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

Optimal. Leaf size=100 \[ \frac{a^2 \log (a+b x)}{(b c-a d)^3}-\frac{a^2 \log (c+d x)}{(b c-a d)^3}+\frac{c^2}{2 d^2 (c+d x)^2 (b c-a d)}-\frac{c (b c-2 a d)}{d^2 (c+d x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0806473, antiderivative size = 100, 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} \[ \frac{a^2 \log (a+b x)}{(b c-a d)^3}-\frac{a^2 \log (c+d x)}{(b c-a d)^3}+\frac{c^2}{2 d^2 (c+d x)^2 (b c-a d)}-\frac{c (b c-2 a d)}{d^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)*(c + d*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0595153, size = 99, normalized size = 0.99 \[ \frac{-2 a^2 d^2 (c+d x)^2 \log (a+b x)+2 a^2 d^2 (c+d x)^2 \log (c+d x)+c (b c-a d) (b c (c+2 d x)-a d (3 c+4 d x))}{2 d^2 (c+d x)^2 (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 118, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{{c}^{2}}{2\,{d}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+2\,{\frac{ac}{d \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{c}^{2}b}{ \left ( ad-bc \right ) ^{2}{d}^{2} \left ( dx+c \right ) }}-{\frac{{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.17019, size = 304, normalized size = 3.04 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{a^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{b c^{3} - 3 \, a c^{2} d + 2 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.21929, size = 556, normalized size = 5.56 \begin{align*} -\frac{b^{2} c^{4} - 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{3} d - 3 \, a b c^{2} d^{2} + 2 \, a^{2} c d^{3}\right )} x - 2 \,{\left (a^{2} d^{4} x^{2} + 2 \, a^{2} c d^{3} x + a^{2} c^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (a^{2} d^{4} x^{2} + 2 \, a^{2} c d^{3} x + a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5} +{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.05694, size = 408, normalized size = 4.08 \begin{align*} \frac{a^{2} \log{\left (x + \frac{- \frac{a^{6} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{5} b c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{4} b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{3} c^{3} d}{\left (a d - b c\right )^{3}} + a^{3} d - \frac{a^{2} b^{4} c^{4}}{\left (a d - b c\right )^{3}} + a^{2} b c}{2 a^{2} b d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a^{2} \log{\left (x + \frac{\frac{a^{6} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{5} b c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{4} b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{3} c^{3} d}{\left (a d - b c\right )^{3}} + a^{3} d + \frac{a^{2} b^{4} c^{4}}{\left (a d - b c\right )^{3}} + a^{2} b c}{2 a^{2} b d} \right )}}{\left (a d - b c\right )^{3}} + \frac{3 a c^{2} d - b c^{3} + x \left (4 a c d^{2} - 2 b c^{2} d\right )}{2 a^{2} c^{2} d^{4} - 4 a b c^{3} d^{3} + 2 b^{2} c^{4} d^{2} + x^{2} \left (2 a^{2} d^{6} - 4 a b c d^{5} + 2 b^{2} c^{2} d^{4}\right ) + x \left (4 a^{2} c d^{5} - 8 a b c^{2} d^{4} + 4 b^{2} c^{3} d^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.22639, size = 254, normalized size = 2.54 \begin{align*} \frac{a^{2} b \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{a^{2} d \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac{b^{2} c^{4} - 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{3} d - 3 \, a b c^{2} d^{2} + 2 \, a^{2} c d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*b*log(abs(b*x + a))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - a^2*d*log(abs(d*x + c))/(b^3
*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4) - 1/2*(b^2*c^4 - 4*a*b*c^3*d + 3*a^2*c^2*d^2 + 2*(b^2*c^3*
d - 3*a*b*c^2*d^2 + 2*a^2*c*d^3)*x)/((b*c - a*d)^3*(d*x + c)^2*d^2)