Optimal. Leaf size=92 \[ -\frac {2 b \log (c+d x)}{a^3 d e^4}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac {b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^2 d e^4 (c+d x)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {372, 266, 44} \[ -\frac {b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d e^4}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac {1}{3 a^2 d e^4 (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d e^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e^4}\\ &=-\frac {1}{3 a^2 d e^4 (c+d x)^3}-\frac {b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d e^4}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 63, normalized size = 0.68 \[ -\frac {a \left (\frac {b}{a+b (c+d x)^3}+\frac {1}{(c+d x)^3}\right )-2 b \log \left (a+b (c+d x)^3\right )+6 b \log (c+d x)}{3 a^3 d e^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 452, normalized size = 4.91 \[ -\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \, {\left (a^{3} b d^{7} e^{4} x^{6} + 6 \, a^{3} b c d^{6} e^{4} x^{5} + 15 \, a^{3} b c^{2} d^{5} e^{4} x^{4} + {\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} e^{4} x + {\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 158, normalized size = 1.72 \[ \frac {2 \, b e^{\left (-4\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} - \frac {2 \, b e^{\left (-4\right )} \log \left ({\left | d x + c \right |}\right )}{a^{3} d} - \frac {{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2}\right )} e^{\left (-4\right )}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} {\left (d x + c\right )}^{3} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 131, normalized size = 1.42 \[ -\frac {b}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a^{2} d \,e^{4}}-\frac {2 b \ln \left (d x +c \right )}{a^{3} d \,e^{4}}+\frac {2 b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{3} d \,e^{4}}-\frac {1}{3 \left (d x +c \right )^{3} a^{2} d \,e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 249, normalized size = 2.71 \[ -\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \, {\left (a^{2} b d^{7} e^{4} x^{6} + 6 \, a^{2} b c d^{6} e^{4} x^{5} + 15 \, a^{2} b c^{2} d^{5} e^{4} x^{4} + {\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} e^{4} x + {\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d e^{4}\right )}} + \frac {2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e^{4}} - \frac {2 \, b \log \left (d x + c\right )}{a^{3} d e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.98, size = 250, normalized size = 2.72 \[ \frac {2\,b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d\,e^4}-\frac {\frac {2\,b\,c^3+a}{3\,a^2\,d}+\frac {2\,b\,d^2\,x^3}{3\,a^2}+\frac {2\,b\,c^2\,x}{a^2}+\frac {2\,b\,c\,d\,x^2}{a^2}}{x^2\,\left (15\,b\,c^4\,d^2\,e^4+3\,a\,c\,d^2\,e^4\right )+x\,\left (6\,b\,d\,c^5\,e^4+3\,a\,d\,c^2\,e^4\right )+x^3\,\left (20\,b\,c^3\,d^3\,e^4+a\,d^3\,e^4\right )+a\,c^3\,e^4+b\,c^6\,e^4+b\,d^6\,e^4\,x^6+6\,b\,c\,d^5\,e^4\,x^5+15\,b\,c^2\,d^4\,e^4\,x^4}-\frac {2\,b\,\ln \left (c+d\,x\right )}{a^3\,d\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.10, size = 294, normalized size = 3.20 \[ \frac {- a - 2 b c^{3} - 6 b c^{2} d x - 6 b c d^{2} x^{2} - 2 b d^{3} x^{3}}{3 a^{3} c^{3} d e^{4} + 3 a^{2} b c^{6} d e^{4} + 45 a^{2} b c^{2} d^{5} e^{4} x^{4} + 18 a^{2} b c d^{6} e^{4} x^{5} + 3 a^{2} b d^{7} e^{4} x^{6} + x^{3} \left (3 a^{3} d^{4} e^{4} + 60 a^{2} b c^{3} d^{4} e^{4}\right ) + x^{2} \left (9 a^{3} c d^{3} e^{4} + 45 a^{2} b c^{4} d^{3} e^{4}\right ) + x \left (9 a^{3} c^{2} d^{2} e^{4} + 18 a^{2} b c^{5} d^{2} e^{4}\right )} - \frac {2 b \log {\left (\frac {c}{d} + x \right )}}{a^{3} d e^{4}} + \frac {2 b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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