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

Optimal. Leaf size=71 \[ \frac{d x^2 (2 b c-a d)}{2 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^2 x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0530015, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{d x^2 (2 b c-a d)}{2 b^2}+\frac{x (b c-a d)^2}{b^3}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^2 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^2)/(a + b*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x (c+d x)^2}{a+b x} \, dx &=\int \left (\frac{(b c-a d)^2}{b^3}+\frac{d (2 b c-a d) x}{b^2}+\frac{d^2 x^2}{b}-\frac{a (-b c+a d)^2}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^2}{2 b^2}+\frac{d^2 x^3}{3 b}-\frac{a (b c-a d)^2 \log (a+b x)}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.027347, size = 74, normalized size = 1.04 \[ \frac{b x \left (6 a^2 d^2-3 a b d (4 c+d x)+2 b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )-6 a (b c-a d)^2 \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(6*a^2*d^2 - 3*a*b*d*(4*c + d*x) + 2*b^2*(3*c^2 + 3*c*d*x + d^2*x^2)) - 6*a*(b*c - a*d)^2*Log[a + b*x])/(
6*b^4)

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Maple [A]  time = 0.002, size = 110, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,b}}-{\frac{a{x}^{2}{d}^{2}}{2\,{b}^{2}}}+{\frac{c{x}^{2}d}{b}}+{\frac{{a}^{2}{d}^{2}x}{{b}^{3}}}-2\,{\frac{acdx}{{b}^{2}}}+{\frac{{c}^{2}x}{b}}-{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{4}}}+2\,{\frac{{a}^{2}\ln \left ( bx+a \right ) cd}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ){c}^{2}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02429, size = 131, normalized size = 1.85 \begin{align*} \frac{2 \, b^{2} d^{2} x^{3} + 3 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^2*d^2*x^3 + 3*(2*b^2*c*d - a*b*d^2)*x^2 + 6*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^3 - (a*b^2*c^2 - 2*a
^2*b*c*d + a^3*d^2)*log(b*x + a)/b^4

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Fricas [A]  time = 1.94013, size = 205, normalized size = 2.89 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} + 3 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^3*d^2*x^3 + 3*(2*b^3*c*d - a*b^2*d^2)*x^2 + 6*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x - 6*(a*b^2*c^2 -
2*a^2*b*c*d + a^3*d^2)*log(b*x + a))/b^4

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Sympy [A]  time = 0.499274, size = 76, normalized size = 1.07 \begin{align*} - \frac{a \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{4}} + \frac{d^{2} x^{3}}{3 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26777, size = 134, normalized size = 1.89 \begin{align*} \frac{2 \, b^{2} d^{2} x^{3} + 6 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2} + 6 \, b^{2} c^{2} x - 12 \, a b c d x + 6 \, a^{2} d^{2} x}{6 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^2*d^2*x^3 + 6*b^2*c*d*x^2 - 3*a*b*d^2*x^2 + 6*b^2*c^2*x - 12*a*b*c*d*x + 6*a^2*d^2*x)/b^3 - (a*b^2*c^
2 - 2*a^2*b*c*d + a^3*d^2)*log(abs(b*x + a))/b^4