Optimal. Leaf size=101 \[ -\frac {3 b \log (c+d x)}{a^4 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d}-\frac {2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^3 d (c+d x)^3}-\frac {b}{6 a^2 d \left (a+b (c+d x)^3\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 266, 44} \[ -\frac {2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac {b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac {3 b \log (c+d x)}{a^4 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d}-\frac {1}{3 a^3 d (c+d x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=-\frac {1}{3 a^3 d (c+d x)^3}-\frac {b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac {2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac {3 b \log (c+d x)}{a^4 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 80, normalized size = 0.79 \[ \frac {a \left (-\frac {4 b}{a+b (c+d x)^3}-\frac {a b}{\left (a+b (c+d x)^3\right )^2}-\frac {2}{(c+d x)^3}\right )+6 b \log \left (a+b (c+d x)^3\right )-18 b \log (c+d x)}{6 a^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.20, size = 889, normalized size = 8.80 \[ -\frac {6 \, a b^{2} d^{6} x^{6} + 36 \, a b^{2} c d^{5} x^{5} + 90 \, a b^{2} c^{2} d^{4} x^{4} + 6 \, a b^{2} c^{6} + 3 \, {\left (40 \, a b^{2} c^{3} + 3 \, a^{2} b\right )} d^{3} x^{3} + 9 \, a^{2} b c^{3} + 9 \, {\left (10 \, a b^{2} c^{4} + 3 \, a^{2} b c\right )} d^{2} x^{2} + 2 \, a^{3} + 9 \, {\left (4 \, a b^{2} c^{5} + 3 \, a^{2} b c^{2}\right )} d x - 6 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 18 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{6 \, {\left (a^{4} b^{2} d^{10} x^{9} + 9 \, a^{4} b^{2} c d^{9} x^{8} + 36 \, a^{4} b^{2} c^{2} d^{8} x^{7} + 2 \, {\left (42 \, a^{4} b^{2} c^{3} + a^{5} b\right )} d^{7} x^{6} + 6 \, {\left (21 \, a^{4} b^{2} c^{4} + 2 \, a^{5} b c\right )} d^{6} x^{5} + 6 \, {\left (21 \, a^{4} b^{2} c^{5} + 5 \, a^{5} b c^{2}\right )} d^{5} x^{4} + {\left (84 \, a^{4} b^{2} c^{6} + 40 \, a^{5} b c^{3} + a^{6}\right )} d^{4} x^{3} + 3 \, {\left (12 \, a^{4} b^{2} c^{7} + 10 \, a^{5} b c^{4} + a^{6} c\right )} d^{3} x^{2} + 3 \, {\left (3 \, a^{4} b^{2} c^{8} + 4 \, a^{5} b c^{5} + a^{6} c^{2}\right )} d^{2} x + {\left (a^{4} b^{2} c^{9} + 2 \, a^{5} b c^{6} + a^{6} c^{3}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 80, normalized size = 0.79 \[ \frac {b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{a^{4} d} + \frac {5 \, b^{3} + \frac {6 \, a b^{2}}{{\left (d x + c\right )}^{3}}}{6 \, a^{4} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )}^{2} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 311, normalized size = 3.08 \[ -\frac {2 b^{2} d^{2} x^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {2 b^{2} c d \,x^{2}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {2 b^{2} c^{2} x}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3}}-\frac {2 b^{2} c^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{3} d}-\frac {5 b}{6 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} d}-\frac {3 b \ln \left (d x +c \right )}{a^{4} d}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{a^{4} d}-\frac {1}{3 \left (d x +c \right )^{3} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.78, size = 438, normalized size = 4.34 \[ -\frac {6 \, b^{2} d^{6} x^{6} + 36 \, b^{2} c d^{5} x^{5} + 90 \, b^{2} c^{2} d^{4} x^{4} + 6 \, b^{2} c^{6} + 3 \, {\left (40 \, b^{2} c^{3} + 3 \, a b\right )} d^{3} x^{3} + 9 \, a b c^{3} + 9 \, {\left (10 \, b^{2} c^{4} + 3 \, a b c\right )} d^{2} x^{2} + 9 \, {\left (4 \, b^{2} c^{5} + 3 \, a b c^{2}\right )} d x + 2 \, a^{2}}{6 \, {\left (a^{3} b^{2} d^{10} x^{9} + 9 \, a^{3} b^{2} c d^{9} x^{8} + 36 \, a^{3} b^{2} c^{2} d^{8} x^{7} + 2 \, {\left (42 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{7} x^{6} + 6 \, {\left (21 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{6} x^{5} + 6 \, {\left (21 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{5} x^{4} + {\left (84 \, a^{3} b^{2} c^{6} + 40 \, a^{4} b c^{3} + a^{5}\right )} d^{4} x^{3} + 3 \, {\left (12 \, a^{3} b^{2} c^{7} + 10 \, a^{4} b c^{4} + a^{5} c\right )} d^{3} x^{2} + 3 \, {\left (3 \, a^{3} b^{2} c^{8} + 4 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d^{2} x + {\left (a^{3} b^{2} c^{9} + 2 \, a^{4} b c^{6} + a^{5} c^{3}\right )} d\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{a^{4} d} - \frac {3 \, b \log \left (d x + c\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.20, size = 438, normalized size = 4.34 \[ \frac {b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{a^4\,d}-\frac {\frac {2\,a^2+9\,a\,b\,c^3+6\,b^2\,c^6}{6\,a^3\,d}+\frac {3\,x^2\,\left (10\,d\,b^2\,c^4+3\,a\,d\,b\,c\right )}{2\,a^3}+\frac {3\,x\,\left (4\,b^2\,c^5+3\,a\,b\,c^2\right )}{2\,a^3}+\frac {x^3\,\left (40\,b^2\,c^3\,d^2+3\,a\,b\,d^2\right )}{2\,a^3}+\frac {b^2\,d^5\,x^6}{a^3}+\frac {15\,b^2\,c^2\,d^3\,x^4}{a^3}+\frac {6\,b^2\,c\,d^4\,x^5}{a^3}}{x\,\left (3\,d\,a^2\,c^2+12\,d\,a\,b\,c^5+9\,d\,b^2\,c^8\right )+x^6\,\left (84\,b^2\,c^3\,d^6+2\,a\,b\,d^6\right )+x^2\,\left (3\,a^2\,c\,d^2+30\,a\,b\,c^4\,d^2+36\,b^2\,c^7\,d^2\right )+x^5\,\left (126\,b^2\,c^4\,d^5+12\,a\,b\,c\,d^5\right )+x^3\,\left (a^2\,d^3+40\,a\,b\,c^3\,d^3+84\,b^2\,c^6\,d^3\right )+a^2\,c^3+b^2\,c^9+x^4\,\left (126\,b^2\,c^5\,d^4+30\,a\,b\,c^2\,d^4\right )+b^2\,d^9\,x^9+2\,a\,b\,c^6+9\,b^2\,c\,d^8\,x^8+36\,b^2\,c^2\,d^7\,x^7}-\frac {3\,b\,\ln \left (c+d\,x\right )}{a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 7.80, size = 500, normalized size = 4.95 \[ \frac {- 2 a^{2} - 9 a b c^{3} - 6 b^{2} c^{6} - 90 b^{2} c^{2} d^{4} x^{4} - 36 b^{2} c d^{5} x^{5} - 6 b^{2} d^{6} x^{6} + x^{3} \left (- 9 a b d^{3} - 120 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 27 a b c d^{2} - 90 b^{2} c^{4} d^{2}\right ) + x \left (- 27 a b c^{2} d - 36 b^{2} c^{5} d\right )}{6 a^{5} c^{3} d + 12 a^{4} b c^{6} d + 6 a^{3} b^{2} c^{9} d + 216 a^{3} b^{2} c^{2} d^{8} x^{7} + 54 a^{3} b^{2} c d^{9} x^{8} + 6 a^{3} b^{2} d^{10} x^{9} + x^{6} \left (12 a^{4} b d^{7} + 504 a^{3} b^{2} c^{3} d^{7}\right ) + x^{5} \left (72 a^{4} b c d^{6} + 756 a^{3} b^{2} c^{4} d^{6}\right ) + x^{4} \left (180 a^{4} b c^{2} d^{5} + 756 a^{3} b^{2} c^{5} d^{5}\right ) + x^{3} \left (6 a^{5} d^{4} + 240 a^{4} b c^{3} d^{4} + 504 a^{3} b^{2} c^{6} d^{4}\right ) + x^{2} \left (18 a^{5} c d^{3} + 180 a^{4} b c^{4} d^{3} + 216 a^{3} b^{2} c^{7} d^{3}\right ) + x \left (18 a^{5} c^{2} d^{2} + 72 a^{4} b c^{5} d^{2} + 54 a^{3} b^{2} c^{8} d^{2}\right )} - \frac {3 b \log {\left (\frac {c}{d} + x \right )}}{a^{4} d} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________