3.1765 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^4} \, dx\)

Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.100281, antiderivative size = 51, 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 \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^4,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b^{2}}\, dx - \frac{2 d \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} - \frac{\left (a d - b c\right )^{2}}{b^{3} \left (a + b x\right )} \]

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0676694, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{a+b x}+2 d (b c-a d) \log (a+b x)+b d^2 x}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^2/(a + b*x)^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 86, normalized size = 1.7 \[{\frac{{d}^{2}x}{{b}^{2}}}-2\,{\frac{{d}^{2}\ln \left ( bx+a \right ) a}{{b}^{3}}}+2\,{\frac{d\ln \left ( bx+a \right ) c}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}+2\,{\frac{acd}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{c}^{2}}{b \left ( bx+a \right ) }} \]

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)^4,x)

[Out]

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

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Maxima [A]  time = 0.726837, size = 90, normalized size = 1.76 \[ \frac{d^{2} x}{b^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{b^{4} x + a b^{3}} + \frac{2 \,{\left (b c d - a d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]

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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229987, size = 124, normalized size = 2.43 \[ \frac{b^{2} d^{2} x^{2} + a b d^{2} x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]

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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.15723, size = 60, normalized size = 1.18 \[ - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a b^{3} + b^{4} x} + \frac{d^{2} x}{b^{2}} - \frac{2 d \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} \]

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)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21211, size = 88, normalized size = 1.73 \[ \frac{d^{2} x}{b^{2}} + \frac{2 \,{\left (b c d - a d^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (b x + a\right )} b^{3}} \]

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)^4,x, algorithm="giac")

[Out]

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