Optimal. Leaf size=128 \[ -\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} (2 c d-b e)}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^{5/2}}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)^{3/2}}-\frac {3 c (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c^2 \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^4 \sqrt {d+e x}}-\frac {6 c (2 c d-b e) \sqrt {d+e x}}{e^4}+\frac {4 c^2 (d+e x)^{3/2}}{3 e^4}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 108, normalized size = 0.84 \begin {gather*} -\frac {2 \left (c e \left (2 a e (2 d+3 e x)-3 b \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+b e^2 (a e+2 b d+3 b e x)+2 c^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 142, normalized size = 1.11 \begin {gather*} \frac {2 \left (-a b e^3-6 a c e^2 (d+e x)+2 a c d e^2-3 b^2 e^2 (d+e x)+b^2 d e^2-3 b c d^2 e+18 b c d e (d+e x)+9 b c e (d+e x)^2+2 c^2 d^3-18 c^2 d^2 (d+e x)-18 c^2 d (d+e x)^2+2 c^2 (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 137, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (2 \, c^{2} e^{3} x^{3} - 32 \, c^{2} d^{3} + 24 \, b c d^{2} e - a b e^{3} - 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (4 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} - 3 \, {\left (16 \, c^{2} d^{2} e - 12 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 153, normalized size = 1.20 \begin {gather*} \frac {2}{3} \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} e^{8} - 18 \, \sqrt {x e + d} c^{2} d e^{8} + 9 \, \sqrt {x e + d} b c e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (18 \, {\left (x e + d\right )} c^{2} d^{2} - 2 \, c^{2} d^{3} - 18 \, {\left (x e + d\right )} b c d e + 3 \, b c d^{2} e + 3 \, {\left (x e + d\right )} b^{2} e^{2} + 6 \, {\left (x e + d\right )} a c e^{2} - b^{2} d e^{2} - 2 \, a c d e^{2} + a b e^{3}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 122, normalized size = 0.95 \begin {gather*} -\frac {2 \left (-2 c^{2} x^{3} e^{3}-9 b c \,e^{3} x^{2}+12 c^{2} d \,e^{2} x^{2}+6 a c \,e^{3} x +3 b^{2} e^{3} x -36 b c d \,e^{2} x +48 c^{2} d^{2} e x +a b \,e^{3}+4 a c d \,e^{2}+2 b^{2} d \,e^{2}-24 b c \,d^{2} e +32 c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 126, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (\frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} - 9 \, {\left (2 \, c^{2} d - b c e\right )} \sqrt {e x + d}}{e^{3}} + \frac {2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 139, normalized size = 1.09 \begin {gather*} \frac {4\,c^2\,{\left (d+e\,x\right )}^3+4\,c^2\,d^3+2\,b^2\,d\,e^2-6\,b^2\,e^2\,\left (d+e\,x\right )-36\,c^2\,d\,{\left (d+e\,x\right )}^2-36\,c^2\,d^2\,\left (d+e\,x\right )-2\,a\,b\,e^3+4\,a\,c\,d\,e^2-6\,b\,c\,d^2\,e-12\,a\,c\,e^2\,\left (d+e\,x\right )+18\,b\,c\,e\,{\left (d+e\,x\right )}^2+36\,b\,c\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 536, normalized size = 4.19 \begin {gather*} \begin {cases} - \frac {2 a b e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 a c d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 a c e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 b^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 b^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 b c d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {72 b c d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {18 b c e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {64 c^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {96 c^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {24 c^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {4 c^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a b x + a c x^{2} + \frac {b^{2} x^{2}}{2} + b c x^{3} + \frac {c^{2} x^{4}}{2}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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