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

Optimal. Leaf size=61 \[ \frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{a x}+\frac {d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 61, 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} \[ \frac {(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{a x}+\frac {d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^2*(a + b*x)),x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 66, normalized size = 1.08 \[ \frac {b^2 c^2 x \log (x) (3 a d-b c)+a b \left (a d^3 x^2-b c^3\right )+x (b c-a d)^3 \log (a+b x)}{a^2 b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 98, normalized size = 1.61 \[ \frac {a^{2} b d^{3} x^{2} - a b^{2} c^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x \log \left (b x + a\right ) - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x \log \relax (x)}{a^{2} b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.08, size = 91, normalized size = 1.49 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 102, normalized size = 1.67 \[ -\frac {a \,d^{3} \ln \left (b x +a \right )}{b^{2}}+\frac {3 c^{2} d \ln \relax (x )}{a}-\frac {3 c^{2} d \ln \left (b x +a \right )}{a}-\frac {b \,c^{3} \ln \relax (x )}{a^{2}}+\frac {b \,c^{3} \ln \left (b x +a \right )}{a^{2}}+\frac {3 c \,d^{2} \ln \left (b x +a \right )}{b}+\frac {d^{3} x}{b}-\frac {c^{3}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 89, normalized size = 1.46 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \relax (x)}{a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.42, size = 88, normalized size = 1.44 \[ \frac {d^3\,x}{b}-\frac {c^3}{a\,x}+\frac {c^2\,\ln \relax (x)\,\left (3\,a\,d-b\,c\right )}{a^2}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^2\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^2*(a + b*x)),x)

[Out]

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

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sympy [B]  time = 1.91, size = 196, normalized size = 3.21 \[ \frac {d^{3} x}{b} - \frac {c^{3}}{a x} + \frac {c^{2} \left (3 a d - b c\right ) \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} - a b c^{2} \left (3 a d - b c\right )}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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