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

Optimal. Leaf size=66 \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}+\frac{x^2 (b c-a d)}{2 b^2}-\frac{a x (b c-a d)}{b^3}+\frac{d x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0498759, antiderivative size = 66, 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} \[ \frac{a^2 (b c-a d) \log (a+b x)}{b^4}+\frac{x^2 (b c-a d)}{2 b^2}-\frac{a x (b c-a d)}{b^3}+\frac{d x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x))/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0204427, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 d-3 a b (2 c+d x)+b^2 x (3 c+2 d x)\right )+6 a^2 (b c-a d) \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 76, normalized size = 1.2 \begin{align*}{\frac{d{x}^{3}}{3\,b}}-{\frac{a{x}^{2}d}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}dx}{{b}^{3}}}-{\frac{acx}{{b}^{2}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) d}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) c}{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01849, size = 93, normalized size = 1.41 \begin{align*} \frac{2 \, b^{2} d x^{3} + 3 \,{\left (b^{2} c - a b d\right )} x^{2} - 6 \,{\left (a b c - a^{2} d\right )} x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90501, size = 149, normalized size = 2.26 \begin{align*} \frac{2 \, b^{3} d x^{3} + 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{2} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x + 6 \,{\left (a^{2} b c - a^{3} d\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^2*(d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.42056, size = 58, normalized size = 0.88 \begin{align*} - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{d x^{3}}{3 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} + \frac{x \left (a^{2} d - a b c\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19542, size = 95, normalized size = 1.44 \begin{align*} \frac{2 \, b^{2} d x^{3} + 3 \, b^{2} c x^{2} - 3 \, a b d x^{2} - 6 \, a b c x + 6 \, a^{2} d x}{6 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\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^2*(d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

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