3.1251 \(\int \frac{(c+d x)^2}{a+b x} \, dx\)

Optimal. Leaf size=49 \[ \frac{d x (b c-a d)}{b^2}+\frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{(c+d x)^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0186455, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{d x (b c-a d)}{b^2}+\frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{(c+d x)^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0160432, size = 43, normalized size = 0.88 \[ \frac{b d x (-2 a d+4 b c+b d x)+2 (b c-a d)^2 \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x),x]

[Out]

(b*d*x*(4*b*c - 2*a*d + b*d*x) + 2*(b*c - a*d)^2*Log[a + b*x])/(2*b^3)

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Maple [A]  time = 0.003, size = 74, normalized size = 1.5 \begin{align*}{\frac{{d}^{2}{x}^{2}}{2\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{2}}{{b}^{3}}}-2\,{\frac{a\ln \left ( bx+a \right ) cd}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.976674, size = 82, normalized size = 1.67 \begin{align*} \frac{b d^{2} x^{2} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.06895, size = 135, normalized size = 2.76 \begin{align*} \frac{b^{2} d^{2} x^{2} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.378542, size = 44, normalized size = 0.9 \begin{align*} \frac{d^{2} x^{2}}{2 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06733, size = 81, normalized size = 1.65 \begin{align*} \frac{b d^{2} x^{2} + 4 \, b c d x - 2 \, a d^{2} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d^2*x^2 + 4*b*c*d*x - 2*a*d^2*x)/b^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(abs(b*x + a))/b^3