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

Optimal. Leaf size=87 \[ \frac {(b c-a d)^2 (a d+2 b c) \log (a+b x)}{a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{a^2 b^2 (a+b x)}-\frac {c^3}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 87, 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} \[ -\frac {(b c-a d)^3}{a^2 b^2 (a+b x)}+\frac {(b c-a d)^2 (a d+2 b c) \log (a+b x)}{a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {c^3}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^2*(a + b*x)^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 79, normalized size = 0.91 \[ \frac {\frac {a (a d-b c)^3}{b^2 (a+b x)}+\frac {(b c-a d)^2 (a d+2 b c) \log (a+b x)}{b^2}+c^2 \log (x) (3 a d-2 b c)-\frac {a c^3}{x}}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.90, size = 198, normalized size = 2.28 \[ -\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{2} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x\right )} \log \left (b x + a\right ) + {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{2} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x\right )} \log \relax (x)}{a^{3} b^{3} x^{2} + a^{4} b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.93, size = 165, normalized size = 1.90 \[ -\frac {d^{3} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2}} + \frac {b c^{3}}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}} - \frac {{\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 141, normalized size = 1.62 \[ \frac {a \,d^{3}}{\left (b x +a \right ) b^{2}}+\frac {3 c^{2} d}{\left (b x +a \right ) a}-\frac {b \,c^{3}}{\left (b x +a \right ) a^{2}}+\frac {3 c^{2} d \ln \relax (x )}{a^{2}}-\frac {3 c^{2} d \ln \left (b x +a \right )}{a^{2}}-\frac {2 b \,c^{3} \ln \relax (x )}{a^{3}}+\frac {2 b \,c^{3} \ln \left (b x +a \right )}{a^{3}}-\frac {3 c \,d^{2}}{\left (b x +a \right ) b}+\frac {d^{3} \ln \left (b x +a \right )}{b^{2}}-\frac {c^{3}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.09, size = 132, normalized size = 1.52 \[ -\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{a^{2} b^{3} x^{2} + a^{3} b^{2} x} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \relax (x)}{a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.49, size = 118, normalized size = 1.36 \[ \ln \left (a+b\,x\right )\,\left (\frac {d^3}{b^2}+\frac {2\,b\,c^3}{a^3}-\frac {3\,c^2\,d}{a^2}\right )-\frac {\frac {c^3}{a}-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{a^2\,b^2}}{b\,x^2+a\,x}+\frac {c^2\,\ln \relax (x)\,\left (3\,a\,d-2\,b\,c\right )}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^2*(a + b*x)^2),x)

[Out]

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

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sympy [B]  time = 1.44, size = 250, normalized size = 2.87 \[ \frac {- a b^{2} c^{3} + x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{a^{3} b^{2} x + a^{2} b^{3} x^{2}} + \frac {c^{2} \left (3 a d - 2 b c\right ) \log {\left (x + \frac {- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + a b c^{2} \left (3 a d - 2 b c\right )}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (x + \frac {- 3 a^{2} b c^{2} d + 2 a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{2} \left (a d + 2 b c\right )}{b}}{a^{3} d^{3} - 6 a b^{2} c^{2} d + 4 b^{3} c^{3}} \right )}}{a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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