3.223 \(\int \frac{x^2 (c+d x)^3}{a+b x} \, dx\)

Optimal. Leaf size=129 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{a^2 (b c-a d)^3 \log (a+b x)}{b^6}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{x^2 (b c-a d)^3}{2 b^4}-\frac{a x (b c-a d)^3}{b^5}+\frac{d^3 x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0975798, antiderivative size = 129, 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} \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{a^2 (b c-a d)^3 \log (a+b x)}{b^6}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{x^2 (b c-a d)^3}{2 b^4}-\frac{a x (b c-a d)^3}{b^5}+\frac{d^3 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^3)/(a + b*x),x]

[Out]

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

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)^3}{a+b x} \, dx &=\int \left (\frac{a (-b c+a d)^3}{b^5}+\frac{(b c-a d)^3 x}{b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^2}{b^3}+\frac{d^2 (3 b c-a d) x^3}{b^2}+\frac{d^3 x^4}{b}-\frac{a^2 (-b c+a d)^3}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{a (b c-a d)^3 x}{b^5}+\frac{(b c-a d)^3 x^2}{2 b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac{d^2 (3 b c-a d) x^4}{4 b^2}+\frac{d^3 x^5}{5 b}+\frac{a^2 (b c-a d)^3 \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0619294, size = 124, normalized size = 0.96 \[ \frac{20 b^3 d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+60 a^2 (b c-a d)^3 \log (a+b x)+15 b^4 d^2 x^4 (3 b c-a d)+30 b^2 x^2 (b c-a d)^3+60 a b x (a d-b c)^3+12 b^5 d^3 x^5}{60 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x)^3)/(a + b*x),x]

[Out]

(60*a*b*(-(b*c) + a*d)^3*x + 30*b^2*(b*c - a*d)^3*x^2 + 20*b^3*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^3 + 15*b^
4*d^2*(3*b*c - a*d)*x^4 + 12*b^5*d^3*x^5 + 60*a^2*(b*c - a*d)^3*Log[a + b*x])/(60*b^6)

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Maple [B]  time = 0.003, size = 244, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{5}}{5\,b}}-{\frac{{x}^{4}a{d}^{3}}{4\,{b}^{2}}}+{\frac{3\,{x}^{4}c{d}^{2}}{4\,b}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{3\,{b}^{3}}}-{\frac{{x}^{3}ac{d}^{2}}{{b}^{2}}}+{\frac{{x}^{3}{c}^{2}d}{b}}-{\frac{{x}^{2}{a}^{3}{d}^{3}}{2\,{b}^{4}}}+{\frac{3\,{a}^{2}{x}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,a{x}^{2}{c}^{2}d}{2\,{b}^{2}}}+{\frac{{c}^{3}{x}^{2}}{2\,b}}+{\frac{{a}^{4}{d}^{3}x}{{b}^{5}}}-3\,{\frac{{a}^{3}c{d}^{2}x}{{b}^{4}}}+3\,{\frac{{a}^{2}{c}^{2}dx}{{b}^{3}}}-{\frac{a{c}^{3}x}{{b}^{2}}}-{\frac{{a}^{5}\ln \left ( bx+a \right ){d}^{3}}{{b}^{6}}}+3\,{\frac{{a}^{4}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{5}}}-3\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{4}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{3}}{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d*x+c)^3/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.01897, size = 289, normalized size = 2.24 \begin{align*} \frac{12 \, b^{4} d^{3} x^{5} + 15 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} + 20 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 30 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 60 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{60 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^3/(b*x+a),x, algorithm="maxima")

[Out]

1/60*(12*b^4*d^3*x^5 + 15*(3*b^4*c*d^2 - a*b^3*d^3)*x^4 + 20*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^3 +
30*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^2 - 60*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*
d^2 - a^4*d^3)*x)/b^5 + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*log(b*x + a)/b^6

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Fricas [A]  time = 2.15909, size = 439, normalized size = 3.4 \begin{align*} \frac{12 \, b^{5} d^{3} x^{5} + 15 \,{\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 20 \,{\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{3} + 30 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} - 60 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x + 60 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^3/(b*x+a),x, algorithm="fricas")

[Out]

1/60*(12*b^5*d^3*x^5 + 15*(3*b^5*c*d^2 - a*b^4*d^3)*x^4 + 20*(3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 +
30*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^2 - 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^
2*c*d^2 - a^4*b*d^3)*x + 60*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*log(b*x + a))/b^6

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Sympy [A]  time = 0.646917, size = 180, normalized size = 1.4 \begin{align*} - \frac{a^{2} \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{6}} + \frac{d^{3} x^{5}}{5 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x^{3} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{3 b^{3}} - \frac{x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 b^{4}} + \frac{x \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d*x+c)**3/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.26582, size = 306, normalized size = 2.37 \begin{align*} \frac{12 \, b^{4} d^{3} x^{5} + 45 \, b^{4} c d^{2} x^{4} - 15 \, a b^{3} d^{3} x^{4} + 60 \, b^{4} c^{2} d x^{3} - 60 \, a b^{3} c d^{2} x^{3} + 20 \, a^{2} b^{2} d^{3} x^{3} + 30 \, b^{4} c^{3} x^{2} - 90 \, a b^{3} c^{2} d x^{2} + 90 \, a^{2} b^{2} c d^{2} x^{2} - 30 \, a^{3} b d^{3} x^{2} - 60 \, a b^{3} c^{3} x + 180 \, a^{2} b^{2} c^{2} d x - 180 \, a^{3} b c d^{2} x + 60 \, a^{4} d^{3} x}{60 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^3/(b*x+a),x, algorithm="giac")

[Out]

1/60*(12*b^4*d^3*x^5 + 45*b^4*c*d^2*x^4 - 15*a*b^3*d^3*x^4 + 60*b^4*c^2*d*x^3 - 60*a*b^3*c*d^2*x^3 + 20*a^2*b^
2*d^3*x^3 + 30*b^4*c^3*x^2 - 90*a*b^3*c^2*d*x^2 + 90*a^2*b^2*c*d^2*x^2 - 30*a^3*b*d^3*x^2 - 60*a*b^3*c^3*x + 1
80*a^2*b^2*c^2*d*x - 180*a^3*b*c*d^2*x + 60*a^4*d^3*x)/b^5 + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 -
a^5*d^3)*log(abs(b*x + a))/b^6