3.1819 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{e^2 (-5 a B e-A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x} (-5 a B e-A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e-A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

-(e*(6*b*B*d - A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(8*b^3*(b*d - a*e)*(a + b*x)) - (
(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) -
((A*b - a*B)*(d + e*x)^(5/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (e^2*(6*b*B*d - A*
b*e - 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(7/2)*(b*d
- a*e)^(3/2))

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Rubi [A]  time = 0.403552, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{e^2 (-5 a B e-A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x} (-5 a B e-A b e+6 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac{(d+e x)^{3/2} (-5 a B e-A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

-(e*(6*b*B*d - A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(8*b^3*(b*d - a*e)*(a + b*x)) - (
(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(3/2))/(12*b^2*(b*d - a*e)*(a + b*x)^2) -
((A*b - a*B)*(d + e*x)^(5/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (e^2*(6*b*B*d - A*
b*e - 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(7/2)*(b*d
- a*e)^(3/2))

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Rubi in Sympy [A]  time = 76.6327, size = 187, normalized size = 0.89 \[ \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b e + 5 B a e - 6 B b d\right )}{12 b^{2} \left (a + b x\right )^{2} \left (a e - b d\right )} - \frac{e \sqrt{d + e x} \left (A b e + 5 B a e - 6 B b d\right )}{8 b^{3} \left (a + b x\right ) \left (a e - b d\right )} + \frac{e^{2} \left (A b e + 5 B a e - 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{7}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]

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)**2,x)

[Out]

(d + e*x)**(5/2)*(A*b - B*a)/(3*b*(a + b*x)**3*(a*e - b*d)) - (d + e*x)**(3/2)*(
A*b*e + 5*B*a*e - 6*B*b*d)/(12*b**2*(a + b*x)**2*(a*e - b*d)) - e*sqrt(d + e*x)*
(A*b*e + 5*B*a*e - 6*B*b*d)/(8*b**3*(a + b*x)*(a*e - b*d)) + e**2*(A*b*e + 5*B*a
*e - 6*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*b**(7/2)*(a*e - b*d
)**(3/2))

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Mathematica [A]  time = 0.581778, size = 168, normalized size = 0.8 \[ \frac{e^2 (5 a B e+A b e-6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} (b d-a e)^{3/2}}+\frac{\sqrt{d+e x} \left (\frac{3 e (a+b x)^2 (-11 a B e+A b e+10 b B d)}{a e-b d}-2 (a+b x) (-13 a B e+7 A b e+6 b B d)-8 (A b-a B) (b d-a e)\right )}{24 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(-8*(A*b - a*B)*(b*d - a*e) - 2*(6*b*B*d + 7*A*b*e - 13*a*B*e)*(a
+ b*x) + (3*e*(10*b*B*d + A*b*e - 11*a*B*e)*(a + b*x)^2)/(-(b*d) + a*e)))/(24*b
^3*(a + b*x)^3) + (e^2*(-6*b*B*d + A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[b*d - a*e]])/(8*b^(7/2)*(b*d - a*e)^(3/2))

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Maple [B]  time = 0.028, size = 487, normalized size = 2.3 \[{\frac{{e}^{3}A}{8\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{11\,a{e}^{3}B}{8\, \left ( bex+ae \right ) ^{3}b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{2}Bd}{4\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}A}{3\, \left ( bex+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a{e}^{3}B}{3\, \left ( bex+ae \right ) ^{3}{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{e}^{2} \left ( ex+d \right ) ^{3/2}Bd}{ \left ( bex+ae \right ) ^{3}b}}-{\frac{{e}^{4}Aa}{8\, \left ( bex+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}+{\frac{{e}^{3}Ad}{8\, \left ( bex+ae \right ) ^{3}b}\sqrt{ex+d}}-{\frac{5\,B{e}^{4}{a}^{2}}{8\, \left ( bex+ae \right ) ^{3}{b}^{3}}\sqrt{ex+d}}+{\frac{11\,{e}^{3}Bda}{8\, \left ( bex+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}-{\frac{3\,{e}^{2}B{d}^{2}}{4\, \left ( bex+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{{e}^{3}A}{8\, \left ( ae-bd \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{5\,a{e}^{3}B}{8\,{b}^{3} \left ( ae-bd \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-{\frac{3\,{e}^{2}Bd}{4\, \left ( ae-bd \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

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)^2,x)

[Out]

1/8*e^3/(b*e*x+a*e)^3/(a*e-b*d)*(e*x+d)^(5/2)*A-11/8*e^3/(b*e*x+a*e)^3/b/(a*e-b*
d)*(e*x+d)^(5/2)*a*B+5/4*e^2/(b*e*x+a*e)^3/(a*e-b*d)*(e*x+d)^(5/2)*B*d-1/3*e^3/(
b*e*x+a*e)^3/b*(e*x+d)^(3/2)*A-5/3*e^3/(b*e*x+a*e)^3/b^2*(e*x+d)^(3/2)*a*B+2*e^2
/(b*e*x+a*e)^3/b*(e*x+d)^(3/2)*B*d-1/8*e^4/(b*e*x+a*e)^3/b^2*(e*x+d)^(1/2)*A*a+1
/8*e^3/(b*e*x+a*e)^3/b*(e*x+d)^(1/2)*A*d-5/8*e^4/(b*e*x+a*e)^3/b^3*(e*x+d)^(1/2)
*a^2*B+11/8*e^3/(b*e*x+a*e)^3/b^2*(e*x+d)^(1/2)*B*d*a-3/4*e^2/(b*e*x+a*e)^3/b*(e
*x+d)^(1/2)*B*d^2+1/8*e^3/b^2/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)
*b/(b*(a*e-b*d))^(1/2))*A+5/8*e^3/b^3/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*arctan((e*x+
d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B-3/4*e^2/b^2/(a*e-b*d)/(b*(a*e-b*d))^(1/2)*ar
ctan((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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308962, 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)^2,x, algorithm="fricas")

[Out]

[-1/48*(2*sqrt(b^2*d - a*b*e)*(4*(B*a*b^2 + 2*A*b^3)*d^2 + 2*(4*B*a^2*b - A*a*b^
2)*d*e - 3*(5*B*a^3 + A*a^2*b)*e^2 + 3*(10*B*b^3*d*e - (11*B*a*b^2 - A*b^3)*e^2)
*x^2 + 2*(6*B*b^3*d^2 + (11*B*a*b^2 + 7*A*b^3)*d*e - 4*(5*B*a^2*b + A*a*b^2)*e^2
)*x)*sqrt(e*x + d) - 3*(6*B*a^3*b*d*e^2 - (5*B*a^4 + A*a^3*b)*e^3 + (6*B*b^4*d*e
^2 - (5*B*a*b^3 + A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2 - (5*B*a^2*b^2 + A*a*b^3)
*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (5*B*a^3*b + A*a^2*b^2)*e^3)*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^3*b^4*d - a^4*b^3*e + (b^7*d - a*b^6*e)*x^3 + 3*(a*b^6*d - a^2*b^5*e)*x^2 +
3*(a^2*b^5*d - a^3*b^4*e)*x)*sqrt(b^2*d - a*b*e)), -1/24*(sqrt(-b^2*d + a*b*e)*
(4*(B*a*b^2 + 2*A*b^3)*d^2 + 2*(4*B*a^2*b - A*a*b^2)*d*e - 3*(5*B*a^3 + A*a^2*b)
*e^2 + 3*(10*B*b^3*d*e - (11*B*a*b^2 - A*b^3)*e^2)*x^2 + 2*(6*B*b^3*d^2 + (11*B*
a*b^2 + 7*A*b^3)*d*e - 4*(5*B*a^2*b + A*a*b^2)*e^2)*x)*sqrt(e*x + d) + 3*(6*B*a^
3*b*d*e^2 - (5*B*a^4 + A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (5*B*a*b^3 + A*b^4)*e^3)*
x^3 + 3*(6*B*a*b^3*d*e^2 - (5*B*a^2*b^2 + A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e
^2 - (5*B*a^3*b + A*a^2*b^2)*e^3)*x)*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*s
qrt(e*x + d))))/((a^3*b^4*d - a^4*b^3*e + (b^7*d - a*b^6*e)*x^3 + 3*(a*b^6*d - a
^2*b^5*e)*x^2 + 3*(a^2*b^5*d - a^3*b^4*e)*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.294887, size = 514, normalized size = 2.46 \[ \frac{{\left (6 \, B b d e^{2} - 5 \, B a e^{3} - A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \,{\left (b^{4} d - a b^{3} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{30 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{2} - 48 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} + 18 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} - 33 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{3} + 3 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{3} + 88 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} - 51 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} - 3 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} + 48 \, \sqrt{x e + d} B a^{2} b d e^{4} + 6 \, \sqrt{x e + d} A a b^{2} d e^{4} - 15 \, \sqrt{x e + d} B a^{3} e^{5} - 3 \, \sqrt{x e + d} A a^{2} b e^{5}}{24 \,{\left (b^{4} d - a b^{3} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

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)^2,x, algorithm="giac")

[Out]

1/8*(6*B*b*d*e^2 - 5*B*a*e^3 - A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b
*e))/((b^4*d - a*b^3*e)*sqrt(-b^2*d + a*b*e)) - 1/24*(30*(x*e + d)^(5/2)*B*b^3*d
*e^2 - 48*(x*e + d)^(3/2)*B*b^3*d^2*e^2 + 18*sqrt(x*e + d)*B*b^3*d^3*e^2 - 33*(x
*e + d)^(5/2)*B*a*b^2*e^3 + 3*(x*e + d)^(5/2)*A*b^3*e^3 + 88*(x*e + d)^(3/2)*B*a
*b^2*d*e^3 + 8*(x*e + d)^(3/2)*A*b^3*d*e^3 - 51*sqrt(x*e + d)*B*a*b^2*d^2*e^3 -
3*sqrt(x*e + d)*A*b^3*d^2*e^3 - 40*(x*e + d)^(3/2)*B*a^2*b*e^4 - 8*(x*e + d)^(3/
2)*A*a*b^2*e^4 + 48*sqrt(x*e + d)*B*a^2*b*d*e^4 + 6*sqrt(x*e + d)*A*a*b^2*d*e^4
- 15*sqrt(x*e + d)*B*a^3*e^5 - 3*sqrt(x*e + d)*A*a^2*b*e^5)/((b^4*d - a*b^3*e)*(
(x*e + d)*b - b*d + a*e)^3)