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

Optimal. Leaf size=114 \[ \frac{3 d^2 x (b c-a d)}{b^4}-\frac{(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac{a (b c-a d)^3}{2 b^5 (a+b x)^2}+\frac{3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{d^3 x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.106411, antiderivative size = 114, 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} \[ \frac{3 d^2 x (b c-a d)}{b^4}-\frac{(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac{a (b c-a d)^3}{2 b^5 (a+b x)^2}+\frac{3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{d^3 x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^3)/(a + b*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0561213, size = 165, normalized size = 1.45 \[ \frac{a^2 b^2 d \left (9 c^2-12 c d x-11 d^2 x^2\right )+6 d (a+b x)^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (a+b x)+a^3 b d^2 (2 d x-15 c)+7 a^4 d^3-a b^3 \left (-12 c^2 d x+c^3-12 c d^2 x^2+4 d^3 x^3\right )+b^4 x \left (-2 c^3+6 c d^2 x^2+d^3 x^3\right )}{2 b^5 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*d^3 + a^3*b*d^2*(-15*c + 2*d*x) + a^2*b^2*d*(9*c^2 - 12*c*d*x - 11*d^2*x^2) + b^4*x*(-2*c^3 + 6*c*d^2*x
^2 + d^3*x^3) - a*b^3*(c^3 - 12*c^2*d*x - 12*c*d^2*x^2 + 4*d^3*x^3) + 6*d*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*(a
+ b*x)^2*Log[a + b*x])/(2*b^5*(a + b*x)^2)

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Maple [B]  time = 0.006, size = 222, normalized size = 2. \begin{align*}{\frac{{d}^{3}{x}^{2}}{2\,{b}^{3}}}-3\,{\frac{a{d}^{3}x}{{b}^{4}}}+3\,{\frac{{d}^{2}xc}{{b}^{3}}}+4\,{\frac{{a}^{3}{d}^{3}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{{a}^{2}c{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{a{c}^{2}d}{{b}^{3} \left ( bx+a \right ) }}-{\frac{{c}^{3}}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{a}^{4}{d}^{3}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{a}^{3}c{d}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}{c}^{2}d}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{a{c}^{3}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{d}^{3}\ln \left ( bx+a \right ){a}^{2}}{{b}^{5}}}-9\,{\frac{{d}^{2}\ln \left ( bx+a \right ) ac}{{b}^{4}}}+3\,{\frac{d\ln \left ( bx+a \right ){c}^{2}}{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d^3*x^2/b^3-3*d^3/b^4*a*x+3*d^2/b^3*x*c+4/b^5/(b*x+a)*a^3*d^3-9/b^4/(b*x+a)*a^2*c*d^2+6/b^3/(b*x+a)*a*c^2*
d-1/b^2/(b*x+a)*c^3-1/2*a^4/b^5/(b*x+a)^2*d^3+3/2*a^3/b^4/(b*x+a)^2*c*d^2-3/2*a^2/b^3/(b*x+a)^2*c^2*d+1/2*a/b^
2/(b*x+a)^2*c^3+6/b^5*d^3*ln(b*x+a)*a^2-9/b^4*d^2*ln(b*x+a)*a*c+3/b^3*d*ln(b*x+a)*c^2

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Maxima [A]  time = 1.20832, size = 235, normalized size = 2.06 \begin{align*} -\frac{a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \,{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac{b d^{3} x^{2} + 6 \,{\left (b c d^{2} - a d^{3}\right )} x}{2 \, b^{4}} + \frac{3 \,{\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} 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)^3,x, algorithm="maxima")

[Out]

-1/2*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + 2*(b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b^2*c*d^2
- 4*a^3*b*d^3)*x)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5) + 1/2*(b*d^3*x^2 + 6*(b*c*d^2 - a*d^3)*x)/b^4 + 3*(b^2*c^2*d
- 3*a*b*c*d^2 + 2*a^2*d^3)*log(b*x + a)/b^5

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Fricas [B]  time = 2.9395, size = 555, normalized size = 4.87 \begin{align*} \frac{b^{4} d^{3} x^{4} - a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 15 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} + 2 \,{\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{3} +{\left (12 \, a b^{3} c d^{2} - 11 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \,{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x + 6 \,{\left (a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} +{\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*d^3*x^4 - a*b^3*c^3 + 9*a^2*b^2*c^2*d - 15*a^3*b*c*d^2 + 7*a^4*d^3 + 2*(3*b^4*c*d^2 - 2*a*b^3*d^3)*x^
3 + (12*a*b^3*c*d^2 - 11*a^2*b^2*d^3)*x^2 - 2*(b^4*c^3 - 6*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - a^3*b*d^3)*x + 6*(a
^2*b^2*c^2*d - 3*a^3*b*c*d^2 + 2*a^4*d^3 + (b^4*c^2*d - 3*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*x^2 + 2*(a*b^3*c^2*d -
3*a^2*b^2*c*d^2 + 2*a^3*b*d^3)*x)*log(b*x + a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

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Sympy [A]  time = 2.06241, size = 173, normalized size = 1.52 \begin{align*} \frac{7 a^{4} d^{3} - 15 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + x \left (8 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 2 b^{4} c^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac{d^{3} x^{2}}{2 b^{3}} - \frac{x \left (3 a d^{3} - 3 b c d^{2}\right )}{b^{4}} + \frac{3 d \left (a d - b c\right ) \left (2 a d - b c\right ) \log{\left (a + b x \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)**3,x)

[Out]

(7*a**4*d**3 - 15*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - a*b**3*c**3 + x*(8*a**3*b*d**3 - 18*a**2*b**2*c*d**2 +
12*a*b**3*c**2*d - 2*b**4*c**3))/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + d**3*x**2/(2*b**3) - x*(3*a*d**3 -
3*b*c*d**2)/b**4 + 3*d*(a*d - b*c)*(2*a*d - b*c)*log(a + b*x)/b**5

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Giac [A]  time = 1.2204, size = 225, normalized size = 1.97 \begin{align*} \frac{3 \,{\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{b^{3} d^{3} x^{2} + 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6}} - \frac{a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \,{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b^2*c^2*d - 3*a*b*c*d^2 + 2*a^2*d^3)*log(abs(b*x + a))/b^5 + 1/2*(b^3*d^3*x^2 + 6*b^3*c*d^2*x - 6*a*b^2*d^3
*x)/b^6 - 1/2*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + 2*(b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b
^2*c*d^2 - 4*a^3*b*d^3)*x)/((b*x + a)^2*b^5)