Optimal. Leaf size=86 \[ \frac {2}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}-\frac {2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 86, 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 = {693, 618, 206} \begin {gather*} \frac {2}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}-\frac {2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 693
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx &=\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {\int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)}+\frac {\int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^4}\\ &=\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2 d^4}\\ &=\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)}-\frac {2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 83, normalized size = 0.97 \begin {gather*} \frac {2 \left (\frac {b^2-4 a c}{(b+2 c x)^3}+\frac {3 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {3}{b+2 c x}\right )}{3 d^4 \left (b^2-4 a c\right )^2} \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}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 627, normalized size = 7.29 \begin {gather*} \left [\frac {8 \, b^{4} - 40 \, a b^{2} c + 32 \, a^{2} c^{2} + 24 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {b^{2} - 4 \, a c} \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 ) + 24 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{3 \, {\left (8 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{8} c - 12 \, a b^{6} c^{2} + 48 \, a^{2} b^{4} c^{3} - 64 \, a^{3} b^{2} c^{4}\right )} d^{4} x + {\left (b^{9} - 12 \, a b^{7} c + 48 \, a^{2} b^{5} c^{2} - 64 \, a^{3} b^{3} c^{3}\right )} d^{4}\right )}}, \frac {2 \, {\left (4 \, b^{4} - 20 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 12 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )}}{3 \, {\left (8 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{8} c - 12 \, a b^{6} c^{2} + 48 \, a^{2} b^{4} c^{3} - 64 \, a^{3} b^{2} c^{4}\right )} d^{4} x + {\left (b^{9} - 12 \, a b^{7} c + 48 \, a^{2} b^{5} c^{2} - 64 \, a^{3} b^{3} c^{3}\right )} d^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 128, normalized size = 1.49 \begin {gather*} \frac {2 \, \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {8 \, {\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )}}{3 \, {\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} {\left (2 \, c x + b\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 1.03 \begin {gather*} \frac {2 \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {5}{2}} d^{4}}+\frac {2}{\left (4 a c -b^{2}\right )^{2} \left (2 c x +b \right ) d^{4}}-\frac {2}{3 \left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{3} d^{4}} \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 = 0.59, size = 219, normalized size = 2.55 \begin {gather*} \frac {\frac {8\,c^2\,x^2}{{\left (4\,a\,c-b^2\right )}^2}-\frac {8\,\left (a\,c-b^2\right )}{3\,{\left (4\,a\,c-b^2\right )}^2}+\frac {8\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^2}}{b^3\,d^4+6\,b^2\,c\,d^4\,x+12\,b\,c^2\,d^4\,x^2+8\,c^3\,d^4\,x^3}+\frac {2\,\mathrm {atan}\left (\frac {16\,a^2\,b\,c^2\,d^4-8\,a\,b^3\,c\,d^4+b^5\,d^4}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {2\,c\,x\,\left (16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4+b^4\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.74, size = 442, normalized size = 5.14 \begin {gather*} \frac {- 8 a c + 8 b^{2} + 24 b c x + 24 c^{2} x^{2}}{48 a^{2} b^{3} c^{2} d^{4} - 24 a b^{5} c d^{4} + 3 b^{7} d^{4} + x^{3} \left (384 a^{2} c^{5} d^{4} - 192 a b^{2} c^{4} d^{4} + 24 b^{4} c^{3} d^{4}\right ) + x^{2} \left (576 a^{2} b c^{4} d^{4} - 288 a b^{3} c^{3} d^{4} + 36 b^{5} c^{2} d^{4}\right ) + x \left (288 a^{2} b^{2} c^{3} d^{4} - 144 a b^{4} c^{2} d^{4} + 18 b^{6} c d^{4}\right )} - \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} + \frac {\sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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