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

Optimal. Leaf size=156 \[ \frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^3 x^3}{3 b^3} \]

[Out]

(3*d*(b*c - 2*a*d)*(b*c - a*d)*x)/b^5 + (3*d^2*(b*c - a*d)*x^2)/(2*b^4) + (d^3*x^3)/(3*b^3) - (a^2*(b*c - a*d)
^3)/(2*b^6*(a + b*x)^2) + (a*(2*b*c - 5*a*d)*(b*c - a*d)^2)/(b^6*(a + b*x)) + ((b*c - a*d)*(b^2*c^2 - 8*a*b*c*
d + 10*a^2*d^2)*Log[a + b*x])/b^6

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Rubi [A]  time = 0.15293, antiderivative size = 156, 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} \[ \frac{\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{3 d^2 x^2 (b c-a d)}{2 b^4}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^3 x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^3)/(a + b*x)^3,x]

[Out]

(3*d*(b*c - 2*a*d)*(b*c - a*d)*x)/b^5 + (3*d^2*(b*c - a*d)*x^2)/(2*b^4) + (d^3*x^3)/(3*b^3) - (a^2*(b*c - a*d)
^3)/(2*b^6*(a + b*x)^2) + (a*(2*b*c - 5*a*d)*(b*c - a*d)^2)/(b^6*(a + b*x)) + ((b*c - a*d)*(b^2*c^2 - 8*a*b*c*
d + 10*a^2*d^2)*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)^3} \, dx &=\int \left (\frac{3 d (b c-2 a d) (b c-a d)}{b^5}+\frac{3 d^2 (b c-a d) x}{b^4}+\frac{d^3 x^2}{b^3}-\frac{a^2 (-b c+a d)^3}{b^5 (a+b x)^3}+\frac{a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)^2}+\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac{3 d^2 (b c-a d) x^2}{2 b^4}+\frac{d^3 x^3}{3 b^3}-\frac{a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac{a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac{(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0843345, size = 160, normalized size = 1.03 \[ \frac{18 b d x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+6 \left (18 a^2 b c d^2-10 a^3 d^3-9 a b^2 c^2 d+b^3 c^3\right ) \log (a+b x)+\frac{3 a^2 (a d-b c)^3}{(a+b x)^2}+9 b^2 d^2 x^2 (b c-a d)-\frac{6 a (b c-a d)^2 (5 a d-2 b c)}{a+b x}+2 b^3 d^3 x^3}{6 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*b*d*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x + 9*b^2*d^2*(b*c - a*d)*x^2 + 2*b^3*d^3*x^3 + (3*a^2*(-(b*c) + a*d
)^3)/(a + b*x)^2 - (6*a*(b*c - a*d)^2*(-2*b*c + 5*a*d))/(a + b*x) + 6*(b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^
2 - 10*a^3*d^3)*Log[a + b*x])/(6*b^6)

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Maple [A]  time = 0.009, size = 280, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,{b}^{3}}}-{\frac{3\,{d}^{3}{x}^{2}a}{2\,{b}^{4}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{3}x}{{b}^{5}}}-9\,{\frac{ac{d}^{2}x}{{b}^{4}}}+3\,{\frac{{c}^{2}dx}{{b}^{3}}}-5\,{\frac{{a}^{4}{d}^{3}}{{b}^{6} \left ( bx+a \right ) }}+12\,{\frac{{a}^{3}c{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}-9\,{\frac{{a}^{2}{c}^{2}d}{{b}^{4} \left ( bx+a \right ) }}+2\,{\frac{a{c}^{3}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{{a}^{5}{d}^{3}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{4}c{d}^{2}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{a}^{3}{c}^{2}d}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{c}^{3}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}-10\,{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{6}}}+18\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{5}}}-9\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{4}}}+{\frac{\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)^3,x)

[Out]

1/3*d^3*x^3/b^3-3/2*d^3/b^4*x^2*a+3/2*d^2/b^3*x^2*c+6*d^3/b^5*a^2*x-9*d^2/b^4*a*c*x+3*d/b^3*c^2*x-5*a^4/b^6/(b
*x+a)*d^3+12*a^3/b^5/(b*x+a)*c*d^2-9*a^2/b^4/(b*x+a)*c^2*d+2*a/b^3/(b*x+a)*c^3+1/2*a^5/b^6/(b*x+a)^2*d^3-3/2*a
^4/b^5/(b*x+a)^2*c*d^2+3/2*a^3/b^4/(b*x+a)^2*c^2*d-1/2*a^2/b^3/(b*x+a)^2*c^3-10/b^6*ln(b*x+a)*a^3*d^3+18/b^5*l
n(b*x+a)*a^2*c*d^2-9/b^4*ln(b*x+a)*a*c^2*d+1/b^3*ln(b*x+a)*c^3

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Maxima [A]  time = 1.20467, size = 306, normalized size = 1.96 \begin{align*} \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\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 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, b^{2} d^{3} x^{3} + 9 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 18 \,{\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x}{6 \, b^{5}} + \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

1/2*(3*a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 21*a^4*b*c*d^2 - 9*a^5*d^3 + 2*(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)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) + 1/6*(2*b^2*d^3*x^3 + 9*(b^2*c*d^2 - a*b*d^3)*x^
2 + 18*(b^2*c^2*d - 3*a*b*c*d^2 + 2*a^2*d^3)*x)/b^5 + (b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^3)*
log(b*x + a)/b^6

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Fricas [B]  time = 2.99063, size = 745, normalized size = 4.78 \begin{align*} \frac{2 \, b^{5} d^{3} x^{5} + 9 \, a^{2} b^{3} c^{3} - 45 \, a^{3} b^{2} c^{2} d + 63 \, a^{4} b c d^{2} - 27 \, a^{5} d^{3} +{\left (9 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \,{\left (9 \, b^{5} c^{2} d - 18 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{3} + 9 \,{\left (4 \, a b^{4} c^{2} d - 11 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{2} + 6 \,{\left (2 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x + 6 \,{\left (a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2} - 10 \, a^{5} d^{3} +{\left (b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 18 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2} - 10 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^5*d^3*x^5 + 9*a^2*b^3*c^3 - 45*a^3*b^2*c^2*d + 63*a^4*b*c*d^2 - 27*a^5*d^3 + (9*b^5*c*d^2 - 5*a*b^4*d
^3)*x^4 + 2*(9*b^5*c^2*d - 18*a*b^4*c*d^2 + 10*a^2*b^3*d^3)*x^3 + 9*(4*a*b^4*c^2*d - 11*a^2*b^3*c*d^2 + 7*a^3*
b^2*d^3)*x^2 + 6*(2*a*b^4*c^3 - 6*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 + a^4*b*d^3)*x + 6*(a^2*b^3*c^3 - 9*a^3*b^2*
c^2*d + 18*a^4*b*c*d^2 - 10*a^5*d^3 + (b^5*c^3 - 9*a*b^4*c^2*d + 18*a^2*b^3*c*d^2 - 10*a^3*b^2*d^3)*x^2 + 2*(a
*b^4*c^3 - 9*a^2*b^3*c^2*d + 18*a^3*b^2*c*d^2 - 10*a^4*b*d^3)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

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Sympy [A]  time = 1.98321, size = 230, normalized size = 1.47 \begin{align*} - \frac{9 a^{5} d^{3} - 21 a^{4} b c d^{2} + 15 a^{3} b^{2} c^{2} d - 3 a^{2} b^{3} c^{3} + x \left (10 a^{4} b d^{3} - 24 a^{3} b^{2} c d^{2} + 18 a^{2} b^{3} c^{2} d - 4 a b^{4} c^{3}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac{d^{3} x^{3}}{3 b^{3}} - \frac{x^{2} \left (3 a d^{3} - 3 b c d^{2}\right )}{2 b^{4}} + \frac{x \left (6 a^{2} d^{3} - 9 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{5}} - \frac{\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \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)**3,x)

[Out]

-(9*a**5*d**3 - 21*a**4*b*c*d**2 + 15*a**3*b**2*c**2*d - 3*a**2*b**3*c**3 + x*(10*a**4*b*d**3 - 24*a**3*b**2*c
*d**2 + 18*a**2*b**3*c**2*d - 4*a*b**4*c**3))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) + d**3*x**3/(3*b**3) -
x**2*(3*a*d**3 - 3*b*c*d**2)/(2*b**4) + x*(6*a**2*d**3 - 9*a*b*c*d**2 + 3*b**2*c**2*d)/b**5 - (a*d - b*c)*(10*
a**2*d**2 - 8*a*b*c*d + b**2*c**2)*log(a + b*x)/b**6

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Giac [A]  time = 1.16637, size = 300, normalized size = 1.92 \begin{align*} \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \,{\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 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} d^{3} x^{3} + 9 \, b^{6} c d^{2} x^{2} - 9 \, a b^{5} d^{3} x^{2} + 18 \, b^{6} c^{2} d x - 54 \, a b^{5} c d^{2} x + 36 \, a^{2} b^{4} d^{3} x}{6 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^3 - 9*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 10*a^3*d^3)*log(abs(b*x + a))/b^6 + 1/2*(3*a^2*b^3*c^3 - 15*a^3*b^
2*c^2*d + 21*a^4*b*c*d^2 - 9*a^5*d^3 + 2*(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)/(
(b*x + a)^2*b^6) + 1/6*(2*b^6*d^3*x^3 + 9*b^6*c*d^2*x^2 - 9*a*b^5*d^3*x^2 + 18*b^6*c^2*d*x - 54*a*b^5*c*d^2*x
+ 36*a^2*b^4*d^3*x)/b^9