Optimal. Leaf size=104 \[ -\frac {2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac {6 b^2 x (b c-a d)^2}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5} \]
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Rubi [A] time = 0.10, antiderivative size = 104, 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} \[ -\frac {2 b^3 (c+d x)^2 (b c-a d)}{d^5}+\frac {6 b^2 x (b c-a d)^2}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\int \frac {(a+b x)^4}{(c+d x)^2} \, dx\\ &=\int \left (\frac {6 b^2 (b c-a d)^2}{d^4}+\frac {(-b c+a d)^4}{d^4 (c+d x)^2}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)}-\frac {4 b^3 (b c-a d) (c+d x)}{d^4}+\frac {b^4 (c+d x)^2}{d^4}\right ) \, dx\\ &=\frac {6 b^2 (b c-a d)^2 x}{d^4}-\frac {(b c-a d)^4}{d^5 (c+d x)}-\frac {2 b^3 (b c-a d) (c+d x)^2}{d^5}+\frac {b^4 (c+d x)^3}{3 d^5}-\frac {4 b (b c-a d)^3 \log (c+d x)}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 165, normalized size = 1.59 \[ \frac {-3 a^4 d^4+12 a^3 b c d^3+18 a^2 b^2 d^2 \left (-c^2+c d x+d^2 x^2\right )+6 a b^3 d \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )-12 b (c+d x) (b c-a d)^3 \log (c+d x)+b^4 \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )}{3 d^5 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 267, normalized size = 2.57 \[ \frac {b^{4} d^{4} x^{4} - 3 \, b^{4} c^{4} + 12 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} - 2 \, {\left (b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 3 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3}\right )} x - 12 \, {\left (b^{4} c^{4} - 3 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x\right )} \log \left (d x + c\right )}{3 \, {\left (d^{6} x + c d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 188, normalized size = 1.81 \[ -\frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {b^{4} d^{4} x^{3} - 3 \, b^{4} c d^{3} x^{2} + 6 \, a b^{3} d^{4} x^{2} + 9 \, b^{4} c^{2} d^{2} x - 24 \, a b^{3} c d^{3} x + 18 \, a^{2} b^{2} d^{4} x}{3 \, d^{6}} - \frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{{\left (d x + c\right )} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 230, normalized size = 2.21 \[ \frac {b^{4} x^{3}}{3 d^{2}}+\frac {2 a \,b^{3} x^{2}}{d^{2}}-\frac {b^{4} c \,x^{2}}{d^{3}}-\frac {a^{4}}{\left (d x +c \right ) d}+\frac {4 a^{3} b c}{\left (d x +c \right ) d^{2}}+\frac {4 a^{3} b \ln \left (d x +c \right )}{d^{2}}-\frac {6 a^{2} b^{2} c^{2}}{\left (d x +c \right ) d^{3}}-\frac {12 a^{2} b^{2} c \ln \left (d x +c \right )}{d^{3}}+\frac {6 a^{2} b^{2} x}{d^{2}}+\frac {4 a \,b^{3} c^{3}}{\left (d x +c \right ) d^{4}}+\frac {12 a \,b^{3} c^{2} \ln \left (d x +c \right )}{d^{4}}-\frac {8 a \,b^{3} c x}{d^{3}}-\frac {b^{4} c^{4}}{\left (d x +c \right ) d^{5}}-\frac {4 b^{4} c^{3} \ln \left (d x +c \right )}{d^{5}}+\frac {3 b^{4} c^{2} x}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 183, normalized size = 1.76 \[ -\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{d^{6} x + c d^{5}} + \frac {b^{4} d^{2} x^{3} - 3 \, {\left (b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x}{3 \, d^{4}} - \frac {4 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 203, normalized size = 1.95 \[ x^2\,\left (\frac {2\,a\,b^3}{d^2}-\frac {b^4\,c}{d^3}\right )-x\,\left (\frac {2\,c\,\left (\frac {4\,a\,b^3}{d^2}-\frac {2\,b^4\,c}{d^3}\right )}{d}-\frac {6\,a^2\,b^2}{d^2}+\frac {b^4\,c^2}{d^4}\right )+\frac {b^4\,x^3}{3\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{d^5}-\frac {a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}{d\,\left (x\,d^5+c\,d^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 155, normalized size = 1.49 \[ \frac {b^{4} x^{3}}{3 d^{2}} + \frac {4 b \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{5}} + x^{2} \left (\frac {2 a b^{3}}{d^{2}} - \frac {b^{4} c}{d^{3}}\right ) + x \left (\frac {6 a^{2} b^{2}}{d^{2}} - \frac {8 a b^{3} c}{d^{3}} + \frac {3 b^{4} c^{2}}{d^{4}}\right ) + \frac {- a^{4} d^{4} + 4 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d - b^{4} c^{4}}{c d^{5} + d^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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