Optimal. Leaf size=201 \[ -\frac{(d+e x)^4}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{4 e^3 (a+b x) (b d-a e) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e^2 (b d-a e)^2}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^3}{3 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e^4 x (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.388421, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(d+e x)^4}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{4 e^3 (a+b x) (b d-a e) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e^2 (b d-a e)^2}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^3}{3 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 e^4 x (a+b x)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 37.7564, size = 184, normalized size = 0.92 \[ - \frac{\left (d + e x\right )^{4}}{3 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{e \left (2 a + 2 b x\right ) \left (d + e x\right )^{3}}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2 e^{2} \left (d + e x\right )^{2}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{4 e^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{5}} - \frac{4 e^{3} \left (a + b x\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.181819, size = 170, normalized size = 0.85 \[ \frac{-13 a^4 e^4+a^3 b e^3 (22 d-27 e x)-3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a b^3 e \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 (a+b x)^3 (a e-b d) \log (a+b x)+b^4 \left (-\left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )}{3 b^5 \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.022, size = 322, normalized size = 1.6 \[ -{\frac{ \left ( 12\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-12\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}-3\,{x}^{4}{b}^{4}{e}^{4}+36\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-36\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}-9\,{x}^{3}a{b}^{3}{e}^{4}+36\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-36\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}+9\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-36\,{x}^{2}a{b}^{3}d{e}^{3}+18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-12\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+27\,x{a}^{3}b{e}^{4}-54\,x{a}^{2}{b}^{2}d{e}^{3}+18\,xa{b}^{3}{d}^{2}{e}^{2}+6\,x{b}^{4}{d}^{3}e+13\,{a}^{4}{e}^{4}-22\,{a}^{3}bd{e}^{3}+6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+2\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) \left ( bx+a \right ) ^{2}}{3\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.809647, size = 1149, normalized size = 5.72 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287209, size = 394, normalized size = 1.96 \[ \frac{3 \, b^{4} e^{4} x^{4} + 9 \, a b^{3} e^{4} x^{3} - b^{4} d^{4} - 2 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} + 22 \, a^{3} b d e^{3} - 13 \, a^{4} e^{4} - 9 \,{\left (2 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} - 3 \,{\left (2 \, b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 18 \, a^{2} b^{2} d e^{3} + 9 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{4}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]