3.1256 \(\int \frac{(c+d x)^2}{(a+b x)^6} \, dx\)

Optimal. Leaf size=65 \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0764155, 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 \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^2/(a + b*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.7249, size = 54, normalized size = 0.83 \[ - \frac{d^{2}}{3 b^{3} \left (a + b x\right )^{3}} + \frac{d \left (a d - b c\right )}{2 b^{3} \left (a + b x\right )^{4}} - \frac{\left (a d - b c\right )^{2}}{5 b^{3} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**2/(b*x+a)**6,x)

[Out]

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

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Mathematica [A]  time = 0.0415572, size = 57, normalized size = 0.88 \[ -\frac{a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^2/(a + b*x)^6,x]

[Out]

-(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2))/(30*b^3*(
a + b*x)^5)

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{d \left ( ad-bc \right ) }{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^2/(b*x+a)^6,x)

[Out]

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

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Maxima [A]  time = 1.35372, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2/(b*x + a)^6,x, algorithm="maxima")

[Out]

-1/30*(10*b^2*d^2*x^2 + 6*b^2*c^2 + 3*a*b*c*d + a^2*d^2 + 5*(3*b^2*c*d + a*b*d^2
)*x)/(b^8*x^5 + 5*a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^
5*b^3)

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Fricas [A]  time = 0.208306, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2/(b*x + a)^6,x, algorithm="fricas")

[Out]

-1/30*(10*b^2*d^2*x^2 + 6*b^2*c^2 + 3*a*b*c*d + a^2*d^2 + 5*(3*b^2*c*d + a*b*d^2
)*x)/(b^8*x^5 + 5*a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^
5*b^3)

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Sympy [A]  time = 4.2647, size = 116, normalized size = 1.78 \[ - \frac{a^{2} d^{2} + 3 a b c d + 6 b^{2} c^{2} + 10 b^{2} d^{2} x^{2} + x \left (5 a b d^{2} + 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**2/(b*x+a)**6,x)

[Out]

-(a**2*d**2 + 3*a*b*c*d + 6*b**2*c**2 + 10*b**2*d**2*x**2 + x*(5*a*b*d**2 + 15*b
**2*c*d))/(30*a**5*b**3 + 150*a**4*b**4*x + 300*a**3*b**5*x**2 + 300*a**2*b**6*x
**3 + 150*a*b**7*x**4 + 30*b**8*x**5)

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GIAC/XCAS [A]  time = 0.222472, size = 82, normalized size = 1.26 \[ -\frac{10 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c d x + 5 \, a b d^{2} x + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}}{30 \,{\left (b x + a\right )}^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2/(b*x + a)^6,x, algorithm="giac")

[Out]

-1/30*(10*b^2*d^2*x^2 + 15*b^2*c*d*x + 5*a*b*d^2*x + 6*b^2*c^2 + 3*a*b*c*d + a^2
*d^2)/((b*x + a)^5*b^3)