3.3.28 \(\int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx\) [228]

Optimal. Leaf size=85 \[ -\frac {c^3}{2 a x^2}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)}{a^3}-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 85, 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*} -\frac {(b c-a d)^3 \log (a+b x)}{a^3 b}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}-\frac {c^3}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^3*(a + b*x)),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 78, normalized size = 0.92 \begin {gather*} -\frac {\frac {a c^2 (-2 b c x+a (c+6 d x))}{x^2}-2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)+\frac {2 (b c-a d)^3 \log (a+b x)}{b}}{2 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 111, normalized size = 1.31

method	result	size
default	\(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b}-\frac {c^{3}}{2 a \,x^{2}}+\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right )}{a^{2} x}\)	\(111\)
norman	\(\frac {-\frac {c^{3}}{2 a}-\frac {c^{2} \left (3 a d -b c \right ) x}{a^{2}}}{x^{2}}+\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{3} b}\)	\(111\)
risch	\(\frac {-\frac {c^{3}}{2 a}-\frac {c^{2} \left (3 a d -b c \right ) x}{a^{2}}}{x^{2}}+\frac {3 c \ln \left (x \right ) d^{2}}{a}-\frac {3 c^{2} \ln \left (x \right ) b d}{a^{2}}+\frac {c^{3} \ln \left (x \right ) b^{2}}{a^{3}}+\frac {\ln \left (-b x -a \right ) d^{3}}{b}-\frac {3 \ln \left (-b x -a \right ) c \,d^{2}}{a}+\frac {3 b \ln \left (-b x -a \right ) c^{2} d}{a^{2}}-\frac {b^{2} \ln \left (-b x -a \right ) c^{3}}{a^{3}}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 112, normalized size = 1.32 \begin {gather*} \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (x\right )}{a^{3}} - \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^{3} b} - \frac {a c^{3} - 2 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.92, size = 124, normalized size = 1.46 \begin {gather*} -\frac {a^{2} b c^{3} + 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 )} x^{2} \log \left (b x + a\right ) - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{2} \log \left (x\right ) - 2 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d\right )} x}{2 \, a^{3} b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (78) = 156\).
time = 0.72, size = 257, normalized size = 3.02 \begin {gather*} \frac {- a c^{3} + x \left (- 6 a c^{2} d + 2 b c^{3}\right )}{2 a^{2} x^{2}} + \frac {c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + a c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- 3 a^{3} c d^{2} + 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} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.38, size = 119, normalized size = 1.40 \begin {gather*} \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \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^{3} b} - \frac {a^{2} c^{3} - 2 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.46, size = 84, normalized size = 0.99 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a^3\,b}-\frac {\frac {c^3}{2\,a}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{a^2}}{x^2}+\frac {c\,\ln \left (x\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^3*(a + b*x)),x)

[Out]

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

________________________________________________________________________________________