Optimal. Leaf size=96 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^2 (a+b x)} \]
[Out]
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Rubi [A] time = 0.115236, antiderivative size = 96, 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 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 11.8818, size = 80, normalized size = 0.83 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 e} + \frac{4 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{2} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0620831, size = 48, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} (7 a e-2 b d+5 b e x)}{35 e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 43, normalized size = 0.5 \[{\frac{10\,bex+14\,ae-4\,bd}{35\,{e}^{2} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.760082, size = 93, normalized size = 0.97 \[ \frac{2 \,{\left (5 \, b e^{3} x^{3} - 2 \, b d^{3} + 7 \, a d^{2} e +{\left (8 \, b d e^{2} + 7 \, a e^{3}\right )} x^{2} +{\left (b d^{2} e + 14 \, a d e^{2}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205527, size = 93, normalized size = 0.97 \[ \frac{2 \,{\left (5 \, b e^{3} x^{3} - 2 \, b d^{3} + 7 \, a d^{2} e +{\left (8 \, b d e^{2} + 7 \, a e^{3}\right )} x^{2} +{\left (b d^{2} e + 14 \, a d e^{2}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^(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((e*x+d)**(3/2)*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216423, size = 188, normalized size = 1.96 \[ \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b d e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a d{\rm sign}\left (b x + a\right ) +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b e^{\left (-13\right )}{\rm sign}\left (b x + a\right ) + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]