3.318 \(\int \frac{1}{x^2 (a+b x)^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=242 \[ \frac{3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac{3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}-\frac{b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac{b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac{3 \log (x) (a d+b c)}{a^4 c^4}-\frac{1}{a^3 c^3 x}+\frac{d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac{d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.325448, antiderivative size = 242, 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} \[ \frac{3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac{3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}-\frac{b^4 (2 b c-5 a d)}{a^3 (a+b x) (b c-a d)^4}-\frac{b^4}{2 a^2 (a+b x)^2 (b c-a d)^3}-\frac{3 \log (x) (a d+b c)}{a^4 c^4}-\frac{1}{a^3 c^3 x}+\frac{d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac{d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^3*(c + d*x)^3),x]

[Out]

-(1/(a^3*c^3*x)) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x)^2) - (b^4*(2*b*c - 5*a*d))/(a^3*(b*c - a*d)^4*(a + b*x))
+ d^4/(2*c^2*(b*c - a*d)^3*(c + d*x)^2) + (d^4*(5*b*c - 2*a*d))/(c^3*(b*c - a*d)^4*(c + d*x)) - (3*(b*c + a*d
)*Log[x])/(a^4*c^4) + (3*b^4*(b^2*c^2 - 4*a*b*c*d + 5*a^2*d^2)*Log[a + b*x])/(a^4*(b*c - a*d)^5) - (3*d^4*(5*b
^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + d*x])/(c^4*(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{1}{x^2 (a+b x)^3 (c+d x)^3} \, dx &=\int \left (\frac{1}{a^3 c^3 x^2}-\frac{3 (b c+a d)}{a^4 c^4 x}-\frac{b^5}{a^2 (-b c+a d)^3 (a+b x)^3}-\frac{b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)^2}-\frac{3 b^5 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right )}{a^4 (-b c+a d)^5 (a+b x)}-\frac{d^5}{c^2 (b c-a d)^3 (c+d x)^3}-\frac{d^5 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)^2}-\frac{3 d^5 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right )}{c^4 (b c-a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac{1}{a^3 c^3 x}-\frac{b^4}{2 a^2 (b c-a d)^3 (a+b x)^2}-\frac{b^4 (2 b c-5 a d)}{a^3 (b c-a d)^4 (a+b x)}+\frac{d^4}{2 c^2 (b c-a d)^3 (c+d x)^2}+\frac{d^4 (5 b c-2 a d)}{c^3 (b c-a d)^4 (c+d x)}-\frac{3 (b c+a d) \log (x)}{a^4 c^4}+\frac{3 b^4 \left (b^2 c^2-4 a b c d+5 a^2 d^2\right ) \log (a+b x)}{a^4 (b c-a d)^5}-\frac{3 d^4 \left (5 b^2 c^2-4 a b c d+a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}\\ \end{align*}

Mathematica [A]  time = 0.366203, size = 241, normalized size = 1. \[ -\frac{3 b^4 \left (5 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)}{a^4 (a d-b c)^5}-\frac{3 d^4 \left (a^2 d^2-4 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^5}+\frac{b^4 (5 a d-2 b c)}{a^3 (a+b x) (b c-a d)^4}+\frac{b^4}{2 a^2 (a+b x)^2 (a d-b c)^3}-\frac{3 \log (x) (a d+b c)}{a^4 c^4}-\frac{1}{a^3 c^3 x}+\frac{d^4 (5 b c-2 a d)}{c^3 (c+d x) (b c-a d)^4}+\frac{d^4}{2 c^2 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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Maple [A]  time = 0.017, size = 349, normalized size = 1.4 \begin{align*} -{\frac{{d}^{4}}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}-2\,{\frac{{d}^{5}a}{{c}^{3} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+5\,{\frac{{d}^{4}b}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+3\,{\frac{{d}^{6}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{5}}}-12\,{\frac{{d}^{5}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{5}}}+15\,{\frac{{d}^{4}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{5}}}-{\frac{1}{{a}^{3}{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{3}{c}^{4}}}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}{c}^{3}}}+{\frac{{b}^{4}}{2\, \left ( ad-bc \right ) ^{3}{a}^{2} \left ( bx+a \right ) ^{2}}}+5\,{\frac{{b}^{4}d}{ \left ( ad-bc \right ) ^{4}{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{{b}^{5}c}{ \left ( ad-bc \right ) ^{4}{a}^{3} \left ( bx+a \right ) }}-15\,{\frac{{b}^{4}\ln \left ( bx+a \right ){d}^{2}}{ \left ( ad-bc \right ) ^{5}{a}^{2}}}+12\,{\frac{{b}^{5}\ln \left ( bx+a \right ) cd}{ \left ( ad-bc \right ) ^{5}{a}^{3}}}-3\,{\frac{{b}^{6}\ln \left ( bx+a \right ){c}^{2}}{ \left ( ad-bc \right ) ^{5}{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^3/(d*x+c)^3,x)

[Out]

-1/2*d^4/c^2/(a*d-b*c)^3/(d*x+c)^2-2*d^5/c^3/(a*d-b*c)^4/(d*x+c)*a+5*d^4/c^2/(a*d-b*c)^4/(d*x+c)*b+3*d^6/c^4/(
a*d-b*c)^5*ln(d*x+c)*a^2-12*d^5/c^3/(a*d-b*c)^5*ln(d*x+c)*a*b+15*d^4/c^2/(a*d-b*c)^5*ln(d*x+c)*b^2-1/a^3/c^3/x
-3/a^3/c^4*ln(x)*d-3/a^4/c^3*ln(x)*b+1/2*b^4/(a*d-b*c)^3/a^2/(b*x+a)^2+5*b^4/(a*d-b*c)^4/a^2/(b*x+a)*d-2*b^5/(
a*d-b*c)^4/a^3/(b*x+a)*c-15*b^4/(a*d-b*c)^5/a^2*ln(b*x+a)*d^2+12*b^5/(a*d-b*c)^5/a^3*ln(b*x+a)*c*d-3*b^6/(a*d-
b*c)^5/a^4*ln(b*x+a)*c^2

________________________________________________________________________________________

Maxima [B]  time = 1.29577, size = 1264, normalized size = 5.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

3*(b^6*c^2 - 4*a*b^5*c*d + 5*a^2*b^4*d^2)*log(b*x + a)/(a^4*b^5*c^5 - 5*a^5*b^4*c^4*d + 10*a^6*b^3*c^3*d^2 - 1
0*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^9*d^5) - 3*(5*b^2*c^2*d^4 - 4*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/(b^5*c^9
- 5*a*b^4*c^8*d + 10*a^2*b^3*c^7*d^2 - 10*a^3*b^2*c^6*d^3 + 5*a^4*b*c^5*d^4 - a^5*c^4*d^5) - 1/2*(2*a^2*b^4*c
^6 - 8*a^3*b^3*c^5*d + 12*a^4*b^2*c^4*d^2 - 8*a^5*b*c^3*d^3 + 2*a^6*c^2*d^4 + 6*(b^6*c^4*d^2 - 3*a*b^5*c^3*d^3
+ 2*a^2*b^4*c^2*d^4 - 3*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 3*(4*b^6*c^5*d - 9*a*b^5*c^4*d^2 - a^2*b^4*c^3*d^3
- a^3*b^3*c^2*d^4 - 9*a^4*b^2*c*d^5 + 4*a^5*b*d^6)*x^3 + 2*(3*b^6*c^6 - 20*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d
^3 - 20*a^4*b^2*c^2*d^4 + 3*a^6*d^6)*x^2 + (9*a*b^5*c^6 - 23*a^2*b^4*c^5*d + 8*a^3*b^3*c^4*d^2 + 8*a^4*b^2*c^3
*d^3 - 23*a^5*b*c^2*d^4 + 9*a^6*c*d^5)*x)/((a^3*b^6*c^7*d^2 - 4*a^4*b^5*c^6*d^3 + 6*a^5*b^4*c^5*d^4 - 4*a^6*b^
3*c^4*d^5 + a^7*b^2*c^3*d^6)*x^5 + 2*(a^3*b^6*c^8*d - 3*a^4*b^5*c^7*d^2 + 2*a^5*b^4*c^6*d^3 + 2*a^6*b^3*c^5*d^
4 - 3*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6)*x^4 + (a^3*b^6*c^9 - 9*a^5*b^4*c^7*d^2 + 16*a^6*b^3*c^6*d^3 - 9*a^7*b^2
*c^5*d^4 + a^9*c^3*d^6)*x^3 + 2*(a^4*b^5*c^9 - 3*a^5*b^4*c^8*d + 2*a^6*b^3*c^7*d^2 + 2*a^7*b^2*c^6*d^3 - 3*a^8
*b*c^5*d^4 + a^9*c^4*d^5)*x^2 + (a^5*b^4*c^9 - 4*a^6*b^3*c^8*d + 6*a^7*b^2*c^7*d^2 - 4*a^8*b*c^6*d^3 + a^9*c^5
*d^4)*x) - 3*(b*c + a*d)*log(x)/(a^4*c^4)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError