Optimal. Leaf size=78 \[ \frac {3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac {d^3 x}{b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ \frac {3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac {d^3 x}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^6} \, dx &=\int \frac {(c+d x)^3}{(a+b x)^3} \, dx\\ &=\int \left (\frac {d^3}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)^3}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^2}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {d^3 x}{b^3}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}+\frac {3 d^2 (b c-a d) \log (a+b x)}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 114, normalized size = 1.46 \[ \frac {-5 a^3 d^3+a^2 b d^2 (9 c-4 d x)+a b^2 d \left (-3 c^2+12 c d x+4 d^2 x^2\right )-6 d^2 (a+b x)^2 (a d-b c) \log (a+b x)-\left (b^3 \left (c^3+6 c^2 d x-2 d^3 x^3\right )\right )}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.75, size = 188, normalized size = 2.41 \[ \frac {2 \, b^{3} d^{3} x^{3} + 4 \, a b^{2} d^{3} x^{2} - b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} - 2 \, {\left (3 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x + 6 \, {\left (a^{2} b c d^{2} - a^{3} d^{3} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 112, normalized size = 1.44 \[ \frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b c d^{2} - a d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.17, size = 160, normalized size = 2.05 \[ \frac {a^{3} d^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {3 a^{2} c \,d^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {3 a \,c^{2} d}{2 \left (b x +a \right )^{2} b^{2}}-\frac {c^{3}}{2 \left (b x +a \right )^{2} b}-\frac {3 a^{2} d^{3}}{\left (b x +a \right ) b^{4}}+\frac {6 a c \,d^{2}}{\left (b x +a \right ) b^{3}}-\frac {3 a \,d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 c^{2} d}{\left (b x +a \right ) b^{2}}+\frac {3 c \,d^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {d^{3} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.09, size = 125, normalized size = 1.60 \[ \frac {d^{3} x}{b^{3}} - \frac {b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {3 \, {\left (b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.64, size = 130, normalized size = 1.67 \[ \frac {d^3\,x}{b^3}-\frac {\ln \left (a+b\,x\right )\,\left (3\,a\,d^3-3\,b\,c\,d^2\right )}{b^4}-\frac {\frac {5\,a^3\,d^3-9\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3}{2\,b}+x\,\left (3\,a^2\,d^3-6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.97, size = 128, normalized size = 1.64 \[ \frac {- 5 a^{3} d^{3} + 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - b^{3} c^{3} + x \left (- 6 a^{2} b d^{3} + 12 a b^{2} c d^{2} - 6 b^{3} c^{2} d\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {d^{3} x}{b^{3}} - \frac {3 d^{2} \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________