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

Optimal. Leaf size=136 \[ \frac{a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac{a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}+\frac{2 d x^3 (b c-a d)}{3 b^3}+\frac{x^2 (b c-3 a d) (b c-a d)}{2 b^4}-\frac{2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^2 x^4}{4 b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.126591, antiderivative size = 136, 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} \[ \frac{a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac{a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}+\frac{2 d x^3 (b c-a d)}{3 b^3}+\frac{x^2 (b c-3 a d) (b c-a d)}{2 b^4}-\frac{2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac{d^2 x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0783288, size = 149, normalized size = 1.1 \[ \frac{6 b^2 x^2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-24 a b x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+12 a^2 \left (5 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (a+b x)+\frac{12 a^3 (b c-a d)^2}{a+b x}+8 b^3 d x^3 (b c-a d)+3 b^4 d^2 x^4}{12 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*b*(b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x + 6*b^2*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x^2 + 8*b^3*d*(b*c - a*
d)*x^3 + 3*b^4*d^2*x^4 + (12*a^3*(b*c - a*d)^2)/(a + b*x) + 12*a^2*(3*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*Log[a +
b*x])/(12*b^6)

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 205, normalized size = 1.5 \begin{align*}{\frac{{d}^{2}{x}^{4}}{4\,{b}^{2}}}-{\frac{2\,{x}^{3}a{d}^{2}}{3\,{b}^{3}}}+{\frac{2\,c{x}^{3}d}{3\,{b}^{2}}}+{\frac{3\,{a}^{2}{x}^{2}{d}^{2}}{2\,{b}^{4}}}-2\,{\frac{a{x}^{2}cd}{{b}^{3}}}+{\frac{{x}^{2}{c}^{2}}{2\,{b}^{2}}}-4\,{\frac{{a}^{3}{d}^{2}x}{{b}^{5}}}+6\,{\frac{{a}^{2}cdx}{{b}^{4}}}-2\,{\frac{a{c}^{2}x}{{b}^{3}}}+{\frac{{a}^{5}{d}^{2}}{{b}^{6} \left ( bx+a \right ) }}-2\,{\frac{{a}^{4}cd}{{b}^{5} \left ( bx+a \right ) }}+{\frac{{a}^{3}{c}^{2}}{{b}^{4} \left ( bx+a \right ) }}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{2}}{{b}^{6}}}-8\,{\frac{{a}^{3}\ln \left ( bx+a \right ) cd}{{b}^{5}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}}{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01531, size = 236, normalized size = 1.74 \begin{align*} \frac{a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}}{b^{7} x + a b^{6}} + \frac{3 \, b^{3} d^{2} x^{4} + 8 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{2} - 24 \,{\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x}{12 \, b^{5}} + \frac{{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)/(b^7*x + a*b^6) + 1/12*(3*b^3*d^2*x^4 + 8*(b^3*c*d - a*b^2*d^2)*x^3 + 6*
(b^3*c^2 - 4*a*b^2*c*d + 3*a^2*b*d^2)*x^2 - 24*(a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x)/b^5 + (3*a^2*b^2*c^2 -
8*a^3*b*c*d + 5*a^4*d^2)*log(b*x + a)/b^6

________________________________________________________________________________________

Fricas [A]  time = 2.29109, size = 513, normalized size = 3.77 \begin{align*} \frac{3 \, b^{5} d^{2} x^{5} + 12 \, a^{3} b^{2} c^{2} - 24 \, a^{4} b c d + 12 \, a^{5} d^{2} +{\left (8 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{4} + 2 \,{\left (3 \, b^{5} c^{2} - 8 \, a b^{4} c d + 5 \, a^{2} b^{3} d^{2}\right )} x^{3} - 6 \,{\left (3 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 5 \, a^{3} b^{2} d^{2}\right )} x^{2} - 24 \,{\left (a^{2} b^{3} c^{2} - 3 \, a^{3} b^{2} c d + 2 \, a^{4} b d^{2}\right )} x + 12 \,{\left (3 \, a^{3} b^{2} c^{2} - 8 \, a^{4} b c d + 5 \, a^{5} d^{2} +{\left (3 \, a^{2} b^{3} c^{2} - 8 \, a^{3} b^{2} c d + 5 \, a^{4} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^5*d^2*x^5 + 12*a^3*b^2*c^2 - 24*a^4*b*c*d + 12*a^5*d^2 + (8*b^5*c*d - 5*a*b^4*d^2)*x^4 + 2*(3*b^5*c^
2 - 8*a*b^4*c*d + 5*a^2*b^3*d^2)*x^3 - 6*(3*a*b^4*c^2 - 8*a^2*b^3*c*d + 5*a^3*b^2*d^2)*x^2 - 24*(a^2*b^3*c^2 -
3*a^3*b^2*c*d + 2*a^4*b*d^2)*x + 12*(3*a^3*b^2*c^2 - 8*a^4*b*c*d + 5*a^5*d^2 + (3*a^2*b^3*c^2 - 8*a^3*b^2*c*d
+ 5*a^4*b*d^2)*x)*log(b*x + a))/(b^7*x + a*b^6)

________________________________________________________________________________________

Sympy [A]  time = 1.01399, size = 167, normalized size = 1.23 \begin{align*} \frac{a^{2} \left (a d - b c\right ) \left (5 a d - 3 b c\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{a^{5} d^{2} - 2 a^{4} b c d + a^{3} b^{2} c^{2}}{a b^{6} + b^{7} x} + \frac{d^{2} x^{4}}{4 b^{2}} - \frac{x^{3} \left (2 a d^{2} - 2 b c d\right )}{3 b^{3}} + \frac{x^{2} \left (3 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{2 b^{4}} - \frac{x \left (4 a^{3} d^{2} - 6 a^{2} b c d + 2 a b^{2} c^{2}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.18383, size = 319, normalized size = 2.35 \begin{align*} \frac{{\left (3 \, d^{2} + \frac{4 \,{\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{6 \,{\left (b^{4} c^{2} - 8 \, a b^{3} c d + 10 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{12 \,{\left (3 \, a b^{5} c^{2} - 12 \, a^{2} b^{4} c d + 10 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}{\left (b x + a\right )}^{4}}{12 \, b^{6}} - \frac{{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} + \frac{\frac{a^{3} b^{6} c^{2}}{b x + a} - \frac{2 \, a^{4} b^{5} c d}{b x + a} + \frac{a^{5} b^{4} d^{2}}{b x + a}}{b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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