Optimal. Leaf size=59 \[ -\frac {4 e (b+2 c x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 59, 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 = {768, 613} \begin {gather*} -\frac {4 e (b+2 c x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} (2 e) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (d+e x)}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e (b+2 c 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 = 0.20, size = 64, normalized size = 1.08 \begin {gather*} -\frac {2 \left (2 b e \left (a+3 c x^2\right )+4 c \left (c e x^3-a d\right )+b^2 (d+3 e x)\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 = 1.05, size = 68, normalized size = 1.15 \begin {gather*} -\frac {2 \left (2 a b e-4 a c d+b^2 d+3 b^2 e x+6 b c e x^2+4 c^2 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 = 1.16, size = 145, normalized size = 2.46 \begin {gather*} -\frac {2 \, {\left (4 \, c^{2} e x^{3} + 6 \, b c e x^{2} + 3 \, b^{2} e x + 2 \, a b e + {\left (b^{2} - 4 \, a c\right )} d\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 = 0.28, size = 205, normalized size = 3.47 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, {\left (\frac {2 \, {\left (b^{2} c^{2} e - 4 \, a c^{3} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (b^{3} c e - 4 \, a b c^{2} e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (b^{4} e - 4 \, a b^{2} c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d + 2 \, a b^{3} e - 8 \, a^{2} b c e}{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 = 0.05, size = 67, normalized size = 1.14 \begin {gather*} \frac {\frac {8}{3} c^{2} e \,x^{3}+4 b c e \,x^{2}+2 b^{2} e x +\frac {4}{3} a b e -\frac {8}{3} a c d +\frac {2}{3} b^{2} d}{\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 = 2.06, size = 66, normalized size = 1.12 \begin {gather*} \frac {2\,\left (3\,e\,b^2\,x+d\,b^2+6\,e\,b\,c\,x^2+2\,a\,e\,b+4\,e\,c^2\,x^3-4\,a\,d\,c\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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