Optimal. Leaf size=401 \[ \frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]
[Out]
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Rubi [A] time = 0.912346, antiderivative size = 401, 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 \[ \frac{9 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{11/2} d^{11/2} e^{5/2}}-\frac{9 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^5 d^5 e^2}+\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{128 c^4 d^4 e}+\frac{3 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{40 c^3 d^3}+\frac{3 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{28 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 120.151, size = 386, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 c d} - \frac{3 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{28 c^{2} d^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{40 c^{3} d^{3}} - \frac{3 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{128 c^{4} d^{4} e} + \frac{9 \left (a e^{2} - c d^{2}\right )^{5} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{1024 c^{5} d^{5} e^{2}} - \frac{9 \left (a e^{2} - c d^{2}\right )^{7} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2048 c^{\frac{11}{2}} d^{\frac{11}{2}} e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 1.38356, size = 450, normalized size = 1.12 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{2 \left (315 a^6 e^{12}-210 a^5 c d e^{10} (10 d+e x)+21 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )-12 a^3 c^3 d^3 e^6 \left (768 d^3+323 d^2 e x+92 d e^2 x^2+12 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (8393 d^4+5924 d^3 e x+3072 d^2 e^2 x^2+944 d e^3 x^3+128 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (1050 d^5+13643 d^4 e x+30248 d^3 e^2 x^2+31272 d^2 e^3 x^3+15872 d e^4 x^4+3200 e^5 x^5\right )+c^6 d^6 \left (-315 d^6+210 d^5 e x+14168 d^4 e^2 x^2+39056 d^3 e^3 x^3+44928 d^2 e^4 x^4+24320 d e^5 x^5+5120 e^6 x^6\right )\right )}{35 c^5 d^5 e^2 (d+e x) (a e+c d x)}+\frac{9 \left (c d^2-a e^2\right )^7 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{11/2} d^{11/2} e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}\right )}{2048} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.018, size = 1586, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.316669, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3,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)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.259345, size = 824, normalized size = 2.05 \[ \frac{1}{35840} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c d x e^{4} + \frac{{\left (19 \, c^{7} d^{8} e^{9} + 5 \, a c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (351 \, c^{7} d^{9} e^{8} + 248 \, a c^{6} d^{7} e^{10} + a^{2} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (2441 \, c^{7} d^{10} e^{7} + 3909 \, a c^{6} d^{8} e^{9} + 59 \, a^{2} c^{5} d^{6} e^{11} - 9 \, a^{3} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (1771 \, c^{7} d^{11} e^{6} + 7562 \, a c^{6} d^{9} e^{8} + 384 \, a^{2} c^{5} d^{7} e^{10} - 138 \, a^{3} c^{4} d^{5} e^{12} + 21 \, a^{4} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac{{\left (105 \, c^{7} d^{12} e^{5} + 13643 \, a c^{6} d^{10} e^{7} + 2962 \, a^{2} c^{5} d^{8} e^{9} - 1938 \, a^{3} c^{4} d^{6} e^{11} + 693 \, a^{4} c^{3} d^{4} e^{13} - 105 \, a^{5} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac{{\left (315 \, c^{7} d^{13} e^{4} - 2100 \, a c^{6} d^{11} e^{6} - 8393 \, a^{2} c^{5} d^{9} e^{8} + 9216 \, a^{3} c^{4} d^{7} e^{10} - 5943 \, a^{4} c^{3} d^{5} e^{12} + 2100 \, a^{5} c^{2} d^{3} e^{14} - 315 \, a^{6} c d e^{16}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} - \frac{9 \,{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2048 \, c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3,x, algorithm="giac")
[Out]