Optimal. Leaf size=79 \[ \frac {4 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rubi [A] time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {693, 682} \begin {gather*} \frac {4 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2 \sqrt {a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 682
Rule 693
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^4 \sqrt {a+b x+c x^2}} \, dx &=\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {2 \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) d^2}\\ &=\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac {4 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {a+x (b+c x)} \left (-4 c \left (a-2 c x^2\right )+3 b^2+8 b c x\right )}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 62, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {a+b x+c x^2} \left (-4 a c+3 b^2+8 b c x+8 c^2 x^2\right )}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 165, normalized size = 2.09 \begin {gather*} \frac {2 \, {\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x + {\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 134, normalized size = 1.70 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c + b^{2} \sqrt {c} - a c^{\frac {3}{2}}\right )}}{3 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3} c d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 70, normalized size = 0.89 \begin {gather*} -\frac {2 \left (-8 c^{2} x^{2}-8 b c x +4 a c -3 b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{3 \left (2 c x +b \right )^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) 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 = 0.65, size = 60, normalized size = 0.76 \begin {gather*} \frac {2\,\sqrt {c\,x^2+b\,x+a}\,\left (3\,b^2+8\,b\,c\,x+8\,c^2\,x^2-4\,a\,c\right )}{3\,d^4\,{\left (4\,a\,c-b^2\right )}^2\,{\left (b+2\,c\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{b^{4} \sqrt {a + b x + c x^{2}} + 8 b^{3} c x \sqrt {a + b x + c x^{2}} + 24 b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 c^{4} x^{4} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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