Optimal. Leaf size=137 \[ \frac {3 c \log (x) (b c-a d) (2 b c-a d)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}+\frac {3 c^2 (b c-a d)}{a^4 x}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}-\frac {c^3}{2 a^3 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 137, 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} \[ \frac {3 c^2 (b c-a d)}{a^4 x}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}+\frac {3 c \log (x) (b c-a d) (2 b c-a d)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}-\frac {c^3}{2 a^3 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x^3 (a+b x)^3} \, dx &=\int \left (\frac {c^3}{a^3 x^3}+\frac {3 c^2 (-b c+a d)}{a^4 x^2}+\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}+\frac {(-b c+a d)^3}{a^3 (a+b x)^3}-\frac {3 b c (-b c+a d)^2}{a^4 (a+b x)^2}+\frac {3 b c (b c-a d) (-2 b c+a d)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{2 a^3 x^2}+\frac {3 c^2 (b c-a d)}{a^4 x}+\frac {(b c-a d)^3}{2 a^3 b (a+b x)^2}+\frac {3 c (b c-a d)^2}{a^4 (a+b x)}+\frac {3 c (b c-a d) (2 b c-a d) \log (x)}{a^5}-\frac {3 c (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 138, normalized size = 1.01 \[ -\frac {-6 c \log (x) \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )+6 c \left (a^2 d^2-3 a b c d+2 b^2 c^2\right ) \log (a+b x)+\frac {a^2 (a d-b c)^3}{b (a+b x)^2}+\frac {a^2 c^3}{x^2}+\frac {6 a c^2 (a d-b c)}{x}-\frac {6 a c (b c-a d)^2}{a+b x}}{2 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.69, size = 385, normalized size = 2.81 \[ -\frac {a^{4} b c^{3} - 6 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} - {\left (18 \, a^{2} b^{3} c^{3} - 27 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d\right )} x + 6 \, {\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} + {\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d + a^{2} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} c d^{2}\right )} x^{3} + {\left (2 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{5} b^{3} x^{4} + 2 \, a^{6} b^{2} x^{3} + a^{7} b x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.03, size = 219, normalized size = 1.60 \[ \frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {3 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} + \frac {12 \, b^{4} c^{3} x^{3} - 18 \, a b^{3} c^{2} d x^{3} + 6 \, a^{2} b^{2} c d^{2} x^{3} + 18 \, a b^{3} c^{3} x^{2} - 27 \, a^{2} b^{2} c^{2} d x^{2} + 9 \, a^{3} b c d^{2} x^{2} - a^{4} d^{3} x^{2} + 4 \, a^{2} b^{2} c^{3} x - 6 \, a^{3} b c^{2} d x - a^{3} b c^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 238, normalized size = 1.74 \[ \frac {3 c \,d^{2}}{2 \left (b x +a \right )^{2} a}-\frac {3 b \,c^{2} d}{2 \left (b x +a \right )^{2} a^{2}}+\frac {b^{2} c^{3}}{2 \left (b x +a \right )^{2} a^{3}}-\frac {d^{3}}{2 \left (b x +a \right )^{2} b}+\frac {3 c \,d^{2}}{\left (b x +a \right ) a^{2}}-\frac {6 b \,c^{2} d}{\left (b x +a \right ) a^{3}}+\frac {3 c \,d^{2} \ln \relax (x )}{a^{3}}-\frac {3 c \,d^{2} \ln \left (b x +a \right )}{a^{3}}+\frac {3 b^{2} c^{3}}{\left (b x +a \right ) a^{4}}-\frac {9 b \,c^{2} d \ln \relax (x )}{a^{4}}+\frac {9 b \,c^{2} d \ln \left (b x +a \right )}{a^{4}}+\frac {6 b^{2} c^{3} \ln \relax (x )}{a^{5}}-\frac {6 b^{2} c^{3} \ln \left (b x +a \right )}{a^{5}}-\frac {3 c^{2} d}{a^{3} x}+\frac {3 b \,c^{3}}{a^{4} x}-\frac {c^{3}}{2 a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.93, size = 217, normalized size = 1.58 \[ -\frac {a^{3} b c^{3} - 6 \, {\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} - {\left (18 \, a b^{3} c^{3} - 27 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 2 \, {\left (2 \, a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d\right )} x}{2 \, {\left (a^{4} b^{3} x^{4} + 2 \, a^{5} b^{2} x^{3} + a^{6} b x^{2}\right )}} - \frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \relax (x)}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.45, size = 211, normalized size = 1.54 \[ -\frac {\frac {c^3}{2\,a}+\frac {c^2\,x\,\left (3\,a\,d-2\,b\,c\right )}{a^2}+\frac {x^2\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d-18\,b^3\,c^3\right )}{2\,a^3\,b}-\frac {3\,b\,c\,x^3\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^4}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,a^2\,c\,d^2-9\,a\,b\,c^2\,d+6\,b^2\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.84, size = 371, normalized size = 2.71 \[ \frac {- a^{3} b c^{3} + x^{3} \left (6 a^{2} b^{2} c d^{2} - 18 a b^{3} c^{2} d + 12 b^{4} c^{3}\right ) + x^{2} \left (- a^{4} d^{3} + 9 a^{3} b c d^{2} - 27 a^{2} b^{2} c^{2} d + 18 a b^{3} c^{3}\right ) + x \left (- 6 a^{3} b c^{2} d + 4 a^{2} b^{2} c^{3}\right )}{2 a^{6} b x^{2} + 4 a^{5} b^{2} x^{3} + 2 a^{4} b^{3} x^{4}} + \frac {3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} - 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} - \frac {3 c \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {3 a^{3} c d^{2} - 9 a^{2} b c^{2} d + 6 a b^{2} c^{3} + 3 a c \left (a d - 2 b c\right ) \left (a d - b c\right )}{6 a^{2} b c d^{2} - 18 a b^{2} c^{2} d + 12 b^{3} c^{3}} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________