Optimal. Leaf size=124 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.214962, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac{5 b e}{\sqrt{d+e x} (b d-a e)^3}-\frac{1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac{5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]
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Rubi in Sympy [A] time = 51.3248, size = 109, normalized size = 0.88 \[ \frac{5 b^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{7}{2}}} + \frac{5 b e}{\sqrt{d + e x} \left (a e - b d\right )^{3}} - \frac{5 e}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2),x)
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Mathematica [A] time = 0.275198, size = 125, normalized size = 1.01 \[ \frac{5 b^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}}+\sqrt{d+e x} \left (-\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{4 b e}{(d+e x) (b d-a e)^3}-\frac{2 e}{3 (d+e x)^2 (b d-a e)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Maple [A] time = 0.027, size = 125, normalized size = 1. \[ -{\frac{2\,e}{3\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{be}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3} \left ( bex+ae \right ) }\sqrt{ex+d}}+5\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2),x)
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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)*(e*x + d)^(5/2)),x, algorithm="maxima")
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Fricas [A] time = 0.22775, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 28 \, a b d e - 4 \, a^{2} e^{2} + 15 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \,{\left (b d - a e\right )} \sqrt{e x + d} \sqrt{\frac{b}{b d - a e}}}{b x + a}\right ) + 20 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{6 \,{\left (a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} - 15 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d} \sqrt{-\frac{b}{b d - a e}} \arctan \left (-\frac{{\left (b d - a e\right )} \sqrt{-\frac{b}{b d - a e}}}{\sqrt{e x + d} b}\right ) + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(5/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2),x)
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GIAC/XCAS [A] time = 0.215269, size = 302, normalized size = 2.44 \[ -\frac{5 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{\sqrt{x e + d} b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} b e + b d e - a e^{2}\right )}}{3 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^(5/2)),x, algorithm="giac")
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