Optimal. Leaf size=131 \[ \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (b+2 c x) \left (4 a f+\frac {b^2 f}{c}-4 b e+8 c d\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1660, 12, 613} \[ \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (b+2 c x) \left (4 a f+\frac {b^2 f}{c}-4 b e+8 c d\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 613
Rule 1660
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {8 c d-4 b e+4 a f+\frac {b^2 f}{c}}{2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (8 c d-4 b e+4 a f+\frac {b^2 f}{c}\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (c \left (2 a e-b \left (d+\frac {a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 c d-4 b e+4 a f+\frac {b^2 f}{c}\right ) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 147, normalized size = 1.12 \[ \frac {8 b \left (2 a^2 f+3 a c \left (d-e x+f x^2\right )-2 c^2 x^2 (e x-3 d)\right )+16 c \left (-a^2 e+a c x \left (3 d+f x^2\right )+2 c^2 d x^3\right )-4 b^2 \left (a (e-6 f x)-c x \left (3 d-6 e x+f x^2\right )\right )-2 b^3 (d+3 x (e-f x))}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 4.02, size = 286, normalized size = 2.18 \[ \frac {2 \, {\left (8 \, a^{2} b f + 2 \, {\left (8 \, c^{3} d - 4 \, b c^{2} e + {\left (b^{2} c + 4 \, a c^{2}\right )} f\right )} x^{3} + 3 \, {\left (8 \, b c^{2} d - 4 \, b^{2} c e + {\left (b^{3} + 4 \, a b c\right )} f\right )} x^{2} - {\left (b^{3} - 12 \, a b c\right )} d - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \, {\left (4 \, a b^{2} f + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d - {\left (b^{3} + 4 \, a b c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 240, normalized size = 1.83 \[ \frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (8 \, c^{3} d + b^{2} c f + 4 \, a c^{2} f - 4 \, b c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d + b^{3} f + 4 \, a b c f - 4 \, b^{2} c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d + 8 \, a c^{2} d + 4 \, a b^{2} f - b^{3} e - 4 \, a b c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d - 12 \, a b c d - 8 \, a^{2} b f + 2 \, a b^{2} e + 8 \, a^{2} c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 185, normalized size = 1.41 \[ \frac {\frac {16}{3} a \,c^{2} f \,x^{3}+\frac {4}{3} b^{2} c f \,x^{3}-\frac {16}{3} b \,c^{2} e \,x^{3}+\frac {32}{3} c^{3} d \,x^{3}+8 a b c f \,x^{2}+2 b^{3} f \,x^{2}-8 b^{2} c e \,x^{2}+16 b \,c^{2} d \,x^{2}+8 a \,b^{2} f x -8 a b c e x +16 a \,c^{2} d x -2 b^{3} e x +4 b^{2} c d x +\frac {16}{3} a^{2} b f -\frac {16}{3} a^{2} c e -\frac {4}{3} a \,b^{2} e +8 a b c d -\frac {2}{3} b^{3} d}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )} \]
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 = 3.71, size = 175, normalized size = 1.34 \[ \frac {2\,\left (8\,f\,a^2\,b-8\,e\,a^2\,c+12\,f\,a\,b^2\,x-2\,e\,a\,b^2+12\,f\,a\,b\,c\,x^2-12\,e\,a\,b\,c\,x+12\,d\,a\,b\,c+8\,f\,a\,c^2\,x^3+24\,d\,a\,c^2\,x+3\,f\,b^3\,x^2-3\,e\,b^3\,x-d\,b^3+2\,f\,b^2\,c\,x^3-12\,e\,b^2\,c\,x^2+6\,d\,b^2\,c\,x-8\,e\,b\,c^2\,x^3+24\,d\,b\,c^2\,x^2+16\,d\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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