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

Optimal. Leaf size=180 \[ \frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac{a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac{a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac{c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.186215, antiderivative size = 180, 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} \[ \frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac{a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac{a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac{c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.158601, size = 168, normalized size = 0.93 \[ \frac{2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)-2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)-\frac{a^2 (b c-a d)^2}{(a+b x)^2}+\frac{c^2 (b c-a d)^2}{(c+d x)^2}+\frac{2 a (a d+2 b c) (b c-a d)}{a+b x}+\frac{2 c (2 a d+b c) (b c-a d)}{c+d x}}{2 (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 272, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}{d}^{2}}{ \left ( ad-bc \right ) ^{5}}}+4\,{\frac{\ln \left ( dx+c \right ) abcd}{ \left ( ad-bc \right ) ^{5}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{ \left ( ad-bc \right ) ^{5}}}+2\,{\frac{acd}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{c}^{2}b}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{a}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{2}}{ \left ( ad-bc \right ) ^{5}}}-4\,{\frac{\ln \left ( bx+a \right ) abcd}{ \left ( ad-bc \right ) ^{5}}}-{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}}{ \left ( ad-bc \right ) ^{5}}}+{\frac{{a}^{2}d}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+2\,{\frac{abc}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15709, size = 872, normalized size = 4.84 \begin{align*} \frac{{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d + 2 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} + 5 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 2 \,{\left (5 \, a b^{2} c^{3} + 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x}{2 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} 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)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^
3 + 5*a^4*b*c*d^4 - a^5*d^5) - (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*
b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) + 1/2*(6*a^2*b*c^3 + 6*a^3*c^2*d + 2*(b^3*c^2*d +
4*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 3*(b^3*c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3)*x^2 + 2*(5*a*b^2*c^3 +
8*a^2*b*c^2*d + 5*a^3*c*d^2)*x)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2
*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d
- 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^
2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3
*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)

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Fricas [B]  time = 2.82244, size = 1959, normalized size = 10.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*a^2*b^2*c^4 - 6*a^4*c^2*d^2 + 2*(b^4*c^3*d + 3*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^3 + 3*(b^
4*c^4 + 4*a*b^3*c^3*d - 4*a^3*b*c*d^3 - a^4*d^4)*x^2 + 2*(5*a*b^3*c^4 + 3*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 - 5*
a^4*c*d^3)*x + 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4
+ 2*(b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 + 8*a*b^3*c^3*d + 18*a^2*b^2*c^
2*d^2 + 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^4 + 5*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 + a^4*c*d^3)*x)*log(b*
x + a) - 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4 + 2*(b
^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^3 + (b^4*c^4 + 8*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2
+ 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^4 + 5*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 + a^4*c*d^3)*x)*log(d*x + c)
)/(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5 + (
b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 +
2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (
b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*
b*c*d^6 - a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*
d^5 - a^7*c*d^6)*x)

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Sympy [B]  time = 3.65518, size = 1299, normalized size = 7.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*a**3*c**2*d + 6*a**2*b*c**3 + x**3*(2*a**2*b*d**3 + 8*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**2*(3*a**3*d**3 +
15*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 3*b**3*c**3) + x*(10*a**3*c*d**2 + 16*a**2*b*c**2*d + 10*a*b**2*c**3))/(
2*a**6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**2 - 8*a**3*b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*
(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**
3*(4*a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*d**2 +
4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 18*a**4*b**2*c**2*d**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2
+ 2*b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*a**4*b**2*c**3*d**3 + 8*a**3*b**3*c**4*d**2 - 12*
a**2*b**4*c**5*d + 4*a*b**5*c**6)) + (a**2*d**2 + 4*a*b*c*d + b**2*c**2)*log(x + (-a**6*d**6*(a**2*d**2 + 4*a*
b*c*d + b**2*c**2)/(a*d - b*c)**5 + 6*a**5*b*c*d**5*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 15*a*
*4*b**2*c**2*d**4*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 20*a**3*b**3*c**3*d**3*(a**2*d**2 + 4*a
*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a**3*d**3 - 15*a**2*b**4*c**4*d**2*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a
*d - b*c)**5 + 5*a**2*b*c*d**2 + 6*a*b**5*c**5*d*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a*b**2
*c**2*d - b**6*c**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + b**3*c**3)/(2*a**2*b*d**3 + 8*a*b**2*
c*d**2 + 2*b**3*c**2*d))/(a*d - b*c)**5 - (a**2*d**2 + 4*a*b*c*d + b**2*c**2)*log(x + (a**6*d**6*(a**2*d**2 +
4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 6*a**5*b*c*d**5*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 1
5*a**4*b**2*c**2*d**4*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 20*a**3*b**3*c**3*d**3*(a**2*d**2 +
4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a**3*d**3 + 15*a**2*b**4*c**4*d**2*(a**2*d**2 + 4*a*b*c*d + b**2*c**2
)/(a*d - b*c)**5 + 5*a**2*b*c*d**2 - 6*a*b**5*c**5*d*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a*
b**2*c**2*d + b**6*c**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + b**3*c**3)/(2*a**2*b*d**3 + 8*a*b
**2*c*d**2 + 2*b**3*c**2*d))/(a*d - b*c)**5

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError