Optimal. Leaf size=219 \[ \frac{2 b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 b (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac{2 (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
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Rubi [A] time = 0.292409, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{2 b (a+b x)}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}+\frac{2 (a+b x)}{5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}-\frac{2 b^{5/2} (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^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(1/(e*x+d)**(7/2)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.470931, size = 133, normalized size = 0.61 \[ \frac{(a+b x) \left (\frac{6 a^2 e^2-2 a b e (11 d+5 e x)+b^2 \left (46 d^2+70 d e x+30 e^2 x^2\right )}{15 (d+e x)^{5/2} (b d-a e)^3}-\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}\right )}{\sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Maple [A] time = 0.017, size = 202, normalized size = 0.9 \[ -{\frac{2\,bx+2\,a}{15\, \left ( ae-bd \right ) ^{3}} \left ( 15\,{b}^{3}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{5/2}+15\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{2}-5\,\sqrt{b \left ( ae-bd \right ) }xab{e}^{2}+35\,\sqrt{b \left ( ae-bd \right ) }x{b}^{2}de+3\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{2}-11\,\sqrt{b \left ( ae-bd \right ) }abde+23\,\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(7/2)/((b*x+a)^2)^(1/2),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/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="maxima")
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Fricas [A] time = 0.222821, size = 1, normalized size = 0. \[ \left [\frac{30 \, b^{2} e^{2} x^{2} + 46 \, b^{2} d^{2} - 22 \, a b d e + 6 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 10 \,{\left (7 \, b^{2} d e - a b e^{2}\right )} x}{15 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 11 \, a b d e + 3 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 5 \,{\left (7 \, b^{2} d e - a b e^{2}\right )} x\right )}}{15 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),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)**(7/2)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218873, size = 265, normalized size = 1.21 \[ \frac{2}{15} \,{\left (\frac{15 \, b^{3} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} + \frac{15 \,{\left (x e + d\right )}^{2} b^{2} + 5 \,{\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} - 5 \,{\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{5}{2}}}\right )}{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(7/2)),x, algorithm="giac")
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