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

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 31} \begin {gather*} \frac {b^2 \log (c+d x)}{d^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^2 \log (c+d x)}{d^3} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {b c}{d}+b x\right )^2}{(c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (d x + c\right )}{d^{3}} \end {gather*}

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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giac [A]  time = 0.89, size = 14, normalized size = 1.08 \begin {gather*} \frac {b^{2} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end {gather*}

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

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maple [A]  time = 0.00, size = 14, normalized size = 1.08 \begin {gather*} \frac {b^{2} \ln \left (d x +c \right )}{d^{3}} \end {gather*}

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 = 1.37, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^{2} \log \left (d x + c\right )}{d^{3}} \end {gather*}

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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mupad [B]  time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} \frac {b^2\,\ln \left (c+d\,x\right )}{d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + (b*c)/d)^2/(c + d*x)^3,x)

[Out]

(b^2*log(c + d*x))/d^3

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sympy [A]  time = 0.10, size = 17, normalized size = 1.31 \begin {gather*} \frac {b^{2} \log {\left (c d^{2} + d^{3} x \right )}}{d^{3}} \end {gather*}

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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