3.1716 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=104 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(b*d - a*e)^2*(d + e*x)
^2) + ((B*d - A*e)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(3*(b*d - a*e)^2*(d + e*x)^3
)

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Rubi [A]  time = 0.201583, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (B d-A e)}{3 (d+e x)^3 (b d-a e)^2}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 (d+e x)^2 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(b*d - a*e)^2*(d + e*x)
^2) + ((B*d - A*e)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(3*(b*d - a*e)^2*(d + e*x)^3
)

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Rubi in Sympy [A]  time = 38.7099, size = 95, normalized size = 0.91 \[ \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 \left (d + e x\right )^{2} \left (a e - b d\right )^{2}} - \frac{\left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**4,x)

[Out]

(2*a + 2*b*x)*(A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*(d + e*x)**2*(a*e
- b*d)**2) - (A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*(d + e*x)**3*(a*
e - b*d)**2)

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Mathematica [A]  time = 0.0644587, size = 81, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )}{6 e^3 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 87, normalized size = 0.8 \[ -{\frac{6\,B{x}^{2}b{e}^{2}+3\,Ab{e}^{2}x+3\,aB{e}^{2}x+6\,Bbdex+2\,A{e}^{2}a+Abde+aBde+2\,Bb{d}^{2}}{6\, \left ( ex+d \right ) ^{3}{e}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x)

[Out]

-1/6*(6*B*b*e^2*x^2+3*A*b*e^2*x+3*B*a*e^2*x+6*B*b*d*e*x+2*A*a*e^2+A*b*d*e+B*a*d*
e+2*B*b*d^2)*((b*x+a)^2)^(1/2)/(e*x+d)^3/e^3/(b*x+a)

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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(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271439, size = 126, normalized size = 1.21 \[ -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")

[Out]

-1/6*(6*B*b*e^2*x^2 + 2*B*b*d^2 + 2*A*a*e^2 + (B*a + A*b)*d*e + 3*(2*B*b*d*e + (
B*a + A*b)*e^2)*x)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 7.17546, size = 107, normalized size = 1.03 \[ - \frac{2 A a e^{2} + A b d e + B a d e + 2 B b d^{2} + 6 B b e^{2} x^{2} + x \left (3 A b e^{2} + 3 B a e^{2} + 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**4,x)

[Out]

-(2*A*a*e**2 + A*b*d*e + B*a*d*e + 2*B*b*d**2 + 6*B*b*e**2*x**2 + x*(3*A*b*e**2
+ 3*B*a*e**2 + 6*B*b*d*e))/(6*d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 + 6*e*
*6*x**3)

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GIAC/XCAS [A]  time = 0.281859, size = 158, normalized size = 1.52 \[ -\frac{{\left (6 \, B b x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 6 \, B b d x e{\rm sign}\left (b x + a\right ) + 2 \, B b d^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, A b x e^{2}{\rm sign}\left (b x + a\right ) + B a d e{\rm sign}\left (b x + a\right ) + A b d e{\rm sign}\left (b x + a\right ) + 2 \, A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^4,x, algorithm="giac")

[Out]

-1/6*(6*B*b*x^2*e^2*sign(b*x + a) + 6*B*b*d*x*e*sign(b*x + a) + 2*B*b*d^2*sign(b
*x + a) + 3*B*a*x*e^2*sign(b*x + a) + 3*A*b*x*e^2*sign(b*x + a) + B*a*d*e*sign(b
*x + a) + A*b*d*e*sign(b*x + a) + 2*A*a*e^2*sign(b*x + a))*e^(-3)/(x*e + d)^3