3.1829 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ -\frac{3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}+\frac{3 e^3 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac{e^2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac{e \sqrt{d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-(e*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(16*b^3*(b*d - a*e)*(a + b*x)^3) -
(e^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*(a + b*x)^2)
+ (3*e^3*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b^3*(b*d - a*e)^3*(a + b
*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2))/(8*b^2*(b*d - a*e)*(a + b*x)^
4) - ((A*b - a*B)*(d + e*x)^(5/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (3*e^4*(2*b*B
*d - A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/
2)*(b*d - a*e)^(7/2))

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Rubi [A]  time = 0.620281, 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 \[ -\frac{3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}+\frac{3 e^3 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac{e^2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac{e \sqrt{d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(e*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(16*b^3*(b*d - a*e)*(a + b*x)^3) -
(e^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*(a + b*x)^2)
+ (3*e^3*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b^3*(b*d - a*e)^3*(a + b
*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2))/(8*b^2*(b*d - a*e)*(a + b*x)^
4) - ((A*b - a*B)*(d + e*x)^(5/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (3*e^4*(2*b*B
*d - A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/
2)*(b*d - a*e)^(7/2))

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Rubi in Sympy [A]  time = 129.352, size = 284, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b e + B a e - 2 B b d\right )}{8 b^{2} \left (a + b x\right )^{4} \left (a e - b d\right )} + \frac{3 e^{3} \sqrt{d + e x} \left (A b e + B a e - 2 B b d\right )}{128 b^{3} \left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{e^{2} \sqrt{d + e x} \left (A b e + B a e - 2 B b d\right )}{64 b^{3} \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} - \frac{e \sqrt{d + e x} \left (A b e + B a e - 2 B b d\right )}{16 b^{3} \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{3 e^{4} \left (A b e + B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{7}{2}} \left (a e - b d\right )^{\frac{7}{2}}} \]

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)**3,x)

[Out]

(d + e*x)**(5/2)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) - (d + e*x)**(3/2)*(
A*b*e + B*a*e - 2*B*b*d)/(8*b**2*(a + b*x)**4*(a*e - b*d)) + 3*e**3*sqrt(d + e*x
)*(A*b*e + B*a*e - 2*B*b*d)/(128*b**3*(a + b*x)*(a*e - b*d)**3) + e**2*sqrt(d +
e*x)*(A*b*e + B*a*e - 2*B*b*d)/(64*b**3*(a + b*x)**2*(a*e - b*d)**2) - e*sqrt(d
+ e*x)*(A*b*e + B*a*e - 2*B*b*d)/(16*b**3*(a + b*x)**3*(a*e - b*d)) + 3*e**4*(A*
b*e + B*a*e - 2*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(128*b**(7/2)
*(a*e - b*d)**(7/2))

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Mathematica [A]  time = 0.871216, size = 249, normalized size = 0.8 \[ \frac{3 e^4 (a B e+A b e-2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (15 e^3 (a+b x)^4 (a B e+A b e-2 b B d)+10 e^2 (a+b x)^3 (a e-b d) (a B e+A b e-2 b B d)+16 (a+b x) (b d-a e)^3 (-21 a B e+11 A b e+10 b B d)+8 e (a+b x)^2 (b d-a e)^2 (-31 a B e+A b e+30 b B d)+128 (A b-a B) (b d-a e)^4\right )}{640 b^3 (a+b x)^5 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(128*(A*b - a*B)*(b*d - a*e)^4 + 16*(b*d - a*e)^3*(10*b*B*d + 11
*A*b*e - 21*a*B*e)*(a + b*x) + 8*e*(b*d - a*e)^2*(30*b*B*d + A*b*e - 31*a*B*e)*(
a + b*x)^2 + 10*e^2*(-(b*d) + a*e)*(-2*b*B*d + A*b*e + a*B*e)*(a + b*x)^3 + 15*e
^3*(-2*b*B*d + A*b*e + a*B*e)*(a + b*x)^4))/(640*b^3*(b*d - a*e)^3*(a + b*x)^5)
+ (3*e^4*(-2*b*B*d + A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a
*e]])/(128*b^(7/2)*(b*d - a*e)^(7/2))

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Maple [B]  time = 0.038, size = 871, normalized size = 2.8 \[ \text{result too large to display} \]

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)^3,x)

[Out]

3/128*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
)^(9/2)*A+3/128*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)^(9/2)*a*B-3/64*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)^(9/2)*B*d+7/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2
)*(e*x+d)^(7/2)*A*b+7/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(
7/2)*a*B-7/32*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)*B*b*d+
1/5*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(5/2)*A-1/5*e^5/(b*e*x+a*e)^5/b/(a*e-b*d
)*(e*x+d)^(5/2)*B*a-7/64*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*A-7/64*e^5/(b*e*x+a*e
)^5/b^2*(e*x+d)^(3/2)*a*B+7/32*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*B*d-3/128*e^6/(
b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*A*a+3/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A*d-3
/128*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*a^2*B+9/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+
d)^(1/2)*B*d*a-3/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d^2+3/128*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)/(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^3/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3
*d^3)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B-3/64*e
^4/b^2/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(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)*(e*x + d)^(3/2)/(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 = 0.312465, 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)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[-1/1280*(2*(32*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(4*B*a^2*b^3 + 21*A*a*b^4)*d^3*e +
4*(3*B*a^3*b^2 + 62*A*a^2*b^3)*d^2*e^2 + 10*(2*B*a^4*b - A*a^3*b^2)*d*e^3 - 15*(
B*a^5 + A*a^4*b)*e^4 - 15*(2*B*b^5*d*e^3 - (B*a*b^4 + A*b^5)*e^4)*x^4 + 10*(2*B*
b^5*d^2*e^2 - (15*B*a*b^4 + A*b^5)*d*e^3 + 7*(B*a^2*b^3 + A*a*b^4)*e^4)*x^3 + 2*
(120*B*b^5*d^3*e - 2*(167*B*a*b^4 - 2*A*b^5)*d^2*e^2 + (233*B*a^2*b^3 - 23*A*a*b
^4)*d*e^3 - 64*(B*a^3*b^2 - A*a^2*b^3)*e^4)*x^2 + 2*(80*B*b^5*d^4 - 8*(21*B*a*b^
4 - 11*A*b^5)*d^3*e + 2*(23*B*a^2*b^3 - 128*A*a*b^4)*d^2*e^2 + (47*B*a^3*b^2 + 2
33*A*a^2*b^3)*d*e^3 - 35*(B*a^4*b + A*a^3*b^2)*e^4)*x)*sqrt(b^2*d - a*b*e)*sqrt(
e*x + d) - 15*(2*B*a^5*b*d*e^4 - (B*a^6 + A*a^5*b)*e^5 + (2*B*b^6*d*e^4 - (B*a*b
^5 + A*b^6)*e^5)*x^5 + 5*(2*B*a*b^5*d*e^4 - (B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*
(2*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(2*B*a^3*b^3*d*e^4 -
(B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(2*B*a^4*b^2*d*e^4 - (B*a^5*b + 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^3 - 3*a^6*b^5*d^2*e + 3*a^7*b^4*d*e^2 - a^8*b^
3*e^3 + (b^11*d^3 - 3*a*b^10*d^2*e + 3*a^2*b^9*d*e^2 - a^3*b^8*e^3)*x^5 + 5*(a*b
^10*d^3 - 3*a^2*b^9*d^2*e + 3*a^3*b^8*d*e^2 - a^4*b^7*e^3)*x^4 + 10*(a^2*b^9*d^3
- 3*a^3*b^8*d^2*e + 3*a^4*b^7*d*e^2 - a^5*b^6*e^3)*x^3 + 10*(a^3*b^8*d^3 - 3*a^
4*b^7*d^2*e + 3*a^5*b^6*d*e^2 - a^6*b^5*e^3)*x^2 + 5*(a^4*b^7*d^3 - 3*a^5*b^6*d^
2*e + 3*a^6*b^5*d*e^2 - a^7*b^4*e^3)*x)*sqrt(b^2*d - a*b*e)), -1/640*((32*(B*a*b
^4 + 4*A*b^5)*d^4 - 16*(4*B*a^2*b^3 + 21*A*a*b^4)*d^3*e + 4*(3*B*a^3*b^2 + 62*A*
a^2*b^3)*d^2*e^2 + 10*(2*B*a^4*b - A*a^3*b^2)*d*e^3 - 15*(B*a^5 + A*a^4*b)*e^4 -
15*(2*B*b^5*d*e^3 - (B*a*b^4 + A*b^5)*e^4)*x^4 + 10*(2*B*b^5*d^2*e^2 - (15*B*a*
b^4 + A*b^5)*d*e^3 + 7*(B*a^2*b^3 + A*a*b^4)*e^4)*x^3 + 2*(120*B*b^5*d^3*e - 2*(
167*B*a*b^4 - 2*A*b^5)*d^2*e^2 + (233*B*a^2*b^3 - 23*A*a*b^4)*d*e^3 - 64*(B*a^3*
b^2 - A*a^2*b^3)*e^4)*x^2 + 2*(80*B*b^5*d^4 - 8*(21*B*a*b^4 - 11*A*b^5)*d^3*e +
2*(23*B*a^2*b^3 - 128*A*a*b^4)*d^2*e^2 + (47*B*a^3*b^2 + 233*A*a^2*b^3)*d*e^3 -
35*(B*a^4*b + A*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*b*e)*sqrt(e*x + d) + 15*(2*B*a^
5*b*d*e^4 - (B*a^6 + A*a^5*b)*e^5 + (2*B*b^6*d*e^4 - (B*a*b^5 + A*b^6)*e^5)*x^5
+ 5*(2*B*a*b^5*d*e^4 - (B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(2*B*a^2*b^4*d*e^4 -
(B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(2*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + A*a^3*b^
3)*e^5)*x^2 + 5*(2*B*a^4*b^2*d*e^4 - (B*a^5*b + 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^3 - 3*a^6*b^5*d^2*e +
3*a^7*b^4*d*e^2 - a^8*b^3*e^3 + (b^11*d^3 - 3*a*b^10*d^2*e + 3*a^2*b^9*d*e^2 - a
^3*b^8*e^3)*x^5 + 5*(a*b^10*d^3 - 3*a^2*b^9*d^2*e + 3*a^3*b^8*d*e^2 - a^4*b^7*e^
3)*x^4 + 10*(a^2*b^9*d^3 - 3*a^3*b^8*d^2*e + 3*a^4*b^7*d*e^2 - a^5*b^6*e^3)*x^3
+ 10*(a^3*b^8*d^3 - 3*a^4*b^7*d^2*e + 3*a^5*b^6*d*e^2 - a^6*b^5*e^3)*x^2 + 5*(a^
4*b^7*d^3 - 3*a^5*b^6*d^2*e + 3*a^6*b^5*d*e^2 - a^7*b^4*e^3)*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.321954, 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)^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

Done