3.4.5 \(\int \frac {(c+d x)^3}{x (a+b x)^3} \, dx\) [305]

Optimal. Leaf size=93 \[ \frac {(b c-a d)^3}{2 a b^3 (a+b x)^2}+\frac {(b c-a d)^2 (b c+2 a d)}{a^2 b^3 (a+b x)}+\frac {c^3 \log (x)}{a^3}-\left (\frac {c^3}{a^3}-\frac {d^3}{b^3}\right ) \log (a+b x) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 93, 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 = {90} \begin {gather*} -\left (\frac {c^3}{a^3}-\frac {d^3}{b^3}\right ) \log (a+b x)+\frac {c^3 \log (x)}{a^3}+\frac {(b c-a d)^2 (2 a d+b c)}{a^2 b^3 (a+b x)}+\frac {(b c-a d)^3}{2 a b^3 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x*(a + b*x)^3),x]

[Out]

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

Rule 90

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

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Mathematica [A]
time = 0.06, size = 88, normalized size = 0.95 \begin {gather*} \frac {2 c^3 \log (x)+\frac {\frac {a (b c-a d)^2 \left (3 a^2 d+2 b^2 c x+a b (3 c+4 d x)\right )}{(a+b x)^2}+2 \left (-b^3 c^3+a^3 d^3\right ) \log (a+b x)}{b^3}}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 133, normalized size = 1.43

method	result	size
norman	\(\frac {\frac {\left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+b^{3} c^{3}\right ) x}{a^{2} b^{2}}+\frac {3 a^{3} d^{3}-3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +3 b^{3} c^{3}}{2 b^{3} a}}{\left (b x +a \right )^{2}}+\frac {c^{3} \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b^{3}}\)	\(128\)
risch	\(\frac {\frac {\left (2 a^{3} d^{3}-3 a^{2} b c \,d^{2}+b^{3} c^{3}\right ) x}{a^{2} b^{2}}+\frac {\frac {3}{2} a^{3} d^{3}-\frac {3}{2} a^{2} b c \,d^{2}-\frac {3}{2} a \,b^{2} c^{2} d +\frac {3}{2} b^{3} c^{3}}{b^{3} a}}{\left (b x +a \right )^{2}}+\frac {\ln \left (-b x -a \right ) d^{3}}{b^{3}}-\frac {\ln \left (-b x -a \right ) c^{3}}{a^{3}}+\frac {c^{3} \ln \left (x \right )}{a^{3}}\)	\(130\)
default	\(\frac {\left (a^{3} d^{3}-b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b^{3}}-\frac {-2 a^{3} d^{3}+3 a^{2} b c \,d^{2}-b^{3} c^{3}}{a^{2} b^{3} \left (b x +a \right )}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 b^{3} a \left (b x +a \right )^{2}}+\frac {c^{3} \ln \left (x \right )}{a^{3}}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 143, normalized size = 1.54 \begin {gather*} \frac {c^{3} \log \left (x\right )}{a^{3}} + \frac {3 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 3 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x}{2 \, {\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} - \frac {{\left (b^{3} c^{3} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (91) = 182\).
time = 0.74, size = 212, normalized size = 2.28 \begin {gather*} \frac {3 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d - 3 \, a^{4} b c d^{2} + 3 \, a^{5} d^{3} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{3} b^{2} c d^{2} + 2 \, a^{4} b d^{3}\right )} x - 2 \, {\left (a^{2} b^{3} c^{3} - a^{5} d^{3} + {\left (b^{5} c^{3} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{5} c^{3} x^{2} + 2 \, a b^{4} c^{3} x + a^{2} b^{3} c^{3}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{5} x^{2} + 2 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (80) = 160\).
time = 0.80, size = 209, normalized size = 2.25 \begin {gather*} \frac {3 a^{4} d^{3} - 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + 3 a b^{3} c^{3} + x \left (4 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 2 b^{4} c^{3}\right )}{2 a^{4} b^{3} + 4 a^{3} b^{4} x + 2 a^{2} b^{5} x^{2}} + \frac {c^{3} \log {\left (x \right )}}{a^{3}} + \frac {\left (a d - b c\right ) \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- a b^{2} c^{3} + \frac {a \left (a d - b c\right ) \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b}}{a^{3} d^{3} - 2 b^{3} c^{3}} \right )}}{a^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.85, size = 133, normalized size = 1.43 \begin {gather*} \frac {c^{3} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} c^{3} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b^{3}} + \frac {2 \, {\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3}\right )} x + \frac {3 \, {\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d - a^{4} b c d^{2} + a^{5} d^{3}\right )}}{b}}{2 \, {\left (b x + a\right )}^{2} a^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.24, size = 131, normalized size = 1.41 \begin {gather*} \frac {\frac {3\,\left (a^3\,d^3-a^2\,b\,c\,d^2-a\,b^2\,c^2\,d+b^3\,c^3\right )}{2\,a\,b^3}+\frac {x\,\left (2\,a^3\,d^3-3\,a^2\,b\,c\,d^2+b^3\,c^3\right )}{a^2\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2}-\ln \left (a+b\,x\right )\,\left (\frac {c^3}{a^3}-\frac {d^3}{b^3}\right )+\frac {c^3\,\ln \left (x\right )}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x*(a + b*x)^3),x)

[Out]

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

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