Optimal. Leaf size=252 \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.336548, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 56.3197, size = 245, normalized size = 0.97 \[ \frac{128 c^{3} d^{3} e \left (2 a e^{2} + 2 c d^{2} + 4 c d e x\right )}{21 \left (a e^{2} - c d^{2}\right )^{6} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{32 c^{2} d^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right )}{21 \left (a e^{2} - c d^{2}\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} + \frac{4 c d}{7 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{2}{7 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.90154, size = 203, normalized size = 0.81 \[ \frac{2 (d+e x)^3 (a e+c d x)^3 \left (\frac{98 c^4 d^4 e}{a e+c d x}+\frac{7 c^4 d^4 \left (a e^2-c d^2\right )}{(a e+c d x)^2}+\frac{37 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{12 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^3}-\frac{3 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^4}+\frac{158 c^3 d^3 e^2}{d+e x}\right )}{21 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.022, size = 412, normalized size = 1.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}-384\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}-896\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-96\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}-1344\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}-1120\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-336\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}-1680\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}-560\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-6\,{a}^{4}cd{e}^{9}x+56\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-420\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x-840\,a{c}^{4}{d}^{7}{e}^{3}x-70\,{c}^{5}{d}^{9}ex+3\,{a}^{5}{e}^{10}-21\,{a}^{4}c{d}^{2}{e}^{8}+70\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}-105\,a{c}^{4}{d}^{8}{e}^{2}+7\,{c}^{5}{d}^{10} \right ) }{ \left ( 21\,ex+21\,d \right ) \left ({a}^{6}{e}^{12}-6\,{a}^{5}c{d}^{2}{e}^{10}+15\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-20\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+15\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}-6\,a{c}^{5}{d}^{10}{e}^{2}+{c}^{6}{d}^{12} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 17.1878, size = 1428, normalized size = 5.67 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^2),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(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^2),x, algorithm="giac")
[Out]