Optimal. Leaf size=127 \[ \frac{4 c (d+e x)^{7/2} \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{8 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^5}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{9/2}}{9 e^5} \]
[Out]
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Rubi [A] time = 0.136391, antiderivative size = 127, 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 \[ \frac{4 c (d+e x)^{7/2} \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{8 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^5}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{9/2}}{9 e^5} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 25.603, size = 122, normalized size = 0.96 \[ - \frac{8 c^{2} d \left (d + e x\right )^{\frac{9}{2}}}{9 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{5}} - \frac{8 c d \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )}{5 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} + 3 c d^{2}\right )}{7 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{5}} \]
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 = 0.0872798, size = 96, normalized size = 0.76 \[ \frac{2 (d+e x)^{3/2} \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 106, normalized size = 0.8 \[{\frac{630\,{c}^{2}{x}^{4}{e}^{4}-560\,{c}^{2}d{x}^{3}{e}^{3}+1980\,ac{e}^{4}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-1584\,acd{e}^{3}x-384\,{c}^{2}{d}^{3}ex+2310\,{a}^{2}{e}^{4}+1056\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
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 = 0.705124, size = 153, normalized size = 1.2 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} - 1540 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{2} d + 990 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 2772 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229043, size = 193, normalized size = 1.52 \[ \frac{2 \,{\left (315 \, c^{2} e^{5} x^{5} + 35 \, c^{2} d e^{4} x^{4} + 128 \, c^{2} d^{5} + 528 \, a c d^{3} e^{2} + 1155 \, a^{2} d e^{4} - 10 \,{\left (4 \, c^{2} d^{2} e^{3} - 99 \, a c e^{5}\right )} x^{3} + 6 \,{\left (8 \, c^{2} d^{3} e^{2} + 33 \, a c d e^{4}\right )} x^{2} -{\left (64 \, c^{2} d^{4} e + 264 \, a c d^{2} e^{3} - 1155 \, a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \]
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 = 2.4285, size = 148, normalized size = 1.17 \[ \frac{2 \left (- \frac{4 c^{2} d \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 4 a c d e^{2} - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \]
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 = 0.213348, size = 192, normalized size = 1.51 \[ \frac{2}{3465} \,{\left (66 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a c e^{\left (-14\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c^{2} e^{\left (-44\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \]
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]