3.1717 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=158 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]

[Out]

-((b*d - a*e)*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)*(d + e
*x)^4) + ((2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b
*x)*(d + e*x)^3) - (b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^3*(a + b*x)*(d + e*x
)^2)

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Rubi [A]  time = 0.280709, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^3*(a + b*x)*(d + e
*x)^4) + ((2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a + b
*x)*(d + e*x)^3) - (b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^3*(a + b*x)*(d + e*x
)^2)

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Rubi in Sympy [A]  time = 35.2945, size = 162, normalized size = 1.03 \[ - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8 e \left (d + e x\right )^{4} \left (a e - b d\right )} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{4 e^{2} \left (d + e x\right )^{3} \left (a e - b d\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{12 e^{3} \left (a + b x\right ) \left (d + e x\right )^{3}} \]

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)**5,x)

[Out]

-(2*a + 2*b*x)*(A*e - B*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(8*e*(d + e*x)**4*(a
*e - b*d)) + sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e - 2*B*a*e + B*b*d)/(4*e**2*
(d + e*x)**3*(a*e - b*d)) - sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e - 2*B*a*e +
B*b*d)/(12*e**3*(a + b*x)*(d + e*x)**3)

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Mathematica [A]  time = 0.0620082, size = 80, normalized size = 0.51 \[ -\frac{\sqrt{(a+b x)^2} \left (a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(a*e*(3*A*e + B*(d + 4*e*x)) + b*(A*e*(d + 4*e*x) + B*(d^2 +
4*d*e*x + 6*e^2*x^2))))/(12*e^3*(a + b*x)*(d + e*x)^4)

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Maple [A]  time = 0.008, size = 86, normalized size = 0.5 \[ -{\frac{6\,B{x}^{2}b{e}^{2}+4\,Ab{e}^{2}x+4\,aB{e}^{2}x+4\,Bbdex+3\,A{e}^{2}a+Abde+aBde+Bb{d}^{2}}{12\,{e}^{3} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^5,x)

[Out]

-1/12/e^3*(6*B*b*e^2*x^2+4*A*b*e^2*x+4*B*a*e^2*x+4*B*b*d*e*x+3*A*a*e^2+A*b*d*e+B
*a*d*e+B*b*d^2)*((b*x+a)^2)^(1/2)/(e*x+d)^4/(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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267601, size = 138, normalized size = 0.87 \[ -\frac{6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} +{\left (B a + A b\right )} d e + 4 \,{\left (B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")

[Out]

-1/12*(6*B*b*e^2*x^2 + B*b*d^2 + 3*A*a*e^2 + (B*a + A*b)*d*e + 4*(B*b*d*e + (B*a
+ A*b)*e^2)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 11.7765, size = 117, normalized size = 0.74 \[ - \frac{3 A a e^{2} + A b d e + B a d e + B b d^{2} + 6 B b e^{2} x^{2} + x \left (4 A b e^{2} + 4 B a e^{2} + 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**5,x)

[Out]

-(3*A*a*e**2 + A*b*d*e + B*a*d*e + B*b*d**2 + 6*B*b*e**2*x**2 + x*(4*A*b*e**2 +
4*B*a*e**2 + 4*B*b*d*e))/(12*d**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48
*d*e**6*x**3 + 12*e**7*x**4)

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GIAC/XCAS [A]  time = 0.293551, size = 157, normalized size = 0.99 \[ -\frac{{\left (6 \, B b x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, B b d x e{\rm sign}\left (b x + a\right ) + B b d^{2}{\rm sign}\left (b x + a\right ) + 4 \, B a x e^{2}{\rm sign}\left (b x + a\right ) + 4 \, A b x e^{2}{\rm sign}\left (b x + a\right ) + B a d e{\rm sign}\left (b x + a\right ) + A b d e{\rm sign}\left (b x + a\right ) + 3 \, A a e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

-1/12*(6*B*b*x^2*e^2*sign(b*x + a) + 4*B*b*d*x*e*sign(b*x + a) + B*b*d^2*sign(b*
x + a) + 4*B*a*x*e^2*sign(b*x + a) + 4*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) + 3*A*a*e^2*sign(b*x + a))*e^(-3)/(x*e + d)^4