Optimal. Leaf size=125 \[ \frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5} \]
[Out]
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Rubi [A] time = 0.141421, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 25.0862, size = 121, normalized size = 0.97 \[ - \frac{8 c^{2} d \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} - \frac{8 c d \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )}{3 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + 3 c d^{2}\right )}{5 e^{5}} + \frac{2 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{2}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0749551, size = 96, normalized size = 0.77 \[ \frac{2 \sqrt{d+e x} \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^2/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.01, size = 106, normalized size = 0.9 \[{\frac{70\,{c}^{2}{x}^{4}{e}^{4}-80\,{c}^{2}d{x}^{3}{e}^{3}+252\,ac{e}^{4}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-336\,acd{e}^{3}x-128\,{c}^{2}{d}^{3}ex+630\,{a}^{2}{e}^{4}+672\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^2/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.69894, size = 162, normalized size = 1.3 \[ \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} + \frac{42 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} a c}{e^{2}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224643, size = 144, normalized size = 1.15 \[ \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} + 128 \, c^{2} d^{4} + 336 \, a c d^{2} e^{2} + 315 \, a^{2} e^{4} + 6 \,{\left (8 \, c^{2} d^{2} e^{2} + 21 \, a c e^{4}\right )} x^{2} - 8 \,{\left (8 \, c^{2} d^{3} e + 21 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 23.6713, size = 330, normalized size = 2.64 \[ \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 a c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**2/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212837, size = 192, normalized size = 1.54 \[ \frac{2}{315} \,{\left (42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a c e^{\left (-10\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} c^{2} e^{\left (-36\right )} + 315 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/sqrt(e*x + d),x, algorithm="giac")
[Out]