3.1768 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^7} \, dx\)

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)

_______________________________________________________________________________________

Rubi [A]  time = 0.103477, 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 \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^7,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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.267, 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((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**7,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)

_______________________________________________________________________________________

Mathematica [A]  time = 0.036112, 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[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^7,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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+(a*d+b*c)*x+x^2*b*d)^2/(b*x+a)^7,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.741034, 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((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^7,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)

_______________________________________________________________________________________

Fricas [A]  time = 0.227954, 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((b*d*x^2 + a*c + (b*c + a*d)*x)^2/(b*x + a)^7,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)

_______________________________________________________________________________________

Sympy [A]  time = 3.77563, 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((a*c+(a*d+b*c)*x+b*d*x**2)**2/(b*x+a)**7,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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209149, size = 82, normalized size = 1.26 \[ -\frac{6 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, a b d^{2} x + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}}{12 \,{\left (b x + a\right )}^{4} b^{3}} \]

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)^7,x, algorithm="giac")

[Out]

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