3.176 \(\int \frac{x^2 (A+B x)}{a+b x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0467123, 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} \[ \frac{a^2 (A b-a B) \log (a+b x)}{b^4}+\frac{x^2 (A b-a B)}{2 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x))/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0211556, size = 61, normalized size = 0.92 \[ \frac{b x \left (6 a^2 B-3 a b (2 A+B x)+b^2 x (3 A+2 B x)\right )+6 a^2 (A b-a B) \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x))/(a + b*x),x]

[Out]

(b*x*(6*a^2*B - 3*a*b*(2*A + B*x) + b^2*x*(3*A + 2*B*x)) + 6*a^2*(A*b - a*B)*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{B{x}^{3}}{3\,b}}+{\frac{A{x}^{2}}{2\,b}}-{\frac{B{x}^{2}a}{2\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{{a}^{2}Bx}{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ( bx+a \right ) B}{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(B*x+A)/(b*x+a),x)

[Out]

1/3*B*x^3/b+1/2/b*A*x^2-1/2/b^2*B*x^2*a-1/b^2*a*A*x+1/b^3*a^2*B*x+a^2/b^3*ln(b*x+a)*A-a^3/b^4*ln(b*x+a)*B

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Maxima [A]  time = 1.01062, size = 95, normalized size = 1.44 \begin{align*} \frac{2 \, B b^{2} x^{3} - 3 \,{\left (B a b - A b^{2}\right )} x^{2} + 6 \,{\left (B a^{2} - A a b\right )} x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\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*(B*x+A)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43559, size = 149, normalized size = 2.26 \begin{align*} \frac{2 \, B b^{3} x^{3} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x - 6 \,{\left (B a^{3} - A a^{2} b\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*(B*x+A)/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.427378, size = 58, normalized size = 0.88 \begin{align*} \frac{B x^{3}}{3 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(B*x+A)/(b*x+a),x)

[Out]

B*x**3/(3*b) - a**2*(-A*b + B*a)*log(a + b*x)/b**4 - x**2*(-A*b + B*a)/(2*b**2) + x*(-A*a*b + B*a**2)/b**3

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Giac [A]  time = 1.41364, size = 96, normalized size = 1.45 \begin{align*} \frac{2 \, B b^{2} x^{3} - 3 \, B a b x^{2} + 3 \, A b^{2} x^{2} + 6 \, B a^{2} x - 6 \, A a b x}{6 \, b^{3}} - \frac{{\left (B a^{3} - A a^{2} b\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*(B*x+A)/(b*x+a),x, algorithm="giac")

[Out]

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