Optimal. Leaf size=61 \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]
[Out]
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Rubi [A] time = 0.0674393, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 11.391, size = 58, normalized size = 0.95 \[ - \frac{4 c d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.035991, size = 44, normalized size = 0.72 \[ \frac{2 \sqrt{d+e x} \left (15 a e^2+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.006, size = 41, normalized size = 0.7 \[{\frac{6\,c{e}^{2}{x}^{2}-8\,cdex+30\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.694781, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, \sqrt{e x + d} a + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206401, size = 54, normalized size = 0.89 \[ \frac{2 \,{\left (3 \, c e^{2} x^{2} - 4 \, c d e x + 8 \, c d^{2} + 15 \, a e^{2}\right )} \sqrt{e x + d}}{15 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.06693, size = 150, normalized size = 2.46 \[ \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211023, size = 82, normalized size = 1.34 \[ \frac{2}{15} \,{\left ({\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} c e^{\left (-10\right )} + 15 \, \sqrt{x e + d} a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/sqrt(e*x + d),x, algorithm="giac")
[Out]