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

Optimal. Leaf size=64 \[ -\frac {(b c-a d)^3 \log (a+b x)}{a b^3}+\frac {d^2 x (3 b c-a d)}{b^2}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 64, 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} \[ \frac {d^2 x (3 b c-a d)}{b^2}-\frac {(b c-a d)^3 \log (a+b x)}{a b^3}+\frac {c^3 \log (x)}{a}+\frac {d^3 x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x*(a + b*x)),x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 59, normalized size = 0.92 \[ \frac {a b d^2 x (-2 a d+6 b c+b d x)-2 (b c-a d)^3 \log (a+b x)+2 b^3 c^3 \log (x)}{2 a b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 97, normalized size = 1.52 \[ \frac {a b^{2} d^{3} x^{2} + 2 \, b^{3} c^{3} \log \relax (x) + 2 \, {\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.86, size = 91, normalized size = 1.42 \[ \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {b d^{3} x^{2} + 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 103, normalized size = 1.61 \[ \frac {d^{3} x^{2}}{2 b}+\frac {a^{2} d^{3} \ln \left (b x +a \right )}{b^{3}}-\frac {3 a c \,d^{2} \ln \left (b x +a \right )}{b^{2}}-\frac {a \,d^{3} x}{b^{2}}+\frac {c^{3} \ln \relax (x )}{a}-\frac {c^{3} \ln \left (b x +a \right )}{a}+\frac {3 c^{2} d \ln \left (b x +a \right )}{b}+\frac {3 c \,d^{2} x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.07, size = 91, normalized size = 1.42 \[ \frac {c^{3} \log \relax (x)}{a} + \frac {b d^{3} x^{2} + 2 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{2 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 65, normalized size = 1.02 \[ \frac {d^3\,x^2}{2\,b}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {c^3\,\ln \relax (x)}{a}+\frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x*(a + b*x)),x)

[Out]

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

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sympy [B]  time = 1.48, size = 112, normalized size = 1.75 \[ x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) + \frac {d^{3} x^{2}}{2 b} + \frac {c^{3} \log {\relax (x )}}{a} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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