Optimal. Leaf size=198 \[ -\frac{(d+e x)^{5/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
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Rubi [A] time = 0.351281, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^{5/2}}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (a+b x) \sqrt{d+e x} (b d-a e)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
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Mathematica [A] time = 0.310755, size = 121, normalized size = 0.61 \[ \frac{(a+b x) \left (\frac{\sqrt{d+e x} \left (-\frac{3 (b d-a e)^2}{a+b x}+2 e (7 b d-6 a e)+2 b e^2 x\right )}{3 b^3}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\right )}{\sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.024, size = 409, normalized size = 2.1 \[{\frac{ \left ( bx+a \right ) ^{2}}{3\,{b}^{3}} \left ( 2\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}x{b}^{2}e+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{a}^{2}b{e}^{3}-30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) xa{b}^{2}d{e}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) x{b}^{3}{d}^{2}e+2\,\sqrt{b \left ( ae-bd \right ) } \left ( ex+d \right ) ^{3/2}abe-12\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}xab{e}^{2}+12\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}x{b}^{2}de+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{3}{e}^{3}-30\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ){a}^{2}bd{e}^{2}+15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) a{b}^{2}{d}^{2}e-15\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{a}^{2}{e}^{2}+18\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}abde-3\,\sqrt{b \left ( ae-bd \right ) }\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^(5/2)/(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((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.334187, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),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((b*x+a)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.319987, size = 366, normalized size = 1.85 \[ -\frac{5 \,{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{\left (-1\right )}}{\sqrt{-b^{2} d + a b e} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{{\left (\sqrt{x e + d} b^{2} d^{2} e^{2} - 2 \, \sqrt{x e + d} a b d e^{3} + \sqrt{x e + d} a^{2} e^{4}\right )} e^{\left (-1\right )}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{4} + 6 \, \sqrt{x e + d} b^{4} d e^{4} - 6 \, \sqrt{x e + d} a b^{3} e^{5}\right )} e^{\left (-3\right )}}{3 \, b^{6}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]