e + 6*(503*B*a^2*b^3 + 192*A*
a*b^4)*d^2*e^2 - 9*(471*B*a^3*b^2 + 289*A*a^2*b^3)*d*e^3 + 395*(B*a^4*b + 9*A*a^
3*b^2)*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(e*x + d) - 105*(10*B*a^5*b*d*e^4 - (B*a^
6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*(10*B*a*
b^5*d*e^4 - (B*a^2*b^4 + 9*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (B*a^3*b
^3 + 9*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*
e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (B*a^5*b + 9*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^5 - 5*a^6*b^5*d^4*e + 10*a^7*b^4*d^3*e^2 - 10*a^8*b^3*d^2*e^3 + 5*a
^9*b^2*d*e^4 - a^10*b*e^5 + (b^11*d^5 - 5*a*b^10*d^4*e + 10*a^2*b^9*d^3*e^2 - 10
*a^3*b^8*d^2*e^3 + 5*a^4*b^7*d*e^4 - a^5*b^6*e^5)*x^5 + 5*(a*b^10*d^5 - 5*a^2*b^
9*d^4*e + 10*a^3*b^8*d^3*e^2 - 10*a^4*b^7*d^2*e^3 + 5*a^5*b^6*d*e^4 - a^6*b^5*e^
5)*x^4 + 10*(a^2*b^9*d^5 - 5*a^3*b^8*d^4*e + 10*a^4*b^7*d^3*e^2 - 10*a^5*b^6*d^2
*e^3 + 5*a^6*b^5*d*e^4 - a^7*b^4*e^5)*x^3 + 10*(a^3*b^8*d^5 - 5*a^4*b^7*d^4*e +
10*a^5*b^6*d^3*e^2 - 10*a^6*b^5*d^2*e^3 + 5*a^7*b^4*d*e^4 - a^8*b^3*e^5)*x^2 + 5
*(a^4*b^7*d^5 - 5*a^5*b^6*d^4*e + 10*a^6*b^5*d^3*e^2 - 10*a^7*b^4*d^2*e^3 + 5*a^
8*b^3*d*e^4 - a^9*b^2*e^5)*x)*sqrt(b^2*d - a*b*e)), -1/1920*((96*(B*a*b^4 + 4*A*
b^5)*d^4 - 16*(32*B*a^2*b^3 + 123*A*a*b^4)*d^3*e + 4*(289*B*a^3*b^2 + 1026*A*a^2
*b^3)*d^2*e^2 - 10*(158*B*a^4*b + 447*A*a^3*b^2)*d*e^3 - 15*(7*B*a^5 - 193*A*a^4
*b)*e^4 - 105*(10*B*b^5*d*e^3 - (B*a*b^4 + 9*A*b^5)*e^4)*x^4 + 70*(10*B*b^5*d^2*
e^2 - (71*B*a*b^4 + 9*A*b^5)*d*e^3 + 7*(B*a^2*b^3 + 9*A*a*b^4)*e^4)*x^3 - 14*(40
*B*b^5*d^3*e - 18*(13*B*a*b^4 + 2*A*b^5)*d^2*e^2 + 3*(221*B*a^2*b^3 + 69*A*a*b^4
)*d*e^3 - 64*(B*a^3*b^2 + 9*A*a^2*b^3)*e^4)*x^2 + 2*(240*B*b^5*d^4 - 8*(163*B*a*
b^4 + 27*A*b^5)*d^3*e + 6*(503*B*a^2*b^3 + 192*A*a*b^4)*d^2*e^2 - 9*(471*B*a^3*b
^2 + 289*A*a^2*b^3)*d*e^3 + 395*(B*a^4*b + 9*A*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*
b*e)*sqrt(e*x + d) + 105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6
*d*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a
*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)*x^3 + 1
0*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e
^4 - (B*a^5*b + 9*A*a^4*b^2)*e^5)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*s
qrt(e*x + d))))/((a^5*b^6*d^5 - 5*a^6*b^5*d^4*e + 10*a^7*b^4*d^3*e^2 - 10*a^8*b^
3*d^2*e^3 + 5*a^9*b^2*d*e^4 - a^10*b*e^5 + (b^11*d^5 - 5*a*b^10*d^4*e + 10*a^2*b
^9*d^3*e^2 - 10*a^3*b^8*d^2*e^3 + 5*a^4*b^7*d*e^4 - a^5*b^6*e^5)*x^5 + 5*(a*b^10
*d^5 - 5*a^2*b^9*d^4*e + 10*a^3*b^8*d^3*e^2 - 10*a^4*b^7*d^2*e^3 + 5*a^5*b^6*d*e
^4 - a^6*b^5*e^5)*x^4 + 10*(a^2*b^9*d^5 - 5*a^3*b^8*d^4*e + 10*a^4*b^7*d^3*e^2 -
10*a^5*b^6*d^2*e^3 + 5*a^6*b^5*d*e^4 - a^7*b^4*e^5)*x^3 + 10*(a^3*b^8*d^5 - 5*a
^4*b^7*d^4*e + 10*a^5*b^6*d^3*e^2 - 10*a^6*b^5*d^2*e^3 + 5*a^7*b^4*d*e^4 - a^8*b
^3*e^5)*x^2 + 5*(a^4*b^7*d^5 - 5*a^5*b^6*d^4*e + 10*a^6*b^5*d^3*e^2 - 10*a^7*b^4
*d^2*e^3 + 5*a^8*b^3*d*e^4 - a^9*b^2*e^5)*x)*sqrt(-b^2*d + a*b*e))]

_______________________________________________________________________________________

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)/(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300515, 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)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Done