Optimal. Leaf size=64 \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]
[Out]
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Rubi [A] time = 0.0945966, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \sqrt{d+e x} (2 c d-b e)}{e^3}-\frac{2 d (c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 13.351, size = 60, normalized size = 0.94 \[ \frac{2 c \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 d \left (b e - c d\right )}{e^{3} \sqrt{d + e x}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0531498, size = 48, normalized size = 0.75 \[ \frac{2 \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 46, normalized size = 0.7 \[{\frac{2\,c{e}^{2}{x}^{2}+6\,b{e}^{2}x-8\,cdex+12\,bde-16\,c{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.695724, size = 82, normalized size = 1.28 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (c d^{2} - b d e\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217156, size = 63, normalized size = 0.98 \[ \frac{2 \,{\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e -{\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.731, size = 576, normalized size = 9. \[ b \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + c \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.205924, size = 93, normalized size = 1.45 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c e^{6} - 6 \, \sqrt{x e + d} c d e^{6} + 3 \, \sqrt{x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (c d^{2} - b d e\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]