Optimal. Leaf size=76 \[ -12 c d^4 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]
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Rubi [A] time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \[ -12 c d^4 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]
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)^4}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}+\left (6 c d^2\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=12 c d^4 (b+2 c x)-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}+\left (6 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=12 c d^4 (b+2 c x)-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}-\left (12 c \left (b^2-4 a c\right ) d^4\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=12 c d^4 (b+2 c x)-\frac {d^4 (b+2 c x)^3}{a+b x+c x^2}-12 c \sqrt {b^2-4 a c} d^4 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 1.01 \[ d^4 \left (-\frac {\left (b^2-4 a c\right ) (b+2 c x)}{a+x (b+c x)}-12 c \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+16 c^2 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 297, normalized size = 3.91 \[ \left [\frac {16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \, {\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x - {\left (b^{3} - 4 \, a b c\right )} d^{4} + 6 \, {\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{c x^{2} + b x + a}, \frac {16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \, {\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x - {\left (b^{3} - 4 \, a b c\right )} d^{4} - 12 \, {\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{c x^{2} + b x + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 112, normalized size = 1.47 \[ 16 \, c^{2} d^{4} x + \frac {12 \, {\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d^{4} x - 8 \, a c^{2} d^{4} x + b^{3} d^{4} - 4 \, a b c d^{4}}{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 133, normalized size = 1.75 \[ \frac {8 a \,c^{2} d^{4} x}{c \,x^{2}+b x +a}-\frac {2 b^{2} c \,d^{4} x}{c \,x^{2}+b x +a}+\frac {4 a b c \,d^{4}}{c \,x^{2}+b x +a}-\frac {b^{3} d^{4}}{c \,x^{2}+b x +a}+16 c^{2} d^{4} x -12 \sqrt {4 a c -b^{2}}\, c \,d^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 143, normalized size = 1.88 \[ \frac {x\,\left (8\,a\,c^2\,d^4-2\,b^2\,c\,d^4\right )-b^3\,d^4+4\,a\,b\,c\,d^4}{c\,x^2+b\,x+a}+16\,c^2\,d^4\,x-12\,c\,d^4\,\mathrm {atan}\left (\frac {12\,c^2\,d^4\,x\,\sqrt {4\,a\,c-b^2}+6\,b\,c\,d^4\,\sqrt {4\,a\,c-b^2}}{24\,a\,c^2\,d^4-6\,b^2\,c\,d^4}\right )\,\sqrt {4\,a\,c-b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.09, size = 173, normalized size = 2.28 \[ 16 c^{2} d^{4} x + c d^{4} \sqrt {- 144 a c + 36 b^{2}} \log {\left (x + \frac {6 b c d^{4} - c d^{4} \sqrt {- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} - c d^{4} \sqrt {- 144 a c + 36 b^{2}} \log {\left (x + \frac {6 b c d^{4} + c d^{4} \sqrt {- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} + \frac {4 a b c d^{4} - b^{3} d^{4} + x \left (8 a c^{2} d^{4} - 2 b^{2} c d^{4}\right )}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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