3.1756 \(\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]

[Out]

-(b*c - a*d)/(3*b^2*(a + b*x)^3) - d/(2*b^2*(a + b*x)^2)

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Rubi [A]  time = 0.0653632, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^5,x]

[Out]

-(b*c - a*d)/(3*b^2*(a + b*x)^3) - d/(2*b^2*(a + b*x)^2)

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Rubi in Sympy [A]  time = 13.5246, size = 31, normalized size = 0.82 \[ - \frac{d}{2 b^{2} \left (a + b x\right )^{2}} + \frac{a d - b c}{3 b^{2} \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**5,x)

[Out]

-d/(2*b**2*(a + b*x)**2) + (a*d - b*c)/(3*b**2*(a + b*x)**3)

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Mathematica [A]  time = 0.0163681, size = 27, normalized size = 0.71 \[ -\frac{a d+2 b c+3 b d x}{6 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^5,x]

[Out]

-(2*b*c + a*d + 3*b*d*x)/(6*b^2*(a + b*x)^3)

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Maple [A]  time = 0.007, size = 35, normalized size = 0.9 \[ -{\frac{d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{-ad+bc}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+(a*d+b*c)*x+x^2*b*d)/(b*x+a)^5,x)

[Out]

-1/2*d/b^2/(b*x+a)^2-1/3*(-a*d+b*c)/b^2/(b*x+a)^3

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Maxima [A]  time = 0.743517, size = 68, normalized size = 1.79 \[ -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.198843, size = 68, normalized size = 1.79 \[ -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.00526, size = 53, normalized size = 1.39 \[ - \frac{a d + 2 b c + 3 b d x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209569, size = 55, normalized size = 1.45 \[ -\frac{c}{3 \,{\left (b x + a\right )}^{3} b} - \frac{d}{2 \,{\left (b x + a\right )}^{2} b^{2}} + \frac{a d}{3 \,{\left (b x + a\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c/((b*x + a)^3*b) - 1/2*d/((b*x + a)^2*b^2) + 1/3*a*d/((b*x + a)^3*b^2)