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

Optimal. Leaf size=106 \[ \frac{d x^2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{2 b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{a (b c-a d)^3 \log (a+b x)}{b^5}+\frac{d^3 x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0756914, antiderivative size = 106, 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} \[ \frac{d x^2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{2 b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{a (b c-a d)^3 \log (a+b x)}{b^5}+\frac{d^3 x^4}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^3)/(a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0398408, size = 115, normalized size = 1.08 \[ \frac{b x \left (6 a^2 b d^2 (6 c+d x)-12 a^3 d^3-2 a b^2 d \left (18 c^2+9 c d x+2 d^2 x^2\right )+3 b^3 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )+12 a (a d-b c)^3 \log (a+b x)}{12 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(-12*a^3*d^3 + 6*a^2*b*d^2*(6*c + d*x) - 2*a*b^2*d*(18*c^2 + 9*c*d*x + 2*d^2*x^2) + 3*b^3*(4*c^3 + 6*c^2*
d*x + 4*c*d^2*x^2 + d^3*x^3)) + 12*a*(-(b*c) + a*d)^3*Log[a + b*x])/(12*b^5)

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Maple [A]  time = 0.003, size = 186, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,b}}-{\frac{{x}^{3}a{d}^{3}}{3\,{b}^{2}}}+{\frac{c{x}^{3}{d}^{2}}{b}}+{\frac{{a}^{2}{x}^{2}{d}^{3}}{2\,{b}^{3}}}-{\frac{3\,a{x}^{2}c{d}^{2}}{2\,{b}^{2}}}+{\frac{3\,{x}^{2}{c}^{2}d}{2\,b}}-{\frac{{a}^{3}{d}^{3}x}{{b}^{4}}}+3\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{3}}}-3\,{\frac{a{c}^{2}dx}{{b}^{2}}}+{\frac{{c}^{3}x}{b}}+{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{3}}{{b}^{5}}}-3\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{4}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ){c}^{3}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01012, size = 221, normalized size = 2.08 \begin{align*} \frac{3 \, b^{3} d^{3} x^{4} + 4 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{12 \, b^{4}} - \frac{{\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 )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.31471, size = 336, normalized size = 3.17 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} + 4 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{2} + 12 \,{\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 - 12 \,{\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 )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.640157, size = 129, normalized size = 1.22 \begin{align*} \frac{a \left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{5}} + \frac{d^{3} x^{4}}{4 b} - \frac{x^{3} \left (a d^{3} - 3 b c d^{2}\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{3}} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22078, size = 230, normalized size = 2.17 \begin{align*} \frac{3 \, b^{3} d^{3} x^{4} + 12 \, b^{3} c d^{2} x^{3} - 4 \, a b^{2} d^{3} x^{3} + 18 \, b^{3} c^{2} d x^{2} - 18 \, a b^{2} c d^{2} x^{2} + 6 \, a^{2} b d^{3} x^{2} + 12 \, b^{3} c^{3} x - 36 \, a b^{2} c^{2} d x + 36 \, a^{2} b c d^{2} x - 12 \, a^{3} d^{3} x}{12 \, b^{4}} - \frac{{\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 )} \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*(d*x+c)^3/(b*x+a),x, algorithm="giac")

[Out]

1/12*(3*b^3*d^3*x^4 + 12*b^3*c*d^2*x^3 - 4*a*b^2*d^3*x^3 + 18*b^3*c^2*d*x^2 - 18*a*b^2*c*d^2*x^2 + 6*a^2*b*d^3
*x^2 + 12*b^3*c^3*x - 36*a*b^2*c^2*d*x + 36*a^2*b*c*d^2*x - 12*a^3*d^3*x)/b^4 - (a*b^3*c^3 - 3*a^2*b^2*c^2*d +
3*a^3*b*c*d^2 - a^4*d^3)*log(abs(b*x + a))/b^5