3.1821 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.503697, 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 \[ -\frac{e^2 (-a B e-5 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{7/2}}+\frac{e \sqrt{d+e x} (-a B e-5 A b e+6 b B d)}{8 b (a+b x) (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-5 A b e+6 b B d)}{12 b (a+b x)^2 (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 84.9277, size = 187, normalized size = 0.89 \[ \frac{e \sqrt{d + e x} \left (5 A b e + B a e - 6 B b d\right )}{8 b \left (a + b x\right ) \left (a e - b d\right )^{3}} + \frac{\sqrt{d + e x} \left (5 A b e + B a e - 6 B b d\right )}{12 b \left (a + b x\right )^{2} \left (a e - b d\right )^{2}} + \frac{\sqrt{d + e x} \left (A b - B a\right )}{3 b \left (a + b x\right )^{3} \left (a e - b d\right )} + \frac{e^{2} \left (5 A b e + B a e - 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.60277, size = 178, normalized size = 0.85 \[ \frac{e^2 (a B e+5 A b e-6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{7/2}}-\frac{\sqrt{d+e x} \left (2 (a+b x) (b d-a e) (-a B e-5 A b e+6 b B d)+3 e (a+b x)^2 (a B e+5 A b e-6 b B d)+8 (A b-a B) (b d-a e)^2\right )}{24 b (a+b x)^3 (b d-a e)^3} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.032, size = 679, normalized size = 3.3 \[{\frac{5\,A{b}^{2}{e}^{3}}{8\, \left ( bex+ae \right ) ^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{B{e}^{3}ab}{8\, \left ( bex+ae \right ) ^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{3\,{b}^{2}{e}^{2}Bd}{4\, \left ( bex+ae \right ) ^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{3}Ab}{3\, \left ( bex+ae \right ) ^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{B{e}^{3}a}{3\, \left ( bex+ae \right ) ^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{e}^{2} \left ( ex+d \right ) ^{3/2}Bbd}{ \left ( bex+ae \right ) ^{3} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }}+{\frac{11\,{e}^{3}A}{8\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) }\sqrt{ex+d}}-{\frac{B{e}^{3}a}{8\, \left ( bex+ae \right ) ^{3}b \left ( ae-bd \right ) }\sqrt{ex+d}}-{\frac{5\,{e}^{2}Bd}{4\, \left ( bex+ae \right ) ^{3} \left ( ae-bd \right ) }\sqrt{ex+d}}+{\frac{5\,{e}^{3}A}{8\,{a}^{3}{e}^{3}-24\,{a}^{2}bd{e}^{2}+24\,a{b}^{2}{d}^{2}e-8\,{b}^{3}{d}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{B{e}^{3}a}{8\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-{\frac{3\,{e}^{2}Bd}{4\,{a}^{3}{e}^{3}-12\,{a}^{2}bd{e}^{2}+12\,a{b}^{2}{d}^{2}e-4\,{b}^{3}{d}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305993, 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)^2*sqrt(e*x + d)),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*(8*B*a^2*b + 13*A*a
*b^2)*d*e - 3*(B*a^3 - 11*A*a^2*b)*e^2 - 3*(6*B*b^3*d*e - (B*a*b^2 + 5*A*b^3)*e^
2)*x^2 + 2*(6*B*b^3*d^2 - 5*(5*B*a*b^2 + A*b^3)*d*e + 4*(B*a^2*b + 5*A*a*b^2)*e^
2)*x)*sqrt(e*x + d) - 3*(6*B*a^3*b*d*e^2 - (B*a^4 + 5*A*a^3*b)*e^3 + (6*B*b^4*d*
e^2 - (B*a*b^3 + 5*A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2 - (B*a^2*b^2 + 5*A*a*b^3
)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (B*a^3*b + 5*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^3 - 3*a^4*b^3*d^2*e + 3*a^5*b^2*d*e^2 - a^6*b*e^3 + (b^7*d^3 - 3*a
*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*x^3 + 3*(a*b^6*d^3 - 3*a^2*b^5*d^2*e
+ 3*a^3*b^4*d*e^2 - a^4*b^3*e^3)*x^2 + 3*(a^2*b^5*d^3 - 3*a^3*b^4*d^2*e + 3*a^4
*b^3*d*e^2 - a^5*b^2*e^3)*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*(8*B*a^2*b + 13*A*a*b^2)*d*e - 3*(B*a^3 - 11*A*a^2
*b)*e^2 - 3*(6*B*b^3*d*e - (B*a*b^2 + 5*A*b^3)*e^2)*x^2 + 2*(6*B*b^3*d^2 - 5*(5*
B*a*b^2 + A*b^3)*d*e + 4*(B*a^2*b + 5*A*a*b^2)*e^2)*x)*sqrt(e*x + d) + 3*(6*B*a^
3*b*d*e^2 - (B*a^4 + 5*A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (B*a*b^3 + 5*A*b^4)*e^3)*
x^3 + 3*(6*B*a*b^3*d*e^2 - (B*a^2*b^2 + 5*A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e
^2 - (B*a^3*b + 5*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^3 - 3*a^4*b^3*d^2*e + 3*a^5*b^2*d*e^2 - a^6*b*e^3 +
(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*x^3 + 3*(a*b^6*d^3 - 3
*a^2*b^5*d^2*e + 3*a^3*b^4*d*e^2 - a^4*b^3*e^3)*x^2 + 3*(a^2*b^5*d^3 - 3*a^3*b^4
*d^2*e + 3*a^4*b^3*d*e^2 - a^5*b^2*e^3)*x)*sqrt(-b^2*d + a*b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{4} \sqrt{d + e x}}\, dx \]

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)**2/(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)/((a + b*x)**4*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.299525, size = 579, normalized size = 2.77 \[ \frac{{\left (6 \, B b d e^{2} - B a e^{3} - 5 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt{-b^{2} d + a b e}} + \frac{18 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{2} - 48 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} + 30 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} - 3 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{3} - 15 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{3} + 56 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} - 57 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} - 33 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} + 24 \, \sqrt{x e + d} B a^{2} b d e^{4} + 66 \, \sqrt{x e + d} A a b^{2} d e^{4} + 3 \, \sqrt{x e + d} B a^{3} e^{5} - 33 \, \sqrt{x e + d} A a^{2} b e^{5}}{24 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(6*B*b*d*e^2 - B*a*e^3 - 5*A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b
*e))/((b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*sqrt(-b^2*d + a*b*
e)) + 1/24*(18*(x*e + d)^(5/2)*B*b^3*d*e^2 - 48*(x*e + d)^(3/2)*B*b^3*d^2*e^2 +
30*sqrt(x*e + d)*B*b^3*d^3*e^2 - 3*(x*e + d)^(5/2)*B*a*b^2*e^3 - 15*(x*e + d)^(5
/2)*A*b^3*e^3 + 56*(x*e + d)^(3/2)*B*a*b^2*d*e^3 + 40*(x*e + d)^(3/2)*A*b^3*d*e^
3 - 57*sqrt(x*e + d)*B*a*b^2*d^2*e^3 - 33*sqrt(x*e + d)*A*b^3*d^2*e^3 - 8*(x*e +
d)^(3/2)*B*a^2*b*e^4 - 40*(x*e + d)^(3/2)*A*a*b^2*e^4 + 24*sqrt(x*e + d)*B*a^2*
b*d*e^4 + 66*sqrt(x*e + d)*A*a*b^2*d*e^4 + 3*sqrt(x*e + d)*B*a^3*e^5 - 33*sqrt(x
*e + d)*A*a^2*b*e^5)/((b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*((
x*e + d)*b - b*d + a*e)^3)