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

Optimal. Leaf size=134 \[ \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}-\frac {d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac {d}{2 c (c+d x)^2 (b c-a d)}+\frac {\log (x)}{a c^3} \]

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Rubi [A]  time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, 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 = {72} \begin {gather*} \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}-\frac {d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac {d}{2 c (c+d x)^2 (b c-a d)}+\frac {\log (x)}{a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)*(c + d*x)^3),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.30, size = 116, normalized size = 0.87 \begin {gather*} \frac {\frac {d \left (\frac {c (b c-a d) (b c (5 c+4 d x)-a d (3 c+2 d x))}{(c+d x)^2}-2 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)\right )}{c^3}+\frac {2 b^3 \log (a+b x)}{a}}{2 (a d-b c)^3}+\frac {\log (x)}{a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+b x) (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 20.17, size = 506, normalized size = 3.78 \begin {gather*} -\frac {5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \, {\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x + 2 \, {\left (b^{3} c^{3} d^{2} x^{2} + 2 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \log \left (d x + c\right ) - 2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )} \log \relax (x)}{2 \, {\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} + {\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{2} + 2 \, {\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.13, size = 234, normalized size = 1.75 \begin {gather*} -\frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} + \frac {\log \left ({\left | x \right |}\right )}{a c^{3}} - \frac {5 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + 2 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 184, normalized size = 1.37 \begin {gather*} -\frac {a^{2} d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{3}}+\frac {3 a b \,d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{2}}+\frac {b^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a}-\frac {3 b^{2} d \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c}+\frac {a \,d^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c^{2}}-\frac {2 b d}{\left (a d -b c \right )^{2} \left (d x +c \right ) c}+\frac {d}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} c}+\frac {\ln \relax (x )}{a \,c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.23, size = 266, normalized size = 1.99 \begin {gather*} -\frac {b^{3} \log \left (b x + a\right )}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {5 \, b c^{2} d - 3 \, a c d^{2} + 2 \, {\left (2 \, b c d^{2} - a d^{3}\right )} x}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} + \frac {\log \relax (x)}{a c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.87, size = 235, normalized size = 1.75 \begin {gather*} \frac {\frac {3\,a\,d^2-5\,b\,c\,d}{2\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,x\,\left (a\,d-2\,b\,c\right )}{c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x+d^2\,x^2}+\frac {b^3\,\ln \left (a+b\,x\right )}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^3-3\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}+\frac {\ln \relax (x)}{a\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)*(c + d*x)^3),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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