3.1770 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.102345, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 25.2377, size = 56, normalized size = 0.86 \[ - \frac{d^{2}}{4 b^{3} \left (a + b x\right )^{4}} + \frac{2 d \left (a d - b c\right )}{5 b^{3} \left (a + b x\right )^{5}} - \frac{\left (a d - b c\right )^{2}}{6 b^{3} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**9,x)

[Out]

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

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Mathematica [A]  time = 0.0388239, size = 58, normalized size = 0.89 \[ -\frac{a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^9,x]

[Out]

-(a^2*d^2 + 2*a*b*d*(2*c + 3*d*x) + b^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2))/(60*b^
3*(a + b*x)^6)

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Maple [A]  time = 0.008, size = 71, normalized size = 1.1 \[{\frac{ \left ( 2\,ad-2\,bc \right ) d}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{6\,{b}^{3} \left ( bx+a \right ) ^{6}}}-{\frac{{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^9,x)

[Out]

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

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Maxima [A]  time = 0.740962, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^9,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^
2)*x)/(b^9*x^6 + 6*a*b^8*x^5 + 15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2
+ 6*a^5*b^4*x + a^6*b^3)

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Fricas [A]  time = 0.212494, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^9,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^
2)*x)/(b^9*x^6 + 6*a*b^8*x^5 + 15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2
+ 6*a^5*b^4*x + a^6*b^3)

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Sympy [A]  time = 5.33839, size = 128, normalized size = 1.97 \[ - \frac{a^{2} d^{2} + 4 a b c d + 10 b^{2} c^{2} + 15 b^{2} d^{2} x^{2} + x \left (6 a b d^{2} + 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**9,x)

[Out]

-(a**2*d**2 + 4*a*b*c*d + 10*b**2*c**2 + 15*b**2*d**2*x**2 + x*(6*a*b*d**2 + 24*
b**2*c*d))/(60*a**6*b**3 + 360*a**5*b**4*x + 900*a**4*b**5*x**2 + 1200*a**3*b**6
*x**3 + 900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**6)

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GIAC/XCAS [A]  time = 0.210768, size = 82, normalized size = 1.26 \[ -\frac{15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \,{\left (b x + a\right )}^{6} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^9,x, algorithm="giac")

[Out]

-1/60*(15*b^2*d^2*x^2 + 24*b^2*c*d*x + 6*a*b*d^2*x + 10*b^2*c^2 + 4*a*b*c*d + a^
2*d^2)/((b*x + a)^6*b^3)