Optimal. Leaf size=108 \[ -60 c^2 d^6 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+60 c^2 d^6 (b+2 c x) \]
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Rubi [A] time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \begin {gather*} -60 c^2 d^6 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+60 c^2 d^6 (b+2 c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 686
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}+\left (5 c d^2\right ) \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}+\left (30 c^2 d^4\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=60 c^2 d^6 (b+2 c x)-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}+\left (30 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=60 c^2 d^6 (b+2 c x)-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-\left (60 c^2 \left (b^2-4 a c\right ) d^6\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=60 c^2 d^6 (b+2 c x)-\frac {d^6 (b+2 c x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^6 (b+2 c x)^3}{a+b x+c x^2}-60 c^2 \sqrt {b^2-4 a c} 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 = 0.06, size = 113, normalized size = 1.05 \begin {gather*} d^6 \left (-60 c^2 \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+\frac {9 c \left (4 a c-b^2\right ) (b+2 c x)}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^2 (b+2 c x)}{2 (a+x (b+c x))^2}+64 c^3 x\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}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 571, normalized size = 5.29 \begin {gather*} \left [\frac {128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \, {\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \, {\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \, {\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} + 60 \, {\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} x^{2}\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 )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac {128 \, c^{5} d^{6} x^{5} + 256 \, b c^{4} d^{6} x^{4} + 4 \, {\left (23 \, b^{2} c^{3} + 100 \, a c^{4}\right )} d^{6} x^{3} - 2 \, {\left (27 \, b^{3} c^{2} - 236 \, a b c^{3}\right )} d^{6} x^{2} - 4 \, {\left (5 \, b^{4} c - 13 \, a b^{2} c^{2} - 60 \, a^{2} c^{3}\right )} d^{6} x - {\left (b^{5} + 10 \, a b^{3} c - 56 \, a^{2} b c^{2}\right )} d^{6} - 120 \, {\left (c^{4} d^{6} x^{4} + 2 \, b c^{3} d^{6} x^{3} + 2 \, a b c^{2} d^{6} x + a^{2} c^{2} d^{6} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} x^{2}\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 )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 196, normalized size = 1.81 \begin {gather*} 64 \, c^{3} d^{6} x + \frac {60 \, {\left (b^{2} c^{2} d^{6} - 4 \, a 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 {36 \, b^{2} c^{3} d^{6} x^{3} - 144 \, a c^{4} d^{6} x^{3} + 54 \, b^{3} c^{2} d^{6} x^{2} - 216 \, a b c^{3} d^{6} x^{2} + 20 \, b^{4} c d^{6} x - 52 \, a b^{2} c^{2} d^{6} x - 112 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} + 10 \, a b^{3} c d^{6} - 56 \, a^{2} b c^{2} d^{6}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 289, normalized size = 2.68 \begin {gather*} \frac {72 a \,c^{4} d^{6} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {18 b^{2} c^{3} d^{6} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {108 a b \,c^{3} d^{6} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {27 b^{3} c^{2} d^{6} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {56 a^{2} c^{3} d^{6} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {26 a \,b^{2} c^{2} d^{6} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {10 b^{4} c \,d^{6} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {28 a^{2} b \,c^{2} d^{6}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {5 a \,b^{3} c \,d^{6}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {b^{5} d^{6}}{2 \left (c \,x^{2}+b x +a \right )^{2}}+64 c^{3} d^{6} x -60 \sqrt {4 a c -b^{2}}\, c^{2} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) \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 = 0.57, size = 254, normalized size = 2.35 \begin {gather*} \frac {x\,\left (56\,a^2\,c^3\,d^6+26\,a\,b^2\,c^2\,d^6-10\,b^4\,c\,d^6\right )+x^3\,\left (72\,a\,c^4\,d^6-18\,b^2\,c^3\,d^6\right )-\frac {b^5\,d^6}{2}-x^2\,\left (27\,b^3\,c^2\,d^6-108\,a\,b\,c^3\,d^6\right )+28\,a^2\,b\,c^2\,d^6-5\,a\,b^3\,c\,d^6}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+64\,c^3\,d^6\,x-60\,c^2\,d^6\,\mathrm {atan}\left (\frac {30\,b\,c^2\,d^6\,\sqrt {4\,a\,c-b^2}+60\,c^3\,d^6\,x\,\sqrt {4\,a\,c-b^2}}{120\,a\,c^3\,d^6-30\,b^2\,c^2\,d^6}\right )\,\sqrt {4\,a\,c-b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.91, size = 299, normalized size = 2.77 \begin {gather*} 64 c^{3} d^{6} x + c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}} \log {\left (x + \frac {30 b c^{2} d^{6} - c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} - c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}} \log {\left (x + \frac {30 b c^{2} d^{6} + c^{2} d^{6} \sqrt {- 3600 a c + 900 b^{2}}}{60 c^{3} d^{6}} \right )} + \frac {56 a^{2} b c^{2} d^{6} - 10 a b^{3} c d^{6} - b^{5} d^{6} + x^{3} \left (144 a c^{4} d^{6} - 36 b^{2} c^{3} d^{6}\right ) + x^{2} \left (216 a b c^{3} d^{6} - 54 b^{3} c^{2} d^{6}\right ) + x \left (112 a^{2} c^{3} d^{6} + 52 a b^{2} c^{2} d^{6} - 20 b^{4} c d^{6}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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