3.1002 \(\int \frac{(a+b x)^2}{\left (\frac{a d}{b}+d x\right )^3} \, dx\)

Optimal. Leaf size=13 \[ \frac{b^2 \log (a+b x)}{d^3} \]

[Out]

(b^2*Log[a + b*x])/d^3

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Rubi [A]  time = 0.0108081, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b^2 \log (a+b x)}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*Log[a + b*x])/d^3

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Rubi in Sympy [A]  time = 4.31895, size = 12, normalized size = 0.92 \[ \frac{b^{2} \log{\left (a + b x \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(a + b*x)/d**3

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Mathematica [A]  time = 0.0026645, size = 13, normalized size = 1. \[ \frac{b^2 \log (a+b x)}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*Log[a + b*x])/d^3

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Maple [A]  time = 0.002, size = 14, normalized size = 1.1 \[{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.35517, size = 18, normalized size = 1.38 \[ \frac{b^{2} \log \left (b x + a\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*log(b*x + a)/d^3

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Fricas [A]  time = 0.209254, size = 18, normalized size = 1.38 \[ \frac{b^{2} \log \left (b x + a\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*log(b*x + a)/d^3

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Sympy [A]  time = 0.191897, size = 19, normalized size = 1.46 \[ \frac{b^{2} \log{\left (a d^{3} + b d^{3} x \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(a*d**3 + b*d**3*x)/d**3

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GIAC/XCAS [A]  time = 0.215982, size = 19, normalized size = 1.46 \[ \frac{b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*ln(abs(b*x + a))/d^3