3.1785 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.152482, antiderivative size = 92, 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 \[ -\frac{3 d^2 (b c-a d)}{4 b^4 (a+b x)^4}-\frac{3 d (b c-a d)^2}{5 b^4 (a+b x)^5}-\frac{(b c-a d)^3}{6 b^4 (a+b x)^6}-\frac{d^3}{3 b^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 34.2609, size = 82, normalized size = 0.89 \[ - \frac{d^{3}}{3 b^{4} \left (a + b x\right )^{3}} + \frac{3 d^{2} \left (a d - b c\right )}{4 b^{4} \left (a + b x\right )^{4}} - \frac{3 d \left (a d - b c\right )^{2}}{5 b^{4} \left (a + b x\right )^{5}} + \frac{\left (a d - b c\right )^{3}}{6 b^{4} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**10,x)

[Out]

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

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Mathematica [A]  time = 0.0585383, size = 97, normalized size = 1.05 \[ -\frac{a^3 d^3+3 a^2 b d^2 (c+2 d x)+3 a b^2 d \left (2 c^2+6 c d x+5 d^2 x^2\right )+b^3 \left (10 c^3+36 c^2 d x+45 c d^2 x^2+20 d^3 x^3\right )}{60 b^4 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^10,x]

[Out]

-(a^3*d^3 + 3*a^2*b*d^2*(c + 2*d*x) + 3*a*b^2*d*(2*c^2 + 6*c*d*x + 5*d^2*x^2) +
b^3*(10*c^3 + 36*c^2*d*x + 45*c*d^2*x^2 + 20*d^3*x^3))/(60*b^4*(a + b*x)^6)

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Maple [A]  time = 0.009, size = 122, normalized size = 1.3 \[ -{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}c{d}^{2}b-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}}+{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+(a*d+b*c)*x+x^2*b*d)^3/(b*x+a)^10,x)

[Out]

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

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Maxima [A]  time = 0.740714, size = 231, normalized size = 2.51 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.200863, size = 231, normalized size = 2.51 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 15 \,{\left (3 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (6 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 16.8223, size = 182, normalized size = 1.98 \[ - \frac{a^{3} d^{3} + 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d + 10 b^{3} c^{3} + 20 b^{3} d^{3} x^{3} + x^{2} \left (15 a b^{2} d^{3} + 45 b^{3} c d^{2}\right ) + x \left (6 a^{2} b d^{3} + 18 a b^{2} c d^{2} + 36 b^{3} c^{2} d\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**10,x)

[Out]

-(a**3*d**3 + 3*a**2*b*c*d**2 + 6*a*b**2*c**2*d + 10*b**3*c**3 + 20*b**3*d**3*x*
*3 + x**2*(15*a*b**2*d**3 + 45*b**3*c*d**2) + x*(6*a**2*b*d**3 + 18*a*b**2*c*d**
2 + 36*b**3*c**2*d))/(60*a**6*b**4 + 360*a**5*b**5*x + 900*a**4*b**6*x**2 + 1200
*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

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GIAC/XCAS [A]  time = 0.210641, size = 154, normalized size = 1.67 \[ -\frac{20 \, b^{3} d^{3} x^{3} + 45 \, b^{3} c d^{2} x^{2} + 15 \, a b^{2} d^{3} x^{2} + 36 \, b^{3} c^{2} d x + 18 \, a b^{2} c d^{2} x + 6 \, a^{2} b d^{3} x + 10 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}}{60 \,{\left (b x + a\right )}^{6} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3/(b*x + a)^10,x, algorithm="giac")

[Out]

-1/60*(20*b^3*d^3*x^3 + 45*b^3*c*d^2*x^2 + 15*a*b^2*d^3*x^2 + 36*b^3*c^2*d*x + 1
8*a*b^2*c*d^2*x + 6*a^2*b*d^3*x + 10*b^3*c^3 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a
^3*d^3)/((b*x + a)^6*b^4)