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

Optimal. Leaf size=97 \[ \frac {3 c \log (x) (b c-a d)^2}{a^4}-\frac {3 c (b c-a d)^2 \log (a+b x)}{a^4}+\frac {c^2 (2 b c-3 a d)}{a^3 x}+\frac {(b c-a d)^3}{a^3 b (a+b x)}-\frac {c^3}{2 a^2 x^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 97, 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 = {88} \begin {gather*} \frac {c^2 (2 b c-3 a d)}{a^3 x}+\frac {(b c-a d)^3}{a^3 b (a+b x)}+\frac {3 c \log (x) (b c-a d)^2}{a^4}-\frac {3 c (b c-a d)^2 \log (a+b x)}{a^4}-\frac {c^3}{2 a^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^3*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]  time = 0.14, size = 93, normalized size = 0.96 \begin {gather*} -\frac {\frac {a^2 c^3}{x^2}+\frac {2 a c^2 (3 a d-2 b c)}{x}+\frac {2 a (a d-b c)^3}{b (a+b x)}-6 c \log (x) (b c-a d)^2+6 c (b c-a d)^2 \log (a+b x)}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.23, size = 249, normalized size = 2.57 \begin {gather*} -\frac {a^{3} b c^{3} - 2 \, {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 3 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d\right )} x + 6 \, {\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{4} b^{2} x^{3} + a^{5} b x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.02, size = 194, normalized size = 2.00 \begin {gather*} \frac {3 \, {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{a^{3} b^{3}} + \frac {5 \, b^{2} c^{3} - 6 \, a b c^{2} d - \frac {6 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 186, normalized size = 1.92 \begin {gather*} \frac {3 c \,d^{2}}{\left (b x +a \right ) a}-\frac {3 b \,c^{2} d}{\left (b x +a \right ) a^{2}}+\frac {3 c \,d^{2} \ln \relax (x )}{a^{2}}-\frac {3 c \,d^{2} \ln \left (b x +a \right )}{a^{2}}+\frac {b^{2} c^{3}}{\left (b x +a \right ) a^{3}}-\frac {6 b \,c^{2} d \ln \relax (x )}{a^{3}}+\frac {6 b \,c^{2} d \ln \left (b x +a \right )}{a^{3}}+\frac {3 b^{2} c^{3} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} c^{3} \ln \left (b x +a \right )}{a^{4}}-\frac {d^{3}}{\left (b x +a \right ) b}-\frac {3 c^{2} d}{a^{2} x}+\frac {2 b \,c^{3}}{a^{3} x}-\frac {c^{3}}{2 a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 163, normalized size = 1.68 \begin {gather*} -\frac {a^{2} b c^{3} - 2 \, {\left (3 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} - 3 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d\right )} x}{2 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x^{2}\right )}} - \frac {3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \relax (x)}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 156, normalized size = 1.61 \begin {gather*} -\frac {\frac {c^3}{2\,a}+\frac {3\,c^2\,x\,\left (2\,a\,d-b\,c\right )}{2\,a^2}+\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{a^3\,b}}{b\,x^3+a\,x^2}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (3\,a^2\,c\,d^2-6\,a\,b\,c^2\,d+3\,b^2\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^3*(a + b*x)^2),x)

[Out]

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

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sympy [B]  time = 1.61, size = 291, normalized size = 3.00 \begin {gather*} \frac {- a^{2} b c^{3} + x^{2} \left (- 2 a^{3} d^{3} + 6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}\right ) + x \left (- 6 a^{2} b c^{2} d + 3 a b^{2} c^{3}\right )}{2 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {3 c \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} - 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} - \frac {3 c \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} + 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a**2*b*c**3 + x**2*(-2*a**3*d**3 + 6*a**2*b*c*d**2 - 12*a*b**2*c**2*d + 6*b**3*c**3) + x*(-6*a**2*b*c**2*d +
3*a*b**2*c**3))/(2*a**4*b*x**2 + 2*a**3*b**2*x**3) + 3*c*(a*d - b*c)**2*log(x + (3*a**3*c*d**2 - 6*a**2*b*c**
2*d + 3*a*b**2*c**3 - 3*a*c*(a*d - b*c)**2)/(6*a**2*b*c*d**2 - 12*a*b**2*c**2*d + 6*b**3*c**3))/a**4 - 3*c*(a*
d - b*c)**2*log(x + (3*a**3*c*d**2 - 6*a**2*b*c**2*d + 3*a*b**2*c**3 + 3*a*c*(a*d - b*c)**2)/(6*a**2*b*c*d**2
- 12*a*b**2*c**2*d + 6*b**3*c**3))/a**4

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