3.269 \(\int \frac{x^4 (c+d x)^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=200 \[ \frac{3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac{a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac{a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac{d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac{3 d x^4 (b c-a d)^2}{4 b^4}+\frac{x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}-\frac{a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac{d^3 x^6}{6 b^2} \]

[Out]

(3*a^2*(b*c - 2*a*d)*(b*c - a*d)^2*x)/b^7 - (a*(2*b*c - 5*a*d)*(b*c - a*d)^2*x^2)/(2*b^6) + ((b*c - 4*a*d)*(b*
c - a*d)^2*x^3)/(3*b^5) + (3*d*(b*c - a*d)^2*x^4)/(4*b^4) + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^6)/(6*b
^2) - (a^4*(b*c - a*d)^3)/(b^8*(a + b*x)) - (a^3*(4*b*c - 7*a*d)*(b*c - a*d)^2*Log[a + b*x])/b^8

________________________________________________________________________________________

Rubi [A]  time = 0.268016, antiderivative size = 200, 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} \[ \frac{3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac{a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac{a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac{d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac{3 d x^4 (b c-a d)^2}{4 b^4}+\frac{x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}-\frac{a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac{d^3 x^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

(3*a^2*(b*c - 2*a*d)*(b*c - a*d)^2*x)/b^7 - (a*(2*b*c - 5*a*d)*(b*c - a*d)^2*x^2)/(2*b^6) + ((b*c - 4*a*d)*(b*
c - a*d)^2*x^3)/(3*b^5) + (3*d*(b*c - a*d)^2*x^4)/(4*b^4) + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^6)/(6*b
^2) - (a^4*(b*c - a*d)^3)/(b^8*(a + b*x)) - (a^3*(4*b*c - 7*a*d)*(b*c - a*d)^2*Log[a + b*x])/b^8

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^4 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (-\frac{3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^7}+\frac{a (-b c+a d)^2 (-2 b c+5 a d) x}{b^6}+\frac{(b c-4 a d) (b c-a d)^2 x^2}{b^5}+\frac{3 d (b c-a d)^2 x^3}{b^4}+\frac{d^2 (3 b c-2 a d) x^4}{b^3}+\frac{d^3 x^5}{b^2}-\frac{a^4 (-b c+a d)^3}{b^7 (a+b x)^2}+\frac{a^3 (-b c+a d)^2 (-4 b c+7 a d)}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{3 a^2 (b c-2 a d) (b c-a d)^2 x}{b^7}-\frac{a (2 b c-5 a d) (b c-a d)^2 x^2}{2 b^6}+\frac{(b c-4 a d) (b c-a d)^2 x^3}{3 b^5}+\frac{3 d (b c-a d)^2 x^4}{4 b^4}+\frac{d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac{d^3 x^6}{6 b^2}-\frac{a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac{a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.0706099, size = 190, normalized size = 0.95 \[ \frac{-180 a^2 b x (b c-a d)^2 (2 a d-b c)+\frac{60 a^4 (a d-b c)^3}{a+b x}+60 a^3 (b c-a d)^2 (7 a d-4 b c) \log (a+b x)+12 b^5 d^2 x^5 (3 b c-2 a d)+45 b^4 d x^4 (b c-a d)^2+20 b^3 x^3 (b c-4 a d) (b c-a d)^2+30 a b^2 x^2 (b c-a d)^2 (5 a d-2 b c)+10 b^6 d^3 x^6}{60 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

(-180*a^2*b*(b*c - a*d)^2*(-(b*c) + 2*a*d)*x + 30*a*b^2*(b*c - a*d)^2*(-2*b*c + 5*a*d)*x^2 + 20*b^3*(b*c - 4*a
*d)*(b*c - a*d)^2*x^3 + 45*b^4*d*(b*c - a*d)^2*x^4 + 12*b^5*d^2*(3*b*c - 2*a*d)*x^5 + 10*b^6*d^3*x^6 + (60*a^4
*(-(b*c) + a*d)^3)/(a + b*x) + 60*a^3*(b*c - a*d)^2*(-4*b*c + 7*a*d)*Log[a + b*x])/(60*b^8)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 378, normalized size = 1.9 \begin{align*}{\frac{{x}^{3}{c}^{3}}{3\,{b}^{2}}}+3\,{\frac{{x}^{3}{a}^{2}c{d}^{2}}{{b}^{4}}}-{\frac{3\,{x}^{4}ac{d}^{2}}{2\,{b}^{3}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{3}}{{b}^{5}}}-{\frac{2\,{x}^{5}a{d}^{3}}{5\,{b}^{3}}}+{\frac{3\,{x}^{5}c{d}^{2}}{5\,{b}^{2}}}+{\frac{3\,{x}^{4}{a}^{2}{d}^{3}}{4\,{b}^{4}}}+{\frac{3\,{x}^{4}{c}^{2}d}{4\,{b}^{2}}}-{\frac{4\,{x}^{3}{a}^{3}{d}^{3}}{3\,{b}^{5}}}+{\frac{5\,{x}^{2}{a}^{4}{d}^{3}}{2\,{b}^{6}}}-{\frac{a{x}^{2}{c}^{3}}{{b}^{3}}}-6\,{\frac{{a}^{5}{d}^{3}x}{{b}^{7}}}+3\,{\frac{{a}^{2}{c}^{3}x}{{b}^{4}}}+{\frac{{a}^{7}{d}^{3}}{{b}^{8} \left ( bx+a \right ) }}-{\frac{{a}^{4}{c}^{3}}{{b}^{5} \left ( bx+a \right ) }}+7\,{\frac{{a}^{6}\ln \left ( bx+a \right ){d}^{3}}{{b}^{8}}}+{\frac{{d}^{3}{x}^{6}}{6\,{b}^{2}}}-18\,{\frac{{a}^{5}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{7}}}+15\,{\frac{{a}^{4}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{6}}}-2\,{\frac{{x}^{3}a{c}^{2}d}{{b}^{3}}}-6\,{\frac{{x}^{2}{a}^{3}c{d}^{2}}{{b}^{5}}}+{\frac{9\,{a}^{2}{x}^{2}{c}^{2}d}{2\,{b}^{4}}}+15\,{\frac{{a}^{4}c{d}^{2}x}{{b}^{6}}}-12\,{\frac{{a}^{3}{c}^{2}dx}{{b}^{5}}}-3\,{\frac{{a}^{6}c{d}^{2}}{{b}^{7} \left ( bx+a \right ) }}+3\,{\frac{{a}^{5}{c}^{2}d}{{b}^{6} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x+c)^3/(b*x+a)^2,x)

[Out]

1/3/b^2*x^3*c^3+3/b^4*x^3*a^2*c*d^2-3/2/b^3*x^4*a*c*d^2-4*a^3/b^5*ln(b*x+a)*c^3-2/5/b^3*x^5*a*d^3+3/5/b^2*x^5*
c*d^2+3/4/b^4*x^4*a^2*d^3+3/4/b^2*x^4*c^2*d-4/3/b^5*x^3*a^3*d^3+5/2/b^6*x^2*a^4*d^3-1/b^3*x^2*a*c^3-6/b^7*a^5*
d^3*x+3/b^4*a^2*c^3*x+a^7/b^8/(b*x+a)*d^3-a^4/b^5/(b*x+a)*c^3+7*a^6/b^8*ln(b*x+a)*d^3+1/6*d^3*x^6/b^2-18*a^5/b
^7*ln(b*x+a)*c*d^2+15*a^4/b^6*ln(b*x+a)*c^2*d-2/b^3*x^3*a*c^2*d-6/b^5*x^2*a^3*c*d^2+9/2/b^4*x^2*a^2*c^2*d+15/b
^6*a^4*c*d^2*x-12/b^5*a^3*c^2*d*x-3*a^6/b^7/(b*x+a)*c*d^2+3*a^5/b^6/(b*x+a)*c^2*d

________________________________________________________________________________________

Maxima [A]  time = 1.03514, size = 436, normalized size = 2.18 \begin{align*} -\frac{a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}}{b^{9} x + a b^{8}} + \frac{10 \, b^{5} d^{3} x^{6} + 12 \,{\left (3 \, b^{5} c d^{2} - 2 \, a b^{4} d^{3}\right )} x^{5} + 45 \,{\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \,{\left (b^{5} c^{3} - 6 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 4 \, a^{3} b^{2} d^{3}\right )} x^{3} - 30 \,{\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2} + 180 \,{\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 )} x}{60 \, b^{7}} - \frac{{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

-(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)/(b^9*x + a*b^8) + 1/60*(10*b^5*d^3*x^6 + 12*(3*b^5*
c*d^2 - 2*a*b^4*d^3)*x^5 + 45*(b^5*c^2*d - 2*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 + 20*(b^5*c^3 - 6*a*b^4*c^2*d + 9*
a^2*b^3*c*d^2 - 4*a^3*b^2*d^3)*x^3 - 30*(2*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2 +
180*(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)*x)/b^7 - (4*a^3*b^3*c^3 - 15*a^4*b^2*c^2*d +
18*a^5*b*c*d^2 - 7*a^6*d^3)*log(b*x + a)/b^8

________________________________________________________________________________________

Fricas [B]  time = 2.21633, size = 883, normalized size = 4.42 \begin{align*} \frac{10 \, b^{7} d^{3} x^{7} - 60 \, a^{4} b^{3} c^{3} + 180 \, a^{5} b^{2} c^{2} d - 180 \, a^{6} b c d^{2} + 60 \, a^{7} d^{3} + 2 \,{\left (18 \, b^{7} c d^{2} - 7 \, a b^{6} d^{3}\right )} x^{6} + 3 \,{\left (15 \, b^{7} c^{2} d - 18 \, a b^{6} c d^{2} + 7 \, a^{2} b^{5} d^{3}\right )} x^{5} + 5 \,{\left (4 \, b^{7} c^{3} - 15 \, a b^{6} c^{2} d + 18 \, a^{2} b^{5} c d^{2} - 7 \, a^{3} b^{4} d^{3}\right )} x^{4} - 10 \,{\left (4 \, a b^{6} c^{3} - 15 \, a^{2} b^{5} c^{2} d + 18 \, a^{3} b^{4} c d^{2} - 7 \, a^{4} b^{3} d^{3}\right )} x^{3} + 30 \,{\left (4 \, a^{2} b^{5} c^{3} - 15 \, a^{3} b^{4} c^{2} d + 18 \, a^{4} b^{3} c d^{2} - 7 \, a^{5} b^{2} d^{3}\right )} x^{2} + 180 \,{\left (a^{3} b^{4} c^{3} - 4 \, a^{4} b^{3} c^{2} d + 5 \, a^{5} b^{2} c d^{2} - 2 \, a^{6} b d^{3}\right )} x - 60 \,{\left (4 \, a^{4} b^{3} c^{3} - 15 \, a^{5} b^{2} c^{2} d + 18 \, a^{6} b c d^{2} - 7 \, a^{7} d^{3} +{\left (4 \, a^{3} b^{4} c^{3} - 15 \, a^{4} b^{3} c^{2} d + 18 \, a^{5} b^{2} c d^{2} - 7 \, a^{6} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{9} x + a b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*d^3*x^7 - 60*a^4*b^3*c^3 + 180*a^5*b^2*c^2*d - 180*a^6*b*c*d^2 + 60*a^7*d^3 + 2*(18*b^7*c*d^2 - 7
*a*b^6*d^3)*x^6 + 3*(15*b^7*c^2*d - 18*a*b^6*c*d^2 + 7*a^2*b^5*d^3)*x^5 + 5*(4*b^7*c^3 - 15*a*b^6*c^2*d + 18*a
^2*b^5*c*d^2 - 7*a^3*b^4*d^3)*x^4 - 10*(4*a*b^6*c^3 - 15*a^2*b^5*c^2*d + 18*a^3*b^4*c*d^2 - 7*a^4*b^3*d^3)*x^3
+ 30*(4*a^2*b^5*c^3 - 15*a^3*b^4*c^2*d + 18*a^4*b^3*c*d^2 - 7*a^5*b^2*d^3)*x^2 + 180*(a^3*b^4*c^3 - 4*a^4*b^3
*c^2*d + 5*a^5*b^2*c*d^2 - 2*a^6*b*d^3)*x - 60*(4*a^4*b^3*c^3 - 15*a^5*b^2*c^2*d + 18*a^6*b*c*d^2 - 7*a^7*d^3
+ (4*a^3*b^4*c^3 - 15*a^4*b^3*c^2*d + 18*a^5*b^2*c*d^2 - 7*a^6*b*d^3)*x)*log(b*x + a))/(b^9*x + a*b^8)

________________________________________________________________________________________

Sympy [A]  time = 1.54727, size = 308, normalized size = 1.54 \begin{align*} \frac{a^{3} \left (a d - b c\right )^{2} \left (7 a d - 4 b c\right ) \log{\left (a + b x \right )}}{b^{8}} + \frac{a^{7} d^{3} - 3 a^{6} b c d^{2} + 3 a^{5} b^{2} c^{2} d - a^{4} b^{3} c^{3}}{a b^{8} + b^{9} x} + \frac{d^{3} x^{6}}{6 b^{2}} - \frac{x^{5} \left (2 a d^{3} - 3 b c d^{2}\right )}{5 b^{3}} + \frac{x^{4} \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{4 b^{4}} - \frac{x^{3} \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 )}{3 b^{5}} + \frac{x^{2} \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 )}{2 b^{6}} - \frac{x \left (6 a^{5} d^{3} - 15 a^{4} b c d^{2} + 12 a^{3} b^{2} c^{2} d - 3 a^{2} b^{3} c^{3}\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d*x+c)**3/(b*x+a)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.21166, size = 544, normalized size = 2.72 \begin{align*} \frac{{\left (10 \, d^{3} + \frac{12 \,{\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{45 \,{\left (b^{4} c^{2} d - 6 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{20 \,{\left (b^{6} c^{3} - 15 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 35 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac{30 \,{\left (4 \, a b^{7} c^{3} - 30 \, a^{2} b^{6} c^{2} d + 60 \, a^{3} b^{5} c d^{2} - 35 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac{180 \,{\left (2 \, a^{2} b^{8} c^{3} - 10 \, a^{3} b^{7} c^{2} d + 15 \, a^{4} b^{6} c d^{2} - 7 \, a^{5} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )}{\left (b x + a\right )}^{6}}{60 \, b^{8}} + \frac{{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{8}} - \frac{\frac{a^{4} b^{9} c^{3}}{b x + a} - \frac{3 \, a^{5} b^{8} c^{2} d}{b x + a} + \frac{3 \, a^{6} b^{7} c d^{2}}{b x + a} - \frac{a^{7} b^{6} d^{3}}{b x + a}}{b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="giac")

[Out]

1/60*(10*d^3 + 12*(3*b^2*c*d^2 - 7*a*b*d^3)/((b*x + a)*b) + 45*(b^4*c^2*d - 6*a*b^3*c*d^2 + 7*a^2*b^2*d^3)/((b
*x + a)^2*b^2) + 20*(b^6*c^3 - 15*a*b^5*c^2*d + 45*a^2*b^4*c*d^2 - 35*a^3*b^3*d^3)/((b*x + a)^3*b^3) - 30*(4*a
*b^7*c^3 - 30*a^2*b^6*c^2*d + 60*a^3*b^5*c*d^2 - 35*a^4*b^4*d^3)/((b*x + a)^4*b^4) + 180*(2*a^2*b^8*c^3 - 10*a
^3*b^7*c^2*d + 15*a^4*b^6*c*d^2 - 7*a^5*b^5*d^3)/((b*x + a)^5*b^5))*(b*x + a)^6/b^8 + (4*a^3*b^3*c^3 - 15*a^4*
b^2*c^2*d + 18*a^5*b*c*d^2 - 7*a^6*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^8 - (a^4*b^9*c^3/(b*x + a) -
3*a^5*b^8*c^2*d/(b*x + a) + 3*a^6*b^7*c*d^2/(b*x + a) - a^7*b^6*d^3/(b*x + a))/b^14