3.1255 \(\int \frac{(c+d x)^2}{(a+b x)^5} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0780983, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^2/(a + b*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.3452, size = 56, normalized size = 0.86 \[ - \frac{d^{2}}{2 b^{3} \left (a + b x\right )^{2}} + \frac{2 d \left (a d - b c\right )}{3 b^{3} \left (a + b x\right )^{3}} - \frac{\left (a d - b c\right )^{2}}{4 b^{3} \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.029984, size = 56, normalized size = 0.86 \[ -\frac{a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*d^2 + 2*a*b*d*(c + 2*d*x) + b^2*(3*c^2 + 8*c*d*x + 6*d^2*x^2))/(12*b^3*(a
+ b*x)^4)

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Maple [A]  time = 0.009, size = 71, normalized size = 1.1 \[ -{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{2\,d \left ( ad-bc \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^2/(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.35441, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.197798, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.48789, size = 104, normalized size = 1.6 \[ - \frac{a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (4 a b d^{2} + 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a**2*d**2 + 2*a*b*c*d + 3*b**2*c**2 + 6*b**2*d**2*x**2 + x*(4*a*b*d**2 + 8*b**
2*c*d))/(12*a**4*b**3 + 48*a**3*b**4*x + 72*a**2*b**5*x**2 + 48*a*b**6*x**3 + 12
*b**7*x**4)

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GIAC/XCAS [A]  time = 0.226507, size = 126, normalized size = 1.94 \[ -\frac{\frac{3 \, b^{3} c^{2}}{{\left (b x + a\right )}^{4}} + \frac{8 \, b^{2} c d}{{\left (b x + a\right )}^{3}} - \frac{6 \, a b^{2} c d}{{\left (b x + a\right )}^{4}} + \frac{6 \, b d^{2}}{{\left (b x + a\right )}^{2}} - \frac{8 \, a b d^{2}}{{\left (b x + a\right )}^{3}} + \frac{3 \, a^{2} b d^{2}}{{\left (b x + a\right )}^{4}}}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*b^3*c^2/(b*x + a)^4 + 8*b^2*c*d/(b*x + a)^3 - 6*a*b^2*c*d/(b*x + a)^4 +
6*b*d^2/(b*x + a)^2 - 8*a*b*d^2/(b*x + a)^3 + 3*a^2*b*d^2/(b*x + a)^4)/b^4