Optimal. Leaf size=276 \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
[Out]
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Rubi [A] time = 0.41236, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 64.6363, size = 272, normalized size = 0.99 \[ - \frac{6 b^{2} e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{6 b^{2} e^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{\left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{6 b e^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\left (d + e x\right ) \left (a e - b d\right )^{5}} - \frac{3 e^{2} \left (2 a + 2 b x\right )}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 e}{\left (d + e x\right )^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{2 a + 2 b x}{4 \left (d + e x\right )^{2} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
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Mathematica [A] time = 0.176027, size = 163, normalized size = 0.59 \[ \frac{(a+b x) \left (-12 b^2 e^2 (a+b x)^2 \log (d+e x)+6 b^2 e (a+b x) (b d-a e)+b^2 \left (-(b d-a e)^2\right )+12 b^2 e^2 (a+b x)^2 \log (a+b x)+\frac{6 b e^2 (a+b x)^2 (b d-a e)}{d+e x}+\frac{e^2 (a+b x)^2 (b d-a e)^2}{(d+e x)^2}\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.03, size = 508, normalized size = 1.8 \[ -{\frac{ \left ( 12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+24\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}-24\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}-48\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}-24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-24\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+48\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}-24\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}-12\,{x}^{3}a{b}^{3}{e}^{4}+24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}-8\,{a}^{3}bd{e}^{3}-12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+24\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-24\,x{a}^{2}{b}^{2}d{e}^{3}+24\,xa{b}^{3}{d}^{2}{e}^{2}+{a}^{4}{e}^{4}+4\,x{b}^{4}{d}^{3}e+12\,{x}^{3}{b}^{4}d{e}^{3}-18\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-4\,x{a}^{3}b{e}^{4}+24\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}-12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-{b}^{4}{d}^{4}+8\,a{b}^{3}{d}^{3}e-12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4} \right ) \left ( bx+a \right ) }{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^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(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="maxima")
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Fricas [A] time = 0.227974, size = 1026, normalized size = 3.72 \[ -\frac{b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \,{\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} +{\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \,{\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="fricas")
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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)**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.593425, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3),x, algorithm="giac")
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