Optimal. Leaf size=97 \[ \frac {2}{3} d^6 \left (b^2-4 a c\right ) (b+2 c x)^3+2 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)-2 d^6 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{5} d^6 (b+2 c x)^5 \]
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Rubi [A] time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {692, 618, 206} \begin {gather*} \frac {2}{3} d^6 \left (b^2-4 a c\right ) (b+2 c x)^3+2 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)-2 d^6 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{5} d^6 (b+2 c x)^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx &=\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5-\left (2 \left (b^2-4 a c\right )^3 d^6\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right ) d^6 (b+2 c x)^3+\frac {2}{5} d^6 (b+2 c x)^5-2 \left (b^2-4 a c\right )^{5/2} d^6 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 120, normalized size = 1.24 \begin {gather*} d^6 \left (\frac {4}{15} c x \left (16 c^2 \left (15 a^2-5 a c x^2+3 c^2 x^4\right )+20 b^2 c \left (7 c x^2-9 a\right )+120 b c^2 x \left (c x^2-a\right )+45 b^4+90 b^3 c x\right )-2 \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right ) \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 {(b d+2 c d x)^6}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 356, normalized size = 3.67 \begin {gather*} \left [\frac {64}{5} \, c^{5} d^{6} x^{5} + 32 \, b c^{4} d^{6} x^{4} + \frac {16}{3} \, {\left (7 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 8 \, {\left (3 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{2} + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{6} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 4 \, {\left (3 \, b^{4} c - 12 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{6} x, \frac {64}{5} \, c^{5} d^{6} x^{5} + 32 \, b c^{4} d^{6} x^{4} + \frac {16}{3} \, {\left (7 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 8 \, {\left (3 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{2} - 2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} d^{6} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 4 \, {\left (3 \, b^{4} c - 12 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{6} x\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 197, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (b^{6} d^{6} - 12 \, a b^{4} c d^{6} + 48 \, a^{2} b^{2} c^{2} d^{6} - 64 \, a^{3} c^{3} d^{6}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} + \frac {4 \, {\left (48 \, c^{10} d^{6} x^{5} + 120 \, b c^{9} d^{6} x^{4} + 140 \, b^{2} c^{8} d^{6} x^{3} - 80 \, a c^{9} d^{6} x^{3} + 90 \, b^{3} c^{7} d^{6} x^{2} - 120 \, a b c^{8} d^{6} x^{2} + 45 \, b^{4} c^{6} d^{6} x - 180 \, a b^{2} c^{7} d^{6} x + 240 \, a^{2} c^{8} d^{6} x\right )}}{15 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 284, normalized size = 2.93 \begin {gather*} \frac {64 c^{5} d^{6} x^{5}}{5}+32 b \,c^{4} d^{6} x^{4}-\frac {64 a \,c^{4} d^{6} x^{3}}{3}+\frac {112 b^{2} c^{3} d^{6} x^{3}}{3}-\frac {128 a^{3} c^{3} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {96 a^{2} b^{2} c^{2} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {24 a \,b^{4} c \,d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-32 a b \,c^{3} d^{6} x^{2}+\frac {2 b^{6} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+24 b^{3} c^{2} d^{6} x^{2}+64 a^{2} c^{3} d^{6} x -48 a \,b^{2} c^{2} d^{6} x +12 b^{4} c \,d^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 296, normalized size = 3.05 \begin {gather*} x\,\left (60\,b^4\,c\,d^6-\frac {b\,\left (160\,b^3\,c^2\,d^6+\frac {b\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{c}-128\,a\,b\,c^3\,d^6\right )}{c}+\frac {a\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{c}\right )-x^3\,\left (\frac {64\,a\,c^4\,d^6}{3}-\frac {112\,b^2\,c^3\,d^6}{3}\right )+x^2\,\left (80\,b^3\,c^2\,d^6+\frac {b\,\left (64\,a\,c^4\,d^6-112\,b^2\,c^3\,d^6\right )}{2\,c}-64\,a\,b\,c^3\,d^6\right )+2\,d^6\,\mathrm {atan}\left (\frac {b\,d^6\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,c\,d^6\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}}{-64\,a^3\,c^3\,d^6+48\,a^2\,b^2\,c^2\,d^6-12\,a\,b^4\,c\,d^6+b^6\,d^6}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}+\frac {64\,c^5\,d^6\,x^5}{5}+32\,b\,c^4\,d^6\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.77, size = 337, normalized size = 3.47 \begin {gather*} 32 b c^{4} d^{6} x^{4} + \frac {64 c^{5} d^{6} x^{5}}{5} + d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} - d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}} \right )} - d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {16 a^{2} b c^{2} d^{6} - 8 a b^{3} c d^{6} + b^{5} d^{6} + d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{32 a^{2} c^{3} d^{6} - 16 a b^{2} c^{2} d^{6} + 2 b^{4} c d^{6}} \right )} + x^{3} \left (- \frac {64 a c^{4} d^{6}}{3} + \frac {112 b^{2} c^{3} d^{6}}{3}\right ) + x^{2} \left (- 32 a b c^{3} d^{6} + 24 b^{3} c^{2} d^{6}\right ) + x \left (64 a^{2} c^{3} d^{6} - 48 a b^{2} c^{2} d^{6} + 12 b^{4} c d^{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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