Optimal. Leaf size=63 \[ \frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}-\frac{4 c d (d+e x)^{5/2}}{5 e^3} \]
[Out]
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Rubi [A] time = 0.0685973, antiderivative size = 63, 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 \[ \frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}-\frac{4 c d (d+e x)^{5/2}}{5 e^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 11.34, size = 60, normalized size = 0.95 \[ - \frac{4 c d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{3}} + \frac{2 c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )}{3 e^{3}} \]
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 = 0.0429811, size = 44, normalized size = 0.7 \[ \frac{2 (d+e x)^{3/2} \left (35 a e^2+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a + c*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 41, normalized size = 0.7 \[{\frac{30\,c{e}^{2}{x}^{2}-24\,cdex+70\,a{e}^{2}+16\,c{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
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 = 0.686245, size = 63, normalized size = 1. \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 42 \,{\left (e x + d\right )}^{\frac{5}{2}} c d + 35 \,{\left (c d^{2} + a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \]
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 = 0.206429, size = 84, normalized size = 1.33 \[ \frac{2 \,{\left (15 \, c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 8 \, c d^{3} + 35 \, a d e^{2} -{\left (4 \, c d^{2} e - 35 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \]
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 = 2.23284, size = 61, normalized size = 0.97 \[ \frac{2 \left (- \frac{2 c d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{2}} + \frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )}{3 e^{2}}\right )}{e} \]
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 = 0.210609, size = 82, normalized size = 1.3 \[ \frac{2}{105} \,{\left ({\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c e^{\left (-14\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*sqrt(e*x + d),x, algorithm="giac")
[Out]