Optimal. Leaf size=202 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^4 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^4 (a+b x)} \]
[Out]
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Rubi [A] time = 0.205186, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^4 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) \sqrt{d+e x}}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^4 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 22.4303, size = 172, normalized size = 0.85 \[ \frac{4 b \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{2}} + \frac{16 b \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{3}} + \frac{32 b \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e^{4} \left (a + b x\right )} - \frac{2 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.120132, size = 119, normalized size = 0.59 \[ -\frac{2 \sqrt{(a+b x)^2} \left (5 a^3 e^3-15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left (8 d^2+4 d e x-e^2 x^2\right )+b^3 \left (-\left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{5 e^4 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 132, normalized size = 0.7 \[ -{\frac{-2\,{x}^{3}{b}^{3}{e}^{3}-10\,{x}^{2}a{b}^{2}{e}^{3}+4\,{x}^{2}{b}^{3}d{e}^{2}-30\,x{a}^{2}b{e}^{3}+40\,xa{b}^{2}d{e}^{2}-16\,x{b}^{3}{d}^{2}e+10\,{a}^{3}{e}^{3}-60\,{a}^{2}bd{e}^{2}+80\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{5\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.744554, size = 154, normalized size = 0.76 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{5 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20715, size = 154, normalized size = 0.76 \[ \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{5 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219878, size = 284, normalized size = 1.41 \[ \frac{2}{5} \,{\left ({\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{16}{\rm sign}\left (b x + a\right ) - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{16}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} b^{3} d^{2} e^{16}{\rm sign}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{17}{\rm sign}\left (b x + a\right ) - 30 \, \sqrt{x e + d} a b^{2} d e^{17}{\rm sign}\left (b x + a\right ) + 15 \, \sqrt{x e + d} a^{2} b e^{18}{\rm sign}\left (b x + a\right )\right )} e^{\left (-20\right )} + \frac{2 \,{\left (b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]