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

Optimal. Leaf size=164 \[ \frac{a^3 (b c-a d)^3}{b^7 (a+b x)}+\frac{3 a^2 (b c-2 a d) (b c-a d)^2 \log (a+b x)}{b^7}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d x^3 (b c-a d)^2}{b^4}+\frac{x^2 (b c-4 a d) (b c-a d)^2}{2 b^5}-\frac{a x (2 b c-5 a d) (b c-a d)^2}{b^6}+\frac{d^3 x^5}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.164787, antiderivative size = 164, 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} \[ \frac{a^3 (b c-a d)^3}{b^7 (a+b x)}+\frac{3 a^2 (b c-2 a d) (b c-a d)^2 \log (a+b x)}{b^7}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d x^3 (b c-a d)^2}{b^4}+\frac{x^2 (b c-4 a d) (b c-a d)^2}{2 b^5}-\frac{a x (2 b c-5 a d) (b c-a d)^2}{b^6}+\frac{d^3 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

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

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^3 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac{a (-b c+a d)^2 (-2 b c+5 a d)}{b^6}+\frac{(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac{3 d (b c-a d)^2 x^2}{b^4}+\frac{d^2 (3 b c-2 a d) x^3}{b^3}+\frac{d^3 x^4}{b^2}+\frac{a^3 (-b c+a d)^3}{b^6 (a+b x)^2}-\frac{3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac{a (2 b c-5 a d) (b c-a d)^2 x}{b^6}+\frac{(b c-4 a d) (b c-a d)^2 x^2}{2 b^5}+\frac{d (b c-a d)^2 x^3}{b^4}+\frac{d^2 (3 b c-2 a d) x^4}{4 b^3}+\frac{d^3 x^5}{5 b^2}+\frac{a^3 (b c-a d)^3}{b^7 (a+b x)}+\frac{3 a^2 (b c-2 a d) (b c-a d)^2 \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0598227, size = 160, normalized size = 0.98 \[ \frac{-\frac{20 a^3 (a d-b c)^3}{a+b x}-60 a^2 (b c-a d)^2 (2 a d-b c) \log (a+b x)+5 b^4 d^2 x^4 (3 b c-2 a d)+20 b^3 d x^3 (b c-a d)^2+10 b^2 x^2 (b c-4 a d) (b c-a d)^2+20 a b x (b c-a d)^2 (5 a d-2 b c)+4 b^5 d^3 x^5}{20 b^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*a*b*(b*c - a*d)^2*(-2*b*c + 5*a*d)*x + 10*b^2*(b*c - 4*a*d)*(b*c - a*d)^2*x^2 + 20*b^3*d*(b*c - a*d)^2*x^3
+ 5*b^4*d^2*(3*b*c - 2*a*d)*x^4 + 4*b^5*d^3*x^5 - (20*a^3*(-(b*c) + a*d)^3)/(a + b*x) - 60*a^2*(b*c - a*d)^2*
(-(b*c) + 2*a*d)*Log[a + b*x])/(20*b^7)

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Maple [B]  time = 0.006, size = 318, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{5}}{5\,{b}^{2}}}-{\frac{{x}^{4}a{d}^{3}}{2\,{b}^{3}}}+{\frac{3\,{x}^{4}c{d}^{2}}{4\,{b}^{2}}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{{b}^{4}}}-2\,{\frac{{x}^{3}ac{d}^{2}}{{b}^{3}}}+{\frac{{x}^{3}{c}^{2}d}{{b}^{2}}}-2\,{\frac{{x}^{2}{a}^{3}{d}^{3}}{{b}^{5}}}+{\frac{9\,{a}^{2}{x}^{2}c{d}^{2}}{2\,{b}^{4}}}-3\,{\frac{a{x}^{2}{c}^{2}d}{{b}^{3}}}+{\frac{{c}^{3}{x}^{2}}{2\,{b}^{2}}}+5\,{\frac{{a}^{4}{d}^{3}x}{{b}^{6}}}-12\,{\frac{{a}^{3}c{d}^{2}x}{{b}^{5}}}+9\,{\frac{{a}^{2}{c}^{2}dx}{{b}^{4}}}-2\,{\frac{a{c}^{3}x}{{b}^{3}}}-{\frac{{a}^{6}{d}^{3}}{{b}^{7} \left ( bx+a \right ) }}+3\,{\frac{{a}^{5}c{d}^{2}}{{b}^{6} \left ( bx+a \right ) }}-3\,{\frac{{a}^{4}{c}^{2}d}{{b}^{5} \left ( bx+a \right ) }}+{\frac{{a}^{3}{c}^{3}}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ){d}^{3}}{{b}^{7}}}+15\,{\frac{{a}^{4}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{6}}}-12\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{5}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{3}}{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03493, size = 365, normalized size = 2.23 \begin{align*} \frac{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}}{b^{8} x + a b^{7}} + \frac{4 \, b^{4} d^{3} x^{5} + 5 \,{\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{4} + 20 \,{\left (b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 10 \,{\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} - 20 \,{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} x}{20 \, b^{6}} + \frac{3 \,{\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)/(b^8*x + a*b^7) + 1/20*(4*b^4*d^3*x^5 + 5*(3*b^4*c*d
^2 - 2*a*b^3*d^3)*x^4 + 20*(b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3)*x^3 + 10*(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^2 - 20*(2*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 5*a^4*d^3)*x)/b^6 + 3*(a^
2*b^3*c^3 - 4*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 - 2*a^5*d^3)*log(b*x + a)/b^7

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Fricas [B]  time = 2.30563, size = 751, normalized size = 4.58 \begin{align*} \frac{4 \, b^{6} d^{3} x^{6} + 20 \, a^{3} b^{3} c^{3} - 60 \, a^{4} b^{2} c^{2} d + 60 \, a^{5} b c d^{2} - 20 \, a^{6} d^{3} + 3 \,{\left (5 \, b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{5} + 5 \,{\left (4 \, b^{6} c^{2} d - 5 \, a b^{5} c d^{2} + 2 \, a^{2} b^{4} d^{3}\right )} x^{4} + 10 \,{\left (b^{6} c^{3} - 4 \, a b^{5} c^{2} d + 5 \, a^{2} b^{4} c d^{2} - 2 \, a^{3} b^{3} d^{3}\right )} x^{3} - 30 \,{\left (a b^{5} c^{3} - 4 \, a^{2} b^{4} c^{2} d + 5 \, a^{3} b^{3} c d^{2} - 2 \, a^{4} b^{2} d^{3}\right )} x^{2} - 20 \,{\left (2 \, a^{2} b^{4} c^{3} - 9 \, a^{3} b^{3} c^{2} d + 12 \, a^{4} b^{2} c d^{2} - 5 \, a^{5} b d^{3}\right )} x + 60 \,{\left (a^{3} b^{3} c^{3} - 4 \, a^{4} b^{2} c^{2} d + 5 \, a^{5} b c d^{2} - 2 \, a^{6} d^{3} +{\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{20 \,{\left (b^{8} x + a b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b^6*d^3*x^6 + 20*a^3*b^3*c^3 - 60*a^4*b^2*c^2*d + 60*a^5*b*c*d^2 - 20*a^6*d^3 + 3*(5*b^6*c*d^2 - 2*a*b
^5*d^3)*x^5 + 5*(4*b^6*c^2*d - 5*a*b^5*c*d^2 + 2*a^2*b^4*d^3)*x^4 + 10*(b^6*c^3 - 4*a*b^5*c^2*d + 5*a^2*b^4*c*
d^2 - 2*a^3*b^3*d^3)*x^3 - 30*(a*b^5*c^3 - 4*a^2*b^4*c^2*d + 5*a^3*b^3*c*d^2 - 2*a^4*b^2*d^3)*x^2 - 20*(2*a^2*
b^4*c^3 - 9*a^3*b^3*c^2*d + 12*a^4*b^2*c*d^2 - 5*a^5*b*d^3)*x + 60*(a^3*b^3*c^3 - 4*a^4*b^2*c^2*d + 5*a^5*b*c*
d^2 - 2*a^6*d^3 + (a^2*b^4*c^3 - 4*a^3*b^3*c^2*d + 5*a^4*b^2*c*d^2 - 2*a^5*b*d^3)*x)*log(b*x + a))/(b^8*x + a*
b^7)

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Sympy [A]  time = 2.53536, size = 248, normalized size = 1.51 \begin{align*} - \frac{3 a^{2} \left (a d - b c\right )^{2} \left (2 a d - b c\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{a^{6} d^{3} - 3 a^{5} b c d^{2} + 3 a^{4} b^{2} c^{2} d - a^{3} b^{3} c^{3}}{a b^{7} + b^{8} x} + \frac{d^{3} x^{5}}{5 b^{2}} - \frac{x^{4} \left (2 a d^{3} - 3 b c d^{2}\right )}{4 b^{3}} + \frac{x^{3} \left (a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d\right )}{b^{4}} - \frac{x^{2} \left (4 a^{3} d^{3} - 9 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 b^{5}} + \frac{x \left (5 a^{4} d^{3} - 12 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 2 a b^{3} c^{3}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19358, size = 460, normalized size = 2.8 \begin{align*} \frac{{\left (4 \, d^{3} + \frac{15 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{20 \,{\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{10 \,{\left (b^{6} c^{3} - 12 \, a b^{5} c^{2} d + 30 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac{60 \,{\left (a b^{7} c^{3} - 6 \, a^{2} b^{6} c^{2} d + 10 \, a^{3} b^{5} c d^{2} - 5 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}{\left (b x + a\right )}^{5}}{20 \, b^{7}} - \frac{3 \,{\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} + \frac{\frac{a^{3} b^{8} c^{3}}{b x + a} - \frac{3 \, a^{4} b^{7} c^{2} d}{b x + a} + \frac{3 \, a^{5} b^{6} c d^{2}}{b x + a} - \frac{a^{6} b^{5} d^{3}}{b x + a}}{b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*d^3 + 15*(b^2*c*d^2 - 2*a*b*d^3)/((b*x + a)*b) + 20*(b^4*c^2*d - 5*a*b^3*c*d^2 + 5*a^2*b^2*d^3)/((b*x
+ a)^2*b^2) + 10*(b^6*c^3 - 12*a*b^5*c^2*d + 30*a^2*b^4*c*d^2 - 20*a^3*b^3*d^3)/((b*x + a)^3*b^3) - 60*(a*b^7*
c^3 - 6*a^2*b^6*c^2*d + 10*a^3*b^5*c*d^2 - 5*a^4*b^4*d^3)/((b*x + a)^4*b^4))*(b*x + a)^5/b^7 - 3*(a^2*b^3*c^3
- 4*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 - 2*a^5*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 + (a^3*b^8*c^3/(b*x
+ a) - 3*a^4*b^7*c^2*d/(b*x + a) + 3*a^5*b^6*c*d^2/(b*x + a) - a^6*b^5*d^3/(b*x + a))/b^12