3.1010 \(\int \frac{(\frac{b c}{d}+b x)^2}{(c+d x)^3} \, dx\)

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

[Out]

(b^2*Log[c + d*x])/d^3

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Rubi [A]  time = 0.0030361, 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} \[ \frac{b^2 \log (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)^2/(c + d*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0019214, size = 13, normalized size = 1. \[ \frac{b^2 \log (c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*Log[c + d*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 ( dx+c \right ) }{{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c/d+b*x)^2/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.00713, size = 18, normalized size = 1.38 \begin{align*} \frac{b^{2} \log \left (d x + c\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(d*x + c)/d^3

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Fricas [A]  time = 1.55221, size = 30, normalized size = 2.31 \begin{align*} \frac{b^{2} \log \left (d x + c\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(d*x + c)/d^3

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Sympy [A]  time = 0.088541, size = 17, normalized size = 1.31 \begin{align*} \frac{b^{2} \log{\left (c d^{2} + d^{3} x \right )}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)**2/(d*x+c)**3,x)

[Out]

b**2*log(c*d**2 + d**3*x)/d**3

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Giac [A]  time = 1.07414, size = 19, normalized size = 1.46 \begin{align*} \frac{b^{2} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*log(abs(d*x + c))/d^3