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

Optimal. Leaf size=228 \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

[Out]

-1/(2*a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) + b^5/(a^3*(b*c - a*d)^3*(a + b*x)) + d^4/(2*c^3*(b*c - a*d)^
2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d))/(c^4*(b*c - a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Lo
g[x])/(a^4*c^5) - (3*b^5*(b*c - 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 - 6*a*b*c*d + 2*a
^2*d^2)*Log[c + d*x])/(c^5*(b*c - a*d)^4)

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Rubi [A]  time = 0.290479, antiderivative size = 228, 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} \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}+\frac{b^5}{a^3 (a+b x) (b c-a d)^3}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^2*c^3*x^2) + (2*b*c + 3*a*d)/(a^3*c^4*x) + b^5/(a^3*(b*c - a*d)^3*(a + b*x)) + d^4/(2*c^3*(b*c - a*d)^
2*(c + d*x)^2) + (d^4*(5*b*c - 3*a*d))/(c^4*(b*c - a*d)^3*(c + d*x)) + (3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*Lo
g[x])/(a^4*c^5) - (3*b^5*(b*c - 2*a*d)*Log[a + b*x])/(a^4*(b*c - a*d)^4) - (3*d^4*(5*b^2*c^2 - 6*a*b*c*d + 2*a
^2*d^2)*Log[c + d*x])/(c^5*(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{1}{x^3 (a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{1}{a^2 c^3 x^3}+\frac{-2 b c-3 a d}{a^3 c^4 x^2}+\frac{3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right )}{a^4 c^5 x}+\frac{b^6}{a^3 (-b c+a d)^3 (a+b x)^2}+\frac{3 b^6 (-b c+2 a d)}{a^4 (-b c+a d)^4 (a+b x)}-\frac{d^5}{c^3 (b c-a d)^2 (c+d x)^3}-\frac{d^5 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)^2}-\frac{3 d^5 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right )}{c^5 (b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a^2 c^3 x^2}+\frac{2 b c+3 a d}{a^3 c^4 x}+\frac{b^5}{a^3 (b c-a d)^3 (a+b x)}+\frac{d^4}{2 c^3 (b c-a d)^2 (c+d x)^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}+\frac{3 \left (b^2 c^2+2 a b c d+2 a^2 d^2\right ) \log (x)}{a^4 c^5}-\frac{3 b^5 (b c-2 a d) \log (a+b x)}{a^4 (b c-a d)^4}-\frac{3 d^4 \left (5 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.283604, size = 230, normalized size = 1.01 \[ \frac{3 \log (x) \left (2 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^4 c^5}-\frac{3 d^4 \left (2 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \log (c+d x)}{c^5 (b c-a d)^4}-\frac{b^5}{a^3 (a+b x) (a d-b c)^3}+\frac{3 b^5 (2 a d-b c) \log (a+b x)}{a^4 (b c-a d)^4}+\frac{3 a d+2 b c}{a^3 c^4 x}-\frac{1}{2 a^2 c^3 x^2}+\frac{d^4 (5 b c-3 a d)}{c^4 (c+d x) (b c-a d)^3}+\frac{d^4}{2 c^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.018, size = 307, normalized size = 1.4 \begin{align*}{\frac{{d}^{4}}{2\,{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}a}{{c}^{4} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-5\,{\frac{{d}^{4}b}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-6\,{\frac{{d}^{6}\ln \left ( dx+c \right ){a}^{2}}{{c}^{5} \left ( ad-bc \right ) ^{4}}}+18\,{\frac{{d}^{5}\ln \left ( dx+c \right ) ab}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-15\,{\frac{{d}^{4}\ln \left ( dx+c \right ){b}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4}}}-{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}+3\,{\frac{d}{{a}^{2}{c}^{4}x}}+2\,{\frac{b}{{a}^{3}{c}^{3}x}}+6\,{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{5}}}+6\,{\frac{b\ln \left ( x \right ) d}{{a}^{3}{c}^{4}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}{c}^{3}}}-{\frac{{b}^{5}}{ \left ( ad-bc \right ) ^{3}{a}^{3} \left ( bx+a \right ) }}+6\,{\frac{{b}^{5}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{3}}}-3\,{\frac{{b}^{6}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*d^4/c^3/(a*d-b*c)^2/(d*x+c)^2+3*d^5/c^4/(a*d-b*c)^3/(d*x+c)*a-5*d^4/c^3/(a*d-b*c)^3/(d*x+c)*b-6*d^6/c^5/(a
*d-b*c)^4*ln(d*x+c)*a^2+18*d^5/c^4/(a*d-b*c)^4*ln(d*x+c)*a*b-15*d^4/c^3/(a*d-b*c)^4*ln(d*x+c)*b^2-1/2/a^2/c^3/
x^2+3/a^2/c^4/x*d+2/a^3/c^3/x*b+6/a^2/c^5*ln(x)*d^2+6/a^3/c^4*ln(x)*b*d+3/a^4/c^3*ln(x)*b^2-b^5/(a*d-b*c)^3/a^
3/(b*x+a)+6*b^5/(a*d-b*c)^4/a^3*ln(b*x+a)*d-3*b^6/(a*d-b*c)^4/a^4*ln(b*x+a)*c

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Maxima [B]  time = 1.39517, size = 1017, normalized size = 4.46 \begin{align*} -\frac{3 \,{\left (b^{6} c - 2 \, a b^{5} d\right )} \log \left (b x + a\right )}{a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4}} - \frac{3 \,{\left (5 \, b^{2} c^{2} d^{4} - 6 \, a b c d^{5} + 2 \, a^{2} d^{6}\right )} \log \left (d x + c\right )}{b^{4} c^{9} - 4 \, a b^{3} c^{8} d + 6 \, a^{2} b^{2} c^{7} d^{2} - 4 \, a^{3} b c^{6} d^{3} + a^{4} c^{5} d^{4}} - \frac{a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - 6 \,{\left (b^{5} c^{4} d^{2} - a b^{4} c^{3} d^{3} - a^{2} b^{3} c^{2} d^{4} + 4 \, a^{3} b^{2} c d^{5} - 2 \, a^{4} b d^{6}\right )} x^{4} - 3 \,{\left (4 \, b^{5} c^{5} d - 3 \, a b^{4} c^{4} d^{2} - 5 \, a^{2} b^{3} c^{3} d^{3} + 10 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - 4 \, a^{5} d^{6}\right )} x^{3} -{\left (6 \, b^{5} c^{6} - 13 \, a^{2} b^{3} c^{4} d^{2} - a^{3} b^{2} c^{3} d^{3} + 32 \, a^{4} b c^{2} d^{4} - 18 \, a^{5} c d^{5}\right )} x^{2} -{\left (3 \, a b^{4} c^{6} - 5 \, a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 9 \, a^{4} b c^{3} d^{3} - 4 \, a^{5} c^{2} d^{4}\right )} x}{2 \,{\left ({\left (a^{3} b^{4} c^{7} d^{2} - 3 \, a^{4} b^{3} c^{6} d^{3} + 3 \, a^{5} b^{2} c^{5} d^{4} - a^{6} b c^{4} d^{5}\right )} x^{5} +{\left (2 \, a^{3} b^{4} c^{8} d - 5 \, a^{4} b^{3} c^{7} d^{2} + 3 \, a^{5} b^{2} c^{6} d^{3} + a^{6} b c^{5} d^{4} - a^{7} c^{4} d^{5}\right )} x^{4} +{\left (a^{3} b^{4} c^{9} - a^{4} b^{3} c^{8} d - 3 \, a^{5} b^{2} c^{7} d^{2} + 5 \, a^{6} b c^{6} d^{3} - 2 \, a^{7} c^{5} d^{4}\right )} x^{3} +{\left (a^{4} b^{3} c^{9} - 3 \, a^{5} b^{2} c^{8} d + 3 \, a^{6} b c^{7} d^{2} - a^{7} c^{6} d^{3}\right )} x^{2}\right )}} + \frac{3 \,{\left (b^{2} c^{2} + 2 \, a b c d + 2 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(b^6*c - 2*a*b^5*d)*log(b*x + a)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d
^4) - 3*(5*b^2*c^2*d^4 - 6*a*b*c*d^5 + 2*a^2*d^6)*log(d*x + c)/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 -
4*a^3*b*c^6*d^3 + a^4*c^5*d^4) - 1/2*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3 - 6*(b^5*c
^4*d^2 - a*b^4*c^3*d^3 - a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^5 - 2*a^4*b*d^6)*x^4 - 3*(4*b^5*c^5*d - 3*a*b^4*c^4*d
^2 - 5*a^2*b^3*c^3*d^3 + 10*a^3*b^2*c^2*d^4 + 2*a^4*b*c*d^5 - 4*a^5*d^6)*x^3 - (6*b^5*c^6 - 13*a^2*b^3*c^4*d^2
- a^3*b^2*c^3*d^3 + 32*a^4*b*c^2*d^4 - 18*a^5*c*d^5)*x^2 - (3*a*b^4*c^6 - 5*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2
+ 9*a^4*b*c^3*d^3 - 4*a^5*c^2*d^4)*x)/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d
^5)*x^5 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^4 + (a^3*b
^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^3 + (a^4*b^3*c^9 - 3*a^5*b^2*c
^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2) + 3*(b^2*c^2 + 2*a*b*c*d + 2*a^2*d^2)*log(x)/(a^4*c^5)

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.23514, size = 1168, normalized size = 5.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^11/((a^3*b^9*c^3 - 3*a^4*b^8*c^2*d + 3*a^5*b^7*c*d^2 - a^6*b^6*d^3)*(b*x + a)) + 3/2*(b^6*c - 2*a*b^5*d)*log
(abs(-b*c/(b*x + a) + a*b*c/(b*x + a)^2 + 2*a*d/(b*x + a) - a^2*d/(b*x + a)^2 - d))/(a^4*b^4*c^4 - 4*a^5*b^3*c
^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4) - 3/2*(b^8*c^6 - 2*a*b^7*c^5*d + 10*a^4*b^4*c^2*d^4 - 12*a
^5*b^3*c*d^5 + 4*a^6*b^2*d^6)*log(abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*b*d + 2*a^2*b*d/(b*x + a) - b^2*abs(c
))/abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*b*d + 2*a^2*b*d/(b*x + a) + b^2*abs(c)))/((a^4*b^4*c^8 - 4*a^5*b^3*c
^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4)*b^2*abs(c)) + 1/2*(5*b^6*c^5*d^2 - 14*a*b^5*c^4*d^3
+ 6*a^2*b^4*c^3*d^4 + 16*a^3*b^3*c^2*d^5 - 30*a^4*b^2*c*d^6 + 12*a^5*b*d^7 + 2*(5*b^8*c^6*d - 22*a*b^7*c^5*d^2
+ 29*a^2*b^6*c^4*d^3 + 4*a^3*b^5*c^3*d^4 - 47*a^4*b^4*c^2*d^5 + 54*a^5*b^3*c*d^6 - 18*a^6*b^2*d^7)/((b*x + a)
*b) + (5*b^10*c^7 - 36*a*b^9*c^6*d + 87*a^2*b^8*c^5*d^2 - 70*a^3*b^7*c^4*d^3 - 45*a^4*b^6*c^3*d^4 + 144*a^5*b^
5*c^2*d^5 - 126*a^6*b^4*c*d^6 + 36*a^7*b^3*d^7)/((b*x + a)^2*b^2) - 6*(a*b^11*c^7 - 5*a^2*b^10*c^6*d + 9*a^3*b
^9*c^5*d^2 - 5*a^4*b^8*c^4*d^3 - 5*a^5*b^7*c^3*d^4 + 11*a^6*b^6*c^2*d^5 - 8*a^7*b^5*c*d^6 + 2*a^8*b^4*d^7)/((b
*x + a)^3*b^3))/((b*c - a*d)^4*a^4*(b*c/(b*x + a) - a*b*c/(b*x + a)^2 - 2*a*d/(b*x + a) + a^2*d/(b*x + a)^2 +
d)^2*c^4)