Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3} \]
[Out]
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Rubi [A] time = 0.0791865, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 30.8951, size = 65, normalized size = 0.92 \[ \frac{2 b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{4 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )}{7 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}}{5 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0797414, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{5/2} \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.01, size = 63, normalized size = 0.9 \[{\frac{70\,{x}^{2}{b}^{2}{e}^{2}+180\,xab{e}^{2}-40\,x{b}^{2}de+126\,{a}^{2}{e}^{2}-72\,abde+16\,{b}^{2}{d}^{2}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.729302, size = 92, normalized size = 1.3 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{2} - 90 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207659, size = 185, normalized size = 2.61 \[ \frac{2 \,{\left (35 \, b^{2} e^{4} x^{4} + 8 \, b^{2} d^{4} - 36 \, a b d^{3} e + 63 \, a^{2} d^{2} e^{2} + 10 \,{\left (5 \, b^{2} d e^{3} + 9 \, a b e^{4}\right )} x^{3} + 3 \,{\left (b^{2} d^{2} e^{2} + 48 \, a b d e^{3} + 21 \, a^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} d^{3} e - 9 \, a b d^{2} e^{2} - 63 \, a^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.44277, size = 240, normalized size = 3.38 \[ a^{2} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 a b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.224514, size = 315, normalized size = 4.44 \[ \frac{2}{315} \,{\left (42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b d e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b^{2} d e^{\left (-14\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a b e^{\left (-13\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b^{2} e^{\left (-26\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]