Optimal. Leaf size=156 \[ \frac {3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac {3 d^2 (b c-a d) x^2}{2 b^4}+\frac {d^3 x^3}{3 b^3}-\frac {a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac {a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) \log (a+b x)}{b^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 156, 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 {a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac {\left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (b c-a d) \log (a+b x)}{b^6}+\frac {a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac {3 d x (b c-2 a d) (b c-a d)}{b^5}+\frac {3 d^2 x^2 (b c-a d)}{2 b^4}+\frac {d^3 x^3}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^3}{(a+b x)^3} \, dx &=\int \left (\frac {3 d (b c-2 a d) (b c-a d)}{b^5}+\frac {3 d^2 (b c-a d) x}{b^4}+\frac {d^3 x^2}{b^3}-\frac {a^2 (-b c+a d)^3}{b^5 (a+b x)^3}+\frac {a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)^2}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac {3 d^2 (b c-a d) x^2}{2 b^4}+\frac {d^3 x^3}{3 b^3}-\frac {a^2 (b c-a d)^3}{2 b^6 (a+b x)^2}+\frac {a (2 b c-5 a d) (b c-a d)^2}{b^6 (a+b x)}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) \log (a+b x)}{b^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 160, normalized size = 1.03 \begin {gather*} \frac {18 b d \left (b^2 c^2-3 a b c d+2 a^2 d^2\right ) x+9 b^2 d^2 (b c-a d) x^2+2 b^3 d^3 x^3+\frac {3 a^2 (-b c+a d)^3}{(a+b x)^2}-\frac {6 a (b c-a d)^2 (-2 b c+5 a d)}{a+b x}+6 \left (b^3 c^3-9 a b^2 c^2 d+18 a^2 b c d^2-10 a^3 d^3\right ) \log (a+b x)}{6 b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 211, normalized size = 1.35
method | result | size |
default | \(\frac {d \left (\frac {1}{3} d^{2} x^{3} b^{2}-\frac {3}{2} a b \,d^{2} x^{2}+\frac {3}{2} b^{2} c d \,x^{2}+6 a^{2} d^{2} x -9 a b c d x +3 b^{2} c^{2} x \right )}{b^{5}}+\frac {\left (-10 a^{3} d^{3}+18 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {a \left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right )}{b^{6} \left (b x +a \right )}+\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{6} \left (b x +a \right )^{2}}\) | \(211\) |
norman | \(\frac {-\frac {a^{2} \left (30 a^{3} d^{3}-54 a^{2} b c \,d^{2}+27 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right )}{2 b^{6}}+\frac {d^{3} x^{5}}{3 b}-\frac {2 a \left (10 a^{3} d^{3}-18 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{5}}+\frac {d \left (10 a^{2} d^{2}-18 a b c d +9 b^{2} c^{2}\right ) x^{3}}{3 b^{3}}-\frac {d^{2} \left (5 a d -9 b c \right ) x^{4}}{6 b^{2}}}{\left (b x +a \right )^{2}}-\frac {\left (10 a^{3} d^{3}-18 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(211\) |
risch | \(\frac {d^{3} x^{3}}{3 b^{3}}-\frac {3 d^{3} a \,x^{2}}{2 b^{4}}+\frac {3 d^{2} c \,x^{2}}{2 b^{3}}+\frac {6 d^{3} a^{2} x}{b^{5}}-\frac {9 d^{2} a c x}{b^{4}}+\frac {3 d \,c^{2} x}{b^{3}}+\frac {\left (-5 a^{4} d^{3}+12 a^{3} b c \,d^{2}-9 a^{2} b^{2} c^{2} d +2 a \,b^{3} c^{3}\right ) x -\frac {3 a^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b}}{b^{5} \left (b x +a \right )^{2}}-\frac {10 \ln \left (b x +a \right ) a^{3} d^{3}}{b^{6}}+\frac {18 \ln \left (b x +a \right ) a^{2} c \,d^{2}}{b^{5}}-\frac {9 \ln \left (b x +a \right ) a \,c^{2} d}{b^{4}}+\frac {\ln \left (b x +a \right ) c^{3}}{b^{3}}\) | \(233\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 227, normalized size = 1.46 \begin {gather*} \frac {3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, b^{2} d^{3} x^{3} + 9 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 18 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x}{6 \, b^{5}} + \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs.
\(2 (150) = 300\).
time = 1.16, size = 361, normalized size = 2.31 \begin {gather*} \frac {2 \, b^{5} d^{3} x^{5} + 9 \, a^{2} b^{3} c^{3} - 45 \, a^{3} b^{2} c^{2} d + 63 \, a^{4} b c d^{2} - 27 \, a^{5} d^{3} + {\left (9 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \, {\left (9 \, b^{5} c^{2} d - 18 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{3} + 9 \, {\left (4 \, a b^{4} c^{2} d - 11 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{2} + 6 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x + 6 \, {\left (a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 18 \, a^{4} b c d^{2} - 10 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 18 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 18 \, a^{3} b^{2} c d^{2} - 10 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.78, size = 235, normalized size = 1.51 \begin {gather*} x^{2} \left (- \frac {3 a d^{3}}{2 b^{4}} + \frac {3 c d^{2}}{2 b^{3}}\right ) + x \left (\frac {6 a^{2} d^{3}}{b^{5}} - \frac {9 a c d^{2}}{b^{4}} + \frac {3 c^{2} d}{b^{3}}\right ) + \frac {- 9 a^{5} d^{3} + 21 a^{4} b c d^{2} - 15 a^{3} b^{2} c^{2} d + 3 a^{2} b^{3} c^{3} + x \left (- 10 a^{4} b d^{3} + 24 a^{3} b^{2} c d^{2} - 18 a^{2} b^{3} c^{2} d + 4 a b^{4} c^{3}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac {d^{3} x^{3}}{3 b^{3}} - \frac {\left (a d - b c\right ) \left (10 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \log {\left (a + b x \right )}}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.87, size = 222, normalized size = 1.42 \begin {gather*} \frac {{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {3 \, a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 21 \, a^{4} b c d^{2} - 9 \, a^{5} d^{3} + 2 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} d^{3} x^{3} + 9 \, b^{6} c d^{2} x^{2} - 9 \, a b^{5} d^{3} x^{2} + 18 \, b^{6} c^{2} d x - 54 \, a b^{5} c d^{2} x + 36 \, a^{2} b^{4} d^{3} x}{6 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.39, size = 248, normalized size = 1.59 \begin {gather*} x\,\left (\frac {3\,c^2\,d}{b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b}-\frac {3\,a^2\,d^3}{b^5}\right )-x^2\,\left (\frac {3\,a\,d^3}{2\,b^4}-\frac {3\,c\,d^2}{2\,b^3}\right )-\frac {x\,\left (5\,a^4\,d^3-12\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3\right )+\frac {3\,\left (3\,a^5\,d^3-7\,a^4\,b\,c\,d^2+5\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{2\,b}}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}+\frac {d^3\,x^3}{3\,b^3}-\frac {\ln \left (a+b\,x\right )\,\left (10\,a^3\,d^3-18\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________