3.1766 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^5} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]

[Out]

-(b*c - a*d)^2/(2*b^3*(a + b*x)^2) - (2*d*(b*c - a*d))/(b^3*(a + b*x)) + (d^2*Lo
g[a + b*x])/b^3

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Rubi [A]  time = 0.0974697, antiderivative size = 59, 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 \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(b*c - a*d)^2/(2*b^3*(a + b*x)^2) - (2*d*(b*c - a*d))/(b^3*(a + b*x)) + (d^2*Lo
g[a + b*x])/b^3

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Rubi in Sympy [A]  time = 23.5449, size = 51, normalized size = 0.86 \[ \frac{d^{2} \log{\left (a + b x \right )}}{b^{3}} + \frac{2 d \left (a d - b c\right )}{b^{3} \left (a + b x\right )} - \frac{\left (a d - b c\right )^{2}}{2 b^{3} \left (a + b x\right )^{2}} \]

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)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0435558, size = 49, normalized size = 0.83 \[ \frac{2 d^2 \log (a+b x)-\frac{(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 92, normalized size = 1.6 \[ -{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{acd}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}}+2\,{\frac{a{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{cd}{{b}^{2} \left ( bx+a \right ) }} \]

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)^5,x)

[Out]

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

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Maxima [A]  time = 0.729946, size = 107, normalized size = 1.81 \[ -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{d^{2} \log \left (b x + a\right )}{b^{3}} \]

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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.228797, size = 134, normalized size = 2.27 \[ -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.57752, size = 80, normalized size = 1.36 \[ \frac{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{d^{2} \log{\left (a + b x \right )}}{b^{3}} \]

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)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214749, size = 149, normalized size = 2.53 \[ -\frac{d^{2}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{5} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{4 \, b^{4} c d}{b x + a} - \frac{2 \, a b^{4} c d}{{\left (b x + a\right )}^{2}} - \frac{4 \, a b^{3} d^{2}}{b x + a} + \frac{a^{2} b^{3} d^{2}}{{\left (b x + a\right )}^{2}}}{2 \, b^{6}} \]

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)^5,x, algorithm="giac")

[Out]

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