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

Optimal. Leaf size=94 \[ \frac{a^2 (b c-a d)^2 \log (a+b x)}{b^5}+\frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{x^2 (b c-a d)^2}{2 b^3}-\frac{a x (b c-a d)^2}{b^4}+\frac{d^2 x^4}{4 b} \]

[Out]

-((a*(b*c - a*d)^2*x)/b^4) + ((b*c - a*d)^2*x^2)/(2*b^3) + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^4)/(4*b) + (
a^2*(b*c - a*d)^2*Log[a + b*x])/b^5

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Rubi [A]  time = 0.0732267, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{a^2 (b c-a d)^2 \log (a+b x)}{b^5}+\frac{d x^3 (2 b c-a d)}{3 b^2}+\frac{x^2 (b c-a d)^2}{2 b^3}-\frac{a x (b c-a d)^2}{b^4}+\frac{d^2 x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^2)/(a + b*x),x]

[Out]

-((a*(b*c - a*d)^2*x)/b^4) + ((b*c - a*d)^2*x^2)/(2*b^3) + (d*(2*b*c - a*d)*x^3)/(3*b^2) + (d^2*x^4)/(4*b) + (
a^2*(b*c - a*d)^2*Log[a + b*x])/b^5

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^2 (c+d x)^2}{a+b x} \, dx &=\int \left (-\frac{a (-b c+a d)^2}{b^4}+\frac{(b c-a d)^2 x}{b^3}+\frac{d (2 b c-a d) x^2}{b^2}+\frac{d^2 x^3}{b}+\frac{a^2 (-b c+a d)^2}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac{a (b c-a d)^2 x}{b^4}+\frac{(b c-a d)^2 x^2}{2 b^3}+\frac{d (2 b c-a d) x^3}{3 b^2}+\frac{d^2 x^4}{4 b}+\frac{a^2 (b c-a d)^2 \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0373322, size = 103, normalized size = 1.1 \[ \frac{b x \left (6 a^2 b d (4 c+d x)-12 a^3 d^2-4 a b^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^3 x \left (6 c^2+8 c d x+3 d^2 x^2\right )\right )+12 a^2 (b c-a d)^2 \log (a+b x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(-12*a^3*d^2 + 6*a^2*b*d*(4*c + d*x) - 4*a*b^2*(3*c^2 + 3*c*d*x + d^2*x^2) + b^3*x*(6*c^2 + 8*c*d*x + 3*d
^2*x^2)) + 12*a^2*(b*c - a*d)^2*Log[a + b*x])/(12*b^5)

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Maple [A]  time = 0.004, size = 152, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{{x}^{3}a{d}^{2}}{3\,{b}^{2}}}+{\frac{2\,c{x}^{3}d}{3\,b}}+{\frac{{a}^{2}{x}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{2}cd}{{b}^{2}}}+{\frac{{x}^{2}{c}^{2}}{2\,b}}-{\frac{{a}^{3}{d}^{2}x}{{b}^{4}}}+2\,{\frac{{a}^{2}cdx}{{b}^{3}}}-{\frac{a{c}^{2}x}{{b}^{2}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{2}}{{b}^{5}}}-2\,{\frac{{a}^{3}\ln \left ( bx+a \right ) cd}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}}{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d^2*x^4/b-1/3/b^2*x^3*a*d^2+2/3/b*x^3*c*d+1/2/b^3*x^2*a^2*d^2-1/b^2*x^2*a*c*d+1/2/b*x^2*c^2-1/b^4*a^3*d^2*
x+2/b^3*a^2*c*d*x-1/b^2*a*c^2*x+a^4/b^5*ln(b*x+a)*d^2-2*a^3/b^4*ln(b*x+a)*c*d+a^2/b^3*ln(b*x+a)*c^2

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Maxima [A]  time = 1.03451, size = 178, normalized size = 1.89 \begin{align*} \frac{3 \, b^{3} d^{2} x^{4} + 4 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2} - 12 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{12 \, b^{4}} + \frac{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^3*d^2*x^4 + 4*(2*b^3*c*d - a*b^2*d^2)*x^3 + 6*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^2 - 12*(a*b^2*c^
2 - 2*a^2*b*c*d + a^3*d^2)*x)/b^4 + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x + a)/b^5

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Fricas [A]  time = 1.92729, size = 279, normalized size = 2.97 \begin{align*} \frac{3 \, b^{4} d^{2} x^{4} + 4 \,{\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} - 12 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x + 12 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*d^2*x^4 + 4*(2*b^4*c*d - a*b^3*d^2)*x^3 + 6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 - 12*(a*b^3*
c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x + 12*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x + a))/b^5

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Sympy [A]  time = 0.546695, size = 112, normalized size = 1.19 \begin{align*} \frac{a^{2} \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{5}} + \frac{d^{2} x^{4}}{4 b} - \frac{x^{3} \left (a d^{2} - 2 b c d\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 b^{3}} - \frac{x \left (a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(a*d - b*c)**2*log(a + b*x)/b**5 + d**2*x**4/(4*b) - x**3*(a*d**2 - 2*b*c*d)/(3*b**2) + x**2*(a**2*d**2 -
2*a*b*c*d + b**2*c**2)/(2*b**3) - x*(a**3*d**2 - 2*a**2*b*c*d + a*b**2*c**2)/b**4

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Giac [A]  time = 1.16846, size = 188, normalized size = 2. \begin{align*} \frac{3 \, b^{3} d^{2} x^{4} + 8 \, b^{3} c d x^{3} - 4 \, a b^{2} d^{2} x^{3} + 6 \, b^{3} c^{2} x^{2} - 12 \, a b^{2} c d x^{2} + 6 \, a^{2} b d^{2} x^{2} - 12 \, a b^{2} c^{2} x + 24 \, a^{2} b c d x - 12 \, a^{3} d^{2} x}{12 \, b^{4}} + \frac{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^3*d^2*x^4 + 8*b^3*c*d*x^3 - 4*a*b^2*d^2*x^3 + 6*b^3*c^2*x^2 - 12*a*b^2*c*d*x^2 + 6*a^2*b*d^2*x^2 - 1
2*a*b^2*c^2*x + 24*a^2*b*c*d*x - 12*a^3*d^2*x)/b^4 + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(abs(b*x + a))/b
^5