Optimal. Leaf size=170 \[ -\frac {10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac {10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}-\frac {6 b^2 d^2}{(a+b x) (b c-a d)^5}+\frac {3 b^2 d}{2 (a+b x)^2 (b c-a d)^4}-\frac {b^2}{3 (a+b x)^3 (b c-a d)^3}-\frac {4 b d^3}{(c+d x) (b c-a d)^5}-\frac {d^3}{2 (c+d x)^2 (b c-a d)^4} \]
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Rubi [A] time = 0.16, antiderivative size = 170, 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, 44} \begin {gather*} -\frac {6 b^2 d^2}{(a+b x) (b c-a d)^5}-\frac {10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac {10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}+\frac {3 b^2 d}{2 (a+b x)^2 (b c-a d)^4}-\frac {b^2}{3 (a+b x)^3 (b c-a d)^3}-\frac {4 b d^3}{(c+d x) (b c-a d)^5}-\frac {d^3}{2 (c+d x)^2 (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 626
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^4 (c+d x)^3} \, dx\\ &=\int \left (\frac {b^3}{(b c-a d)^3 (a+b x)^4}-\frac {3 b^3 d}{(b c-a d)^4 (a+b x)^3}+\frac {6 b^3 d^2}{(b c-a d)^5 (a+b x)^2}-\frac {10 b^3 d^3}{(b c-a d)^6 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)^3}+\frac {4 b d^4}{(b c-a d)^5 (c+d x)^2}+\frac {10 b^2 d^4}{(b c-a d)^6 (c+d x)}\right ) \, dx\\ &=-\frac {b^2}{3 (b c-a d)^3 (a+b x)^3}+\frac {3 b^2 d}{2 (b c-a d)^4 (a+b x)^2}-\frac {6 b^2 d^2}{(b c-a d)^5 (a+b x)}-\frac {d^3}{2 (b c-a d)^4 (c+d x)^2}-\frac {4 b d^3}{(b c-a d)^5 (c+d x)}-\frac {10 b^2 d^3 \log (a+b x)}{(b c-a d)^6}+\frac {10 b^2 d^3 \log (c+d x)}{(b c-a d)^6}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 154, normalized size = 0.91 \begin {gather*} -\frac {\frac {36 b^2 d^2 (b c-a d)}{a+b x}-\frac {9 b^2 d (b c-a d)^2}{(a+b x)^2}+\frac {2 b^2 (b c-a d)^3}{(a+b x)^3}+60 b^2 d^3 \log (a+b x)+\frac {24 b d^3 (b c-a d)}{c+d x}+\frac {3 d^3 (b c-a d)^2}{(c+d x)^2}-60 b^2 d^3 \log (c+d x)}{6 (b c-a d)^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x) \left (a c+(b c+a d) x+b d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 1151, normalized size = 6.77 \begin {gather*} -\frac {2 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 60 \, a^{2} b^{3} c^{3} d^{2} - 20 \, a^{3} b^{2} c^{2} d^{3} - 30 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5} + 60 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} - 5 \, a^{2} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} c^{3} d^{2} + 21 \, a b^{4} c^{2} d^{3} - 12 \, a^{2} b^{3} c d^{4} - 11 \, a^{3} b^{2} d^{5}\right )} x^{2} - 5 \, {\left (b^{5} c^{4} d - 12 \, a b^{4} c^{3} d^{2} - 24 \, a^{2} b^{3} c^{2} d^{3} + 32 \, a^{3} b^{2} c d^{4} + 3 \, a^{4} b d^{5}\right )} x + 60 \, {\left (b^{5} d^{5} x^{5} + a^{3} b^{2} c^{2} d^{3} + {\left (2 \, b^{5} c d^{4} + 3 \, a b^{4} d^{5}\right )} x^{4} + {\left (b^{5} c^{2} d^{3} + 6 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + {\left (3 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} c^{2} d^{3} + 2 \, a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} d^{5} x^{5} + a^{3} b^{2} c^{2} d^{3} + {\left (2 \, b^{5} c d^{4} + 3 \, a b^{4} d^{5}\right )} x^{4} + {\left (b^{5} c^{2} d^{3} + 6 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + {\left (3 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{3} c^{2} d^{3} + 2 \, a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{6} c^{8} - 6 \, a^{4} b^{5} c^{7} d + 15 \, a^{5} b^{4} c^{6} d^{2} - 20 \, a^{6} b^{3} c^{5} d^{3} + 15 \, a^{7} b^{2} c^{4} d^{4} - 6 \, a^{8} b c^{3} d^{5} + a^{9} c^{2} d^{6} + {\left (b^{9} c^{6} d^{2} - 6 \, a b^{8} c^{5} d^{3} + 15 \, a^{2} b^{7} c^{4} d^{4} - 20 \, a^{3} b^{6} c^{3} d^{5} + 15 \, a^{4} b^{5} c^{2} d^{6} - 6 \, a^{5} b^{4} c d^{7} + a^{6} b^{3} d^{8}\right )} x^{5} + {\left (2 \, b^{9} c^{7} d - 9 \, a b^{8} c^{6} d^{2} + 12 \, a^{2} b^{7} c^{5} d^{3} + 5 \, a^{3} b^{6} c^{4} d^{4} - 30 \, a^{4} b^{5} c^{3} d^{5} + 33 \, a^{5} b^{4} c^{2} d^{6} - 16 \, a^{6} b^{3} c d^{7} + 3 \, a^{7} b^{2} d^{8}\right )} x^{4} + {\left (b^{9} c^{8} - 18 \, a^{2} b^{7} c^{6} d^{2} + 52 \, a^{3} b^{6} c^{5} d^{3} - 60 \, a^{4} b^{5} c^{4} d^{4} + 24 \, a^{5} b^{4} c^{3} d^{5} + 10 \, a^{6} b^{3} c^{2} d^{6} - 12 \, a^{7} b^{2} c d^{7} + 3 \, a^{8} b d^{8}\right )} x^{3} + {\left (3 \, a b^{8} c^{8} - 12 \, a^{2} b^{7} c^{7} d + 10 \, a^{3} b^{6} c^{6} d^{2} + 24 \, a^{4} b^{5} c^{5} d^{3} - 60 \, a^{5} b^{4} c^{4} d^{4} + 52 \, a^{6} b^{3} c^{3} d^{5} - 18 \, a^{7} b^{2} c^{2} d^{6} + a^{9} d^{8}\right )} x^{2} + {\left (3 \, a^{2} b^{7} c^{8} - 16 \, a^{3} b^{6} c^{7} d + 33 \, a^{4} b^{5} c^{6} d^{2} - 30 \, a^{5} b^{4} c^{5} d^{3} + 5 \, a^{6} b^{3} c^{4} d^{4} + 12 \, a^{7} b^{2} c^{3} d^{5} - 9 \, a^{8} b c^{2} d^{6} + 2 \, a^{9} c d^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 458, normalized size = 2.69 \begin {gather*} -\frac {10 \, b^{3} d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} c^{6} - 6 \, a b^{6} c^{5} d + 15 \, a^{2} b^{5} c^{4} d^{2} - 20 \, a^{3} b^{4} c^{3} d^{3} + 15 \, a^{4} b^{3} c^{2} d^{4} - 6 \, a^{5} b^{2} c d^{5} + a^{6} b d^{6}} + \frac {10 \, b^{2} d^{4} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}} - \frac {2 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 60 \, a^{2} b^{3} c^{3} d^{2} - 20 \, a^{3} b^{2} c^{2} d^{3} - 30 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5} + 60 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 30 \, {\left (3 \, b^{5} c^{2} d^{3} + 2 \, a b^{4} c d^{4} - 5 \, a^{2} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} c^{3} d^{2} + 21 \, a b^{4} c^{2} d^{3} - 12 \, a^{2} b^{3} c d^{4} - 11 \, a^{3} b^{2} d^{5}\right )} x^{2} - 5 \, {\left (b^{5} c^{4} d - 12 \, a b^{4} c^{3} d^{2} - 24 \, a^{2} b^{3} c^{2} d^{3} + 32 \, a^{3} b^{2} c d^{4} + 3 \, a^{4} b d^{5}\right )} x}{6 \, {\left (b c - a d\right )}^{6} {\left (b x + a\right )}^{3} {\left (d x + c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 165, normalized size = 0.97 \begin {gather*} -\frac {10 b^{2} d^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{6}}+\frac {10 b^{2} d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{6}}+\frac {6 b^{2} d^{2}}{\left (a d -b c \right )^{5} \left (b x +a \right )}+\frac {4 b \,d^{3}}{\left (a d -b c \right )^{5} \left (d x +c \right )}+\frac {3 b^{2} d}{2 \left (a d -b c \right )^{4} \left (b x +a \right )^{2}}-\frac {d^{3}}{2 \left (a d -b c \right )^{4} \left (d x +c \right )^{2}}+\frac {b^{2}}{3 \left (a d -b c \right )^{3} \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.72, size = 889, normalized size = 5.23 \begin {gather*} -\frac {10 \, b^{2} d^{3} \log \left (b x + a\right )}{b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}} + \frac {10 \, b^{2} d^{3} \log \left (d x + c\right )}{b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}} - \frac {60 \, b^{4} d^{4} x^{4} + 2 \, b^{4} c^{4} - 13 \, a b^{3} c^{3} d + 47 \, a^{2} b^{2} c^{2} d^{2} + 27 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} + 30 \, {\left (3 \, b^{4} c d^{3} + 5 \, a b^{3} d^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{4} c^{2} d^{2} + 23 \, a b^{3} c d^{3} + 11 \, a^{2} b^{2} d^{4}\right )} x^{2} - 5 \, {\left (b^{4} c^{3} d - 11 \, a b^{3} c^{2} d^{2} - 35 \, a^{2} b^{2} c d^{3} - 3 \, a^{3} b d^{4}\right )} x}{6 \, {\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} + {\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} + {\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} + {\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} + {\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} + {\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 797, normalized size = 4.69 \begin {gather*} \frac {\frac {-3\,a^4\,d^4+27\,a^3\,b\,c\,d^3+47\,a^2\,b^2\,c^2\,d^2-13\,a\,b^3\,c^3\,d+2\,b^4\,c^4}{6\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}+\frac {5\,d\,x\,\left (3\,a^3\,b\,d^3+35\,a^2\,b^2\,c\,d^2+11\,a\,b^3\,c^2\,d-b^4\,c^3\right )}{6\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}+\frac {10\,b^4\,d^4\,x^4}{a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5}+\frac {5\,d^2\,x^2\,\left (11\,a^2\,b^2\,d^2+23\,a\,b^3\,c\,d+2\,b^4\,c^2\right )}{3\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}+\frac {5\,b\,d^2\,x^3\,\left (3\,c\,b^3\,d+5\,a\,b^2\,d^2\right )}{a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5}}{x^2\,\left (a^3\,d^2+6\,a^2\,b\,c\,d+3\,a\,b^2\,c^2\right )+x^3\,\left (3\,a^2\,b\,d^2+6\,a\,b^2\,c\,d+b^3\,c^2\right )+x\,\left (2\,d\,a^3\,c+3\,b\,a^2\,c^2\right )+x^4\,\left (2\,c\,b^3\,d+3\,a\,b^2\,d^2\right )+a^3\,c^2+b^3\,d^2\,x^5}-\frac {20\,b^2\,d^3\,\mathrm {atanh}\left (\frac {a^6\,d^6-4\,a^5\,b\,c\,d^5+5\,a^4\,b^2\,c^2\,d^4-5\,a^2\,b^4\,c^4\,d^2+4\,a\,b^5\,c^5\,d-b^6\,c^6}{{\left (a\,d-b\,c\right )}^6}+\frac {2\,b\,d\,x\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{{\left (a\,d-b\,c\right )}^6}\right )}{{\left (a\,d-b\,c\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.54, size = 1217, normalized size = 7.16 \begin {gather*} \frac {10 b^{2} d^{3} \log {\left (x + \frac {- \frac {10 a^{7} b^{2} d^{10}}{\left (a d - b c\right )^{6}} + \frac {70 a^{6} b^{3} c d^{9}}{\left (a d - b c\right )^{6}} - \frac {210 a^{5} b^{4} c^{2} d^{8}}{\left (a d - b c\right )^{6}} + \frac {350 a^{4} b^{5} c^{3} d^{7}}{\left (a d - b c\right )^{6}} - \frac {350 a^{3} b^{6} c^{4} d^{6}}{\left (a d - b c\right )^{6}} + \frac {210 a^{2} b^{7} c^{5} d^{5}}{\left (a d - b c\right )^{6}} - \frac {70 a b^{8} c^{6} d^{4}}{\left (a d - b c\right )^{6}} + 10 a b^{2} d^{4} + \frac {10 b^{9} c^{7} d^{3}}{\left (a d - b c\right )^{6}} + 10 b^{3} c d^{3}}{20 b^{3} d^{4}} \right )}}{\left (a d - b c\right )^{6}} - \frac {10 b^{2} d^{3} \log {\left (x + \frac {\frac {10 a^{7} b^{2} d^{10}}{\left (a d - b c\right )^{6}} - \frac {70 a^{6} b^{3} c d^{9}}{\left (a d - b c\right )^{6}} + \frac {210 a^{5} b^{4} c^{2} d^{8}}{\left (a d - b c\right )^{6}} - \frac {350 a^{4} b^{5} c^{3} d^{7}}{\left (a d - b c\right )^{6}} + \frac {350 a^{3} b^{6} c^{4} d^{6}}{\left (a d - b c\right )^{6}} - \frac {210 a^{2} b^{7} c^{5} d^{5}}{\left (a d - b c\right )^{6}} + \frac {70 a b^{8} c^{6} d^{4}}{\left (a d - b c\right )^{6}} + 10 a b^{2} d^{4} - \frac {10 b^{9} c^{7} d^{3}}{\left (a d - b c\right )^{6}} + 10 b^{3} c d^{3}}{20 b^{3} d^{4}} \right )}}{\left (a d - b c\right )^{6}} + \frac {- 3 a^{4} d^{4} + 27 a^{3} b c d^{3} + 47 a^{2} b^{2} c^{2} d^{2} - 13 a b^{3} c^{3} d + 2 b^{4} c^{4} + 60 b^{4} d^{4} x^{4} + x^{3} \left (150 a b^{3} d^{4} + 90 b^{4} c d^{3}\right ) + x^{2} \left (110 a^{2} b^{2} d^{4} + 230 a b^{3} c d^{3} + 20 b^{4} c^{2} d^{2}\right ) + x \left (15 a^{3} b d^{4} + 175 a^{2} b^{2} c d^{3} + 55 a b^{3} c^{2} d^{2} - 5 b^{4} c^{3} d\right )}{6 a^{8} c^{2} d^{5} - 30 a^{7} b c^{3} d^{4} + 60 a^{6} b^{2} c^{4} d^{3} - 60 a^{5} b^{3} c^{5} d^{2} + 30 a^{4} b^{4} c^{6} d - 6 a^{3} b^{5} c^{7} + x^{5} \left (6 a^{5} b^{3} d^{7} - 30 a^{4} b^{4} c d^{6} + 60 a^{3} b^{5} c^{2} d^{5} - 60 a^{2} b^{6} c^{3} d^{4} + 30 a b^{7} c^{4} d^{3} - 6 b^{8} c^{5} d^{2}\right ) + x^{4} \left (18 a^{6} b^{2} d^{7} - 78 a^{5} b^{3} c d^{6} + 120 a^{4} b^{4} c^{2} d^{5} - 60 a^{3} b^{5} c^{3} d^{4} - 30 a^{2} b^{6} c^{4} d^{3} + 42 a b^{7} c^{5} d^{2} - 12 b^{8} c^{6} d\right ) + x^{3} \left (18 a^{7} b d^{7} - 54 a^{6} b^{2} c d^{6} + 6 a^{5} b^{3} c^{2} d^{5} + 150 a^{4} b^{4} c^{3} d^{4} - 210 a^{3} b^{5} c^{4} d^{3} + 102 a^{2} b^{6} c^{5} d^{2} - 6 a b^{7} c^{6} d - 6 b^{8} c^{7}\right ) + x^{2} \left (6 a^{8} d^{7} + 6 a^{7} b c d^{6} - 102 a^{6} b^{2} c^{2} d^{5} + 210 a^{5} b^{3} c^{3} d^{4} - 150 a^{4} b^{4} c^{4} d^{3} - 6 a^{3} b^{5} c^{5} d^{2} + 54 a^{2} b^{6} c^{6} d - 18 a b^{7} c^{7}\right ) + x \left (12 a^{8} c d^{6} - 42 a^{7} b c^{2} d^{5} + 30 a^{6} b^{2} c^{3} d^{4} + 60 a^{5} b^{3} c^{4} d^{3} - 120 a^{4} b^{4} c^{5} d^{2} + 78 a^{3} b^{5} c^{6} d - 18 a^{2} b^{6} c^{7}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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