Optimal. Leaf size=388 \[ -\frac{21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac{21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac{7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac{21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac{3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]
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Rubi [A] time = 1.08024, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac{21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac{7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac{21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac{3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
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Rubi in Sympy [A] time = 124.803, size = 371, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{6 c d} - \frac{3 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{20 c^{2} d^{2}} + \frac{21 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{160 c^{3} d^{3}} - \frac{7 \left (a e^{2} - c d^{2}\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{64 c^{4} d^{4}} + \frac{21 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 c^{5} d^{5} e} - \frac{21 \left (a e^{2} - c d^{2}\right )^{6} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 c^{\frac{11}{2}} d^{\frac{11}{2}} e^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.768745, size = 393, normalized size = 1.01 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{630 a^5 e^9}{c^5 d^5}-\frac{3570 a^4 e^7}{c^4 d^3}+\frac{8316 a^3 e^5}{c^3 d}+\frac{32 e^2 x^3 \left (-9 a^2 e^4+50 a c d^2 e^2+759 c^2 d^4\right )}{c^2 d^2}-\frac{10116 a^2 d e^3}{c^2}+\frac{16 e x^2 \left (21 a^3 e^6-117 a^2 c d^2 e^4+267 a c^2 d^4 e^2+1429 c^3 d^6\right )}{c^3 d^3}+\frac{4 x \left (-105 a^4 e^8+588 a^3 c d^2 e^6-1350 a^2 c^2 d^4 e^4+1612 a c^3 d^6 e^2+2455 c^4 d^8\right )}{c^4 d^4}-\frac{315 \left (c d^2-a e^2\right )^6 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{11/2} d^{11/2} e^{3/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{6670 a d^3 e}{c}+\frac{256 e^3 x^4 \left (a e^2+49 c d^2\right )}{c d}+\frac{630 d^5}{e}+2560 e^4 x^5\right )}{15360} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
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Maple [B] time = 0.035, size = 1327, normalized size = 3.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289886, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.250624, size = 655, normalized size = 1.69 \[ \frac{1}{7680} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, x e^{4} + \frac{{\left (49 \, c^{5} d^{6} e^{8} + a c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (759 \, c^{5} d^{7} e^{7} + 50 \, a c^{4} d^{5} e^{9} - 9 \, a^{2} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (1429 \, c^{5} d^{8} e^{6} + 267 \, a c^{4} d^{6} e^{8} - 117 \, a^{2} c^{3} d^{4} e^{10} + 21 \, a^{3} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (2455 \, c^{5} d^{9} e^{5} + 1612 \, a c^{4} d^{7} e^{7} - 1350 \, a^{2} c^{3} d^{5} e^{9} + 588 \, a^{3} c^{2} d^{3} e^{11} - 105 \, a^{4} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (315 \, c^{5} d^{10} e^{4} + 3335 \, a c^{4} d^{8} e^{6} - 5058 \, a^{2} c^{3} d^{6} e^{8} + 4158 \, a^{3} c^{2} d^{4} e^{10} - 1785 \, a^{4} c d^{2} e^{12} + 315 \, a^{5} e^{14}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac{21 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^4,x, algorithm="giac")
[Out]