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

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 = 0.0033044, 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, Rules used = {21, 31} \[ \frac{b^2 \log (a+b x)}{d^3} \]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(a+b x)^2}{\left (\frac{a d}{b}+d x\right )^3} \, dx &=\frac{b^3 \int \frac{1}{a+b x} \, dx}{d^3}\\ &=\frac{b^2 \log (a+b x)}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.0015988, 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 = 0.001, size = 14, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{d}^{3}}} \end{align*}

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 = 1.02462, size = 18, normalized size = 1.38 \begin{align*} \frac{b^{2} \log \left (b x + a\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(b*x + a)/d^3

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Fricas [A]  time = 1.48896, size = 30, normalized size = 2.31 \begin{align*} \frac{b^{2} \log \left (b x + a\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(b*x + a)/d^3

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Sympy [A]  time = 0.091023, size = 19, normalized size = 1.46 \begin{align*} \frac{b^{2} \log{\left (a d^{3} + b d^{3} x \right )}}{d^{3}} \end{align*}

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 [A]  time = 1.06317, size = 19, normalized size = 1.46 \begin{align*} \frac{b^{2} \log \left ({\left | b x + a \right |}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(abs(b*x + a))/d^3