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

Optimal. Leaf size=124 \[ -\frac{a^2}{(a+b x) (b c-a d)^3}-\frac{c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac{2 a c}{(c+d x) (b c-a d)^3}-\frac{a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac{a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.103338, antiderivative size = 124, 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} \[ -\frac{a^2}{(a+b x) (b c-a d)^3}-\frac{c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac{2 a c}{(c+d x) (b c-a d)^3}-\frac{a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac{a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.13101, size = 123, normalized size = 0.99 \[ \frac{a^2}{(a+b x) (a d-b c)^3}-\frac{c^2}{2 d (c+d x)^2 (b c-a d)^2}-\frac{2 a c}{(c+d x) (b c-a d)^3}-\frac{a (a d+2 b c) \log (a+b x)}{(b c-a d)^4}+\frac{a (a d+2 b c) \log (c+d x)}{(b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 154, normalized size = 1.2 \begin{align*} -{\frac{{c}^{2}}{2\, \left ( ad-bc \right ) ^{2}d \left ( dx+c \right ) ^{2}}}+{\frac{{a}^{2}\ln \left ( dx+c \right ) d}{ \left ( ad-bc \right ) ^{4}}}+2\,{\frac{a\ln \left ( dx+c \right ) bc}{ \left ( ad-bc \right ) ^{4}}}+2\,{\frac{ac}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{a}^{2}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}}}-2\,{\frac{a\ln \left ( bx+a \right ) bc}{ \left ( ad-bc \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.21345, size = 567, normalized size = 4.57 \begin{align*} -\frac{{\left (2 \, a b c + a^{2} d\right )} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{{\left (2 \, a b c + a^{2} d\right )} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{a b c^{3} + 5 \, a^{2} c^{2} d + 2 \,{\left (2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2} +{\left (b^{2} c^{3} + 3 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} x}{2 \,{\left (a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} +{\left (b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d^{2} - 5 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} +{\left (b^{4} c^{5} d - a b^{3} c^{4} d^{2} - 3 \, a^{2} b^{2} c^{3} d^{3} + 5 \, a^{3} b c^{2} d^{4} - 2 \, a^{4} 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)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.8672, size = 1234, normalized size = 9.95 \begin{align*} -\frac{a b^{2} c^{4} + 4 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2} + 2 \,{\left (2 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{2} +{\left (b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 8 \, a^{3} c d^{3}\right )} x + 2 \,{\left (2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} +{\left (2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} +{\left (4 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2} +{\left (2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} +{\left (2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} +{\left (4 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2} +{\left (2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a b^{4} c^{6} d - 4 \, a^{2} b^{3} c^{5} d^{2} + 6 \, a^{3} b^{2} c^{4} d^{3} - 4 \, a^{4} b c^{3} d^{4} + a^{5} c^{2} d^{5} +{\left (b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}\right )} x^{3} +{\left (2 \, b^{5} c^{5} d^{2} - 7 \, a b^{4} c^{4} d^{3} + 8 \, a^{2} b^{3} c^{3} d^{4} - 2 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6} + a^{5} d^{7}\right )} x^{2} +{\left (b^{5} c^{6} d - 2 \, a b^{4} c^{5} d^{2} - 2 \, a^{2} b^{3} c^{4} d^{3} + 8 \, a^{3} b^{2} c^{3} d^{4} - 7 \, a^{4} b c^{2} d^{5} + 2 \, a^{5} 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)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.77899, size = 787, normalized size = 6.35 \begin{align*} \frac{a \left (a d + 2 b c\right ) \log{\left (x + \frac{- \frac{a^{6} d^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac{5 a^{5} b c d^{4} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{4} b^{2} c^{2} d^{3} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{3} c^{3} d^{2} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + a^{3} d^{2} - \frac{5 a^{2} b^{4} c^{4} d \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 3 a^{2} b c d + \frac{a b^{5} c^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 2 a b^{2} c^{2}}{2 a^{2} b d^{2} + 4 a b^{2} c d} \right )}}{\left (a d - b c\right )^{4}} - \frac{a \left (a d + 2 b c\right ) \log{\left (x + \frac{\frac{a^{6} d^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac{5 a^{5} b c d^{4} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{4} b^{2} c^{2} d^{3} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{3} c^{3} d^{2} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + a^{3} d^{2} + \frac{5 a^{2} b^{4} c^{4} d \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 3 a^{2} b c d - \frac{a b^{5} c^{5} \left (a d + 2 b c\right )}{\left (a d - b c\right )^{4}} + 2 a b^{2} c^{2}}{2 a^{2} b d^{2} + 4 a b^{2} c d} \right )}}{\left (a d - b c\right )^{4}} + \frac{5 a^{2} c^{2} d + a b c^{3} + x^{2} \left (2 a^{2} d^{3} + 4 a b c d^{2}\right ) + x \left (8 a^{2} c d^{2} + 3 a b c^{2} d + b^{2} c^{3}\right )}{2 a^{4} c^{2} d^{4} - 6 a^{3} b c^{3} d^{3} + 6 a^{2} b^{2} c^{4} d^{2} - 2 a b^{3} c^{5} d + x^{3} \left (2 a^{3} b d^{6} - 6 a^{2} b^{2} c d^{5} + 6 a b^{3} c^{2} d^{4} - 2 b^{4} c^{3} d^{3}\right ) + x^{2} \left (2 a^{4} d^{6} - 2 a^{3} b c d^{5} - 6 a^{2} b^{2} c^{2} d^{4} + 10 a b^{3} c^{3} d^{3} - 4 b^{4} c^{4} d^{2}\right ) + x \left (4 a^{4} c d^{5} - 10 a^{3} b c^{2} d^{4} + 6 a^{2} b^{2} c^{3} d^{3} + 2 a b^{3} c^{4} d^{2} - 2 b^{4} c^{5} d\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21244, size = 335, normalized size = 2.7 \begin{align*} -\frac{a^{2} b^{3}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (2 \, a b^{2} c + a^{2} b d\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac{b^{2} c^{2} d + 4 \, a b c d^{2} + \frac{2 \,{\left (b^{4} c^{3} + a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )}}{{\left (b x + a\right )} b}}{2 \,{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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