Optimal. Leaf size=74 \[ -\frac {8 e (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {768, 636} \begin {gather*} -\frac {8 e (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 636
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} (4 e) \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 110, normalized size = 1.49 \begin {gather*} \frac {2 \left (8 a^2 e^2+4 b e \left (c x^2 (e x-3 d)-a (d-3 e x)\right )+4 a c \left (d^2+3 e^2 x^2\right )-b^2 \left (d^2+6 d e x-3 e^2 x^2\right )-8 c^2 d e x^3\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.61, size = 123, normalized size = 1.66 \begin {gather*} -\frac {2 \left (-8 a^2 e^2+4 a b d e-12 a b e^2 x-4 a c d^2-12 a c e^2 x^2+b^2 d^2+6 b^2 d e x-3 b^2 e^2 x^2+12 b c d e x^2-4 b c e^2 x^3+8 c^2 d e x^3\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 194, normalized size = 2.62 \begin {gather*} -\frac {2 \, {\left (4 \, a b d e - 8 \, a^{2} e^{2} + 4 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} + {\left (b^{2} - 4 \, a c\right )} d^{2} + 3 \, {\left (4 \, b c d e - {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} + 6 \, {\left (b^{2} d e - 2 \, a b e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 289, normalized size = 3.91 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {4 \, {\left (2 \, b^{2} c^{2} d e - 8 \, a c^{3} d e - b^{3} c e^{2} + 4 \, a b c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (4 \, b^{3} c d e - 16 \, a b c^{2} d e - b^{4} e^{2} + 16 \, a^{2} c^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {6 \, {\left (b^{4} d e - 4 \, a b^{2} c d e - 2 \, a b^{3} e^{2} + 8 \, a^{2} b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} + 4 \, a b^{3} d e - 16 \, a^{2} b c d e - 8 \, a^{2} b^{2} e^{2} + 32 \, a^{3} c e^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 123, normalized size = 1.66 \begin {gather*} -\frac {2 \left (4 b c \,e^{2} x^{3}-8 c^{2} d e \,x^{3}+12 a c \,e^{2} x^{2}+3 b^{2} e^{2} x^{2}-12 b c d e \,x^{2}+12 a b \,e^{2} x -6 b^{2} d e x +8 a^{2} e^{2}-4 a b d e +4 a c \,d^{2}-b^{2} d^{2}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (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 = 2.27, size = 122, normalized size = 1.65 \begin {gather*} -\frac {2\,\left (8\,a^2\,e^2-4\,a\,b\,d\,e+12\,a\,b\,e^2\,x+4\,a\,c\,d^2+12\,a\,c\,e^2\,x^2-b^2\,d^2-6\,b^2\,d\,e\,x+3\,b^2\,e^2\,x^2-12\,b\,c\,d\,e\,x^2+4\,b\,c\,e^2\,x^3-8\,c^2\,d\,e\,x^3\right )}{3\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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