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3.1828 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=313 \[ \frac{e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac{e^3 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}-\frac{e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-(e^2*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*(a + b*x
)^2) - (e^3*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)^2
*(a + b*x)) - (e*(10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(48*b^3*(b*d -
a*e)*(a + b*x)^3) - ((10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(40*b^2*(b*
d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(7/2))/(5*b*(b*d - a*e)*(a + b*x)
^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))

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Rubi [A]  time = 0.675374, 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 \[ \frac{e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac{e^3 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}-\frac{e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*(a + b*x
)^2) - (e^3*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)^2
*(a + b*x)) - (e*(10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(3/2))/(48*b^3*(b*d -
a*e)*(a + b*x)^3) - ((10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(40*b^2*(b*
d - a*e)*(a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(7/2))/(5*b*(b*d - a*e)*(a + b*x)
^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))

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Rubi in Sympy [A]  time = 124.548, size = 296, normalized size = 0.95 \[ \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b - B a\right )}{5 b \left (a + b x\right )^{5} \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 A b e + 7 B a e - 10 B b d\right )}{40 b^{2} \left (a + b x\right )^{4} \left (a e - b d\right )} - \frac{e \left (d + e x\right )^{\frac{3}{2}} \left (3 A b e + 7 B a e - 10 B b d\right )}{48 b^{3} \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{e^{3} \sqrt{d + e x} \left (3 A b e + 7 B a e - 10 B b d\right )}{128 b^{4} \left (a + b x\right ) \left (a e - b d\right )^{2}} - \frac{e^{2} \sqrt{d + e x} \left (3 A b e + 7 B a e - 10 B b d\right )}{64 b^{4} \left (a + b x\right )^{2} \left (a e - b d\right )} + \frac{e^{4} \left (3 A b e + 7 B a e - 10 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{128 b^{\frac{9}{2}} \left (a e - b d\right )^{\frac{5}{2}}} \]

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)**3,x)

[Out]

(d + e*x)**(7/2)*(A*b - B*a)/(5*b*(a + b*x)**5*(a*e - b*d)) - (d + e*x)**(5/2)*(
3*A*b*e + 7*B*a*e - 10*B*b*d)/(40*b**2*(a + b*x)**4*(a*e - b*d)) - e*(d + e*x)**
(3/2)*(3*A*b*e + 7*B*a*e - 10*B*b*d)/(48*b**3*(a + b*x)**3*(a*e - b*d)) + e**3*s
qrt(d + e*x)*(3*A*b*e + 7*B*a*e - 10*B*b*d)/(128*b**4*(a + b*x)*(a*e - b*d)**2)
- e**2*sqrt(d + e*x)*(3*A*b*e + 7*B*a*e - 10*B*b*d)/(64*b**4*(a + b*x)**2*(a*e -
 b*d)) + e**4*(3*A*b*e + 7*B*a*e - 10*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e
 - b*d))/(128*b**(9/2)*(a*e - b*d)**(5/2))

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Mathematica [A]  time = 0.922241, size = 256, normalized size = 0.82 \[ \frac{e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} \left (-15 e^3 (a+b x)^4 (7 a B e+3 A b e-10 b B d)+10 e^2 (a+b x)^3 (b d-a e) (-121 a B e+3 A b e+118 b B d)+48 (a+b x) (b d-a e)^3 (-31 a B e+21 A b e+10 b B d)+8 e (a+b x)^2 (b d-a e)^2 (-263 a B e+93 A b e+170 b B d)+384 (A b-a B) (b d-a e)^4\right )}{1920 b^4 (a+b x)^5 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^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 + 21
*A*b*e - 31*a*B*e)*(a + b*x) + 8*e*(b*d - a*e)^2*(170*b*B*d + 93*A*b*e - 263*a*B
*e)*(a + b*x)^2 + 10*e^2*(b*d - a*e)*(118*b*B*d + 3*A*b*e - 121*a*B*e)*(a + b*x)
^3 - 15*e^3*(-10*b*B*d + 3*A*b*e + 7*a*B*e)*(a + b*x)^4))/(1920*b^4*(b*d - a*e)^
2*(a + b*x)^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))

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Maple [B]  time = 0.036, size = 872, 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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

3/128*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*A*b+7/128*e^5/
(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*a*B-5/64*e^4/(b*e*x+a*e)
^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*B*b*d+7/64*e^5/(b*e*x+a*e)^5/(a*e-b
*d)*(e*x+d)^(7/2)*A-79/192*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(7/2)*a*B+29/96
*e^4/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)*B*d-1/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(
5/2)*A-7/15*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(5/2)*a*B+2/3*e^4/(b*e*x+a*e)^5/b*(e*x
+d)^(5/2)*B*d-7/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*A*a+7/64*e^5/(b*e*x+a*e)^
5/b*(e*x+d)^(3/2)*A*d-49/192*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*a^2*B+119/192*e
^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*B*d*a-35/96*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)
*B*d^2-3/128*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*A*a^2+3/64*e^6/(b*e*x+a*e)^5/b^
2*(e*x+d)^(1/2)*A*a*d-3/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A*d^2-7/128*e^7/(b
*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*B*a^3+3/16*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*B*a
^2*d-27/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*B*a*d^2+5/64*e^4/(b*e*x+a*e)^5/b
*(e*x+d)^(1/2)*B*d^3+3/128*e^5/b^3/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(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^4/(a^2*e^2-2*a*b*d*
e+b^2*d^2)/(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^3/(a^2*e^2-2*a*b*d*e+b^2*d^2)/(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)^(5/2)/(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 = 0.313245, 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)^3,x, algorithm="fricas")

[Out]

[-1/3840*(2*(96*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(2*B*a^2*b^3 + 33*A*a*b^4)*d^3*e -
4*(11*B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^2 - 10*(8*B*a^4*b - 3*A*a^3*b^2)*d*e^3 + 15
*(7*B*a^5 + 3*A*a^4*b)*e^4 + 15*(10*B*b^5*d*e^3 - (7*B*a*b^4 + 3*A*b^5)*e^4)*x^4
 + 10*(118*B*b^5*d^2*e^2 - (179*B*a*b^4 - 3*A*b^5)*d*e^3 + (79*B*a^2*b^3 - 21*A*
a*b^4)*e^4)*x^3 + 2*(680*B*b^5*d^3*e - 6*(107*B*a*b^4 - 62*A*b^5)*d^2*e^2 - 3*(1
17*B*a^2*b^3 + 233*A*a*b^4)*d*e^3 + 64*(7*B*a^3*b^2 + 3*A*a^2*b^3)*e^4)*x^2 + 2*
(240*B*b^5*d^4 - 8*(13*B*a*b^4 - 63*A*b^5)*d^3*e - 6*(17*B*a^2*b^3 + 128*A*a*b^4
)*d^2*e^2 - 3*(63*B*a^3*b^2 - 23*A*a^2*b^3)*d*e^3 + 35*(7*B*a^4*b + 3*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 - (7*B*a^6 + 3
*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5
*d*e^4 - (7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*
b^3 + 3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^
3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^5*b + 3*A*a^4*b^2)*e^5)*x)*log((sqr
t(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^2 - 2*a^6*b^5*d*e + a^7*b^4*e^2 + (b^11*d^2 - 2*a*b^10*d*e + a
^2*b^9*e^2)*x^5 + 5*(a*b^10*d^2 - 2*a^2*b^9*d*e + a^3*b^8*e^2)*x^4 + 10*(a^2*b^9
*d^2 - 2*a^3*b^8*d*e + a^4*b^7*e^2)*x^3 + 10*(a^3*b^8*d^2 - 2*a^4*b^7*d*e + a^5*
b^6*e^2)*x^2 + 5*(a^4*b^7*d^2 - 2*a^5*b^6*d*e + a^6*b^5*e^2)*x)*sqrt(b^2*d - a*b
*e)), -1/1920*((96*(B*a*b^4 + 4*A*b^5)*d^4 - 16*(2*B*a^2*b^3 + 33*A*a*b^4)*d^3*e
 - 4*(11*B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^2 - 10*(8*B*a^4*b - 3*A*a^3*b^2)*d*e^3 +
 15*(7*B*a^5 + 3*A*a^4*b)*e^4 + 15*(10*B*b^5*d*e^3 - (7*B*a*b^4 + 3*A*b^5)*e^4)*
x^4 + 10*(118*B*b^5*d^2*e^2 - (179*B*a*b^4 - 3*A*b^5)*d*e^3 + (79*B*a^2*b^3 - 21
*A*a*b^4)*e^4)*x^3 + 2*(680*B*b^5*d^3*e - 6*(107*B*a*b^4 - 62*A*b^5)*d^2*e^2 - 3
*(117*B*a^2*b^3 + 233*A*a*b^4)*d*e^3 + 64*(7*B*a^3*b^2 + 3*A*a^2*b^3)*e^4)*x^2 +
 2*(240*B*b^5*d^4 - 8*(13*B*a*b^4 - 63*A*b^5)*d^3*e - 6*(17*B*a^2*b^3 + 128*A*a*
b^4)*d^2*e^2 - 3*(63*B*a^3*b^2 - 23*A*a^2*b^3)*d*e^3 + 35*(7*B*a^4*b + 3*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 - (7*B*a^6
 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5 + 5*(10*B*a
*b^5*d*e^4 - (7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*
a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^
3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^5*b + 3*A*a^4*b^2)*e^5)*x)*arct
an(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a^5*b^6*d^2 - 2*a^6*b^5
*d*e + a^7*b^4*e^2 + (b^11*d^2 - 2*a*b^10*d*e + a^2*b^9*e^2)*x^5 + 5*(a*b^10*d^2
 - 2*a^2*b^9*d*e + a^3*b^8*e^2)*x^4 + 10*(a^2*b^9*d^2 - 2*a^3*b^8*d*e + a^4*b^7*
e^2)*x^3 + 10*(a^3*b^8*d^2 - 2*a^4*b^7*d*e + a^5*b^6*e^2)*x^2 + 5*(a^4*b^7*d^2 -
 2*a^5*b^6*d*e + a^6*b^5*e^2)*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317858, 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)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

Done

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