Optimal. Leaf size=67 \[ -\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x}}{e^3}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} -\frac {4 b \sqrt {d+e x} (b d-a e)}{e^3}-\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^{3/2}}-\frac {2 b (b d-a e)}{e^2 \sqrt {d+e x}}+\frac {b^2 \sqrt {d+e x}}{e^2}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2}{e^3 \sqrt {d+e x}}-\frac {4 b (b d-a e) \sqrt {d+e x}}{e^3}+\frac {2 b^2 (d+e x)^{3/2}}{3 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 59, normalized size = 0.88 \begin {gather*} \frac {2 \left (-3 a^2 e^2+6 a b e (2 d+e x)+b^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 74, normalized size = 1.10
method | result | size |
risch | \(\frac {2 b \left (b e x +6 a e -5 b d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{3} \sqrt {e x +d}}\) | \(61\) |
gosper | \(-\frac {2 \left (-b^{2} x^{2} e^{2}-6 a b \,e^{2} x +4 b^{2} d e x +3 a^{2} e^{2}-12 a b d e +8 b^{2} d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(63\) |
trager | \(-\frac {2 \left (-b^{2} x^{2} e^{2}-6 a b \,e^{2} x +4 b^{2} d e x +3 a^{2} e^{2}-12 a b d e +8 b^{2} d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\) | \(63\) |
derivativedivides | \(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a b e \sqrt {e x +d}-4 b^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(74\) |
default | \(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 a b e \sqrt {e x +d}-4 b^{2} d \sqrt {e x +d}-\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 76, normalized size = 1.13 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{2} - 6 \, {\left (b^{2} d - a b e\right )} \sqrt {x e + d}\right )} e^{\left (-2\right )} - \frac {3 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-2\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 68, normalized size = 1.01 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} d^{2} - {\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )} e^{2} + 4 \, {\left (b^{2} d x - 3 \, a b d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x e^{4} + d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.95, size = 65, normalized size = 0.97 \begin {gather*} \frac {2 b^{2} \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} + \frac {\sqrt {d + e x} \left (4 a b e - 4 b^{2} d\right )}{e^{3}} - \frac {2 \left (a e - b d\right )^{2}}{e^{3} \sqrt {d + e x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 83, normalized size = 1.24 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{6} - 6 \, \sqrt {x e + d} b^{2} d e^{6} + 6 \, \sqrt {x e + d} a b e^{7}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 67, normalized size = 1.00 \begin {gather*} \frac {\frac {2\,b^2\,{\left (d+e\,x\right )}^2}{3}-2\,a^2\,e^2-2\,b^2\,d^2-4\,b^2\,d\,\left (d+e\,x\right )+4\,a\,b\,e\,\left (d+e\,x\right )+4\,a\,b\,d\,e}{e^3\,\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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