3.2124 \(\int \frac{(a+b x) (d+e x)^{3/2}}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(a + b*x)*(d + e*x)^(5/2))/(5*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.11803, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(d + e*x)^(3/2))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(a + b*x)*(d + e*x)^(5/2))/(5*e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 15.9546, size = 37, normalized size = 0.9 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}}{5 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(5/2)/(5*e*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0398465, size = 32, normalized size = 0.78 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(d + e*x)^(3/2))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(a + b*x)*(d + e*x)^(5/2))/(5*e*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.005, size = 27, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{5\,e} \left ( ex+d \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(3/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/5*(b*x+a)*(e*x+d)^(5/2)/e/((b*x+a)^2)^(1/2)

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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)^(3/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279649, size = 38, normalized size = 0.93 \[ \frac{2 \,{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(3/2)/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

2/5*(e^2*x^2 + 2*d*e*x + d^2)*sqrt(e*x + d)/e

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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)**(3/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.291623, size = 69, normalized size = 1.68 \[ \frac{2}{15} \,{\left (5 \,{\left (x e + d\right )}^{\frac{3}{2}} d{\rm sign}\left (b x + a\right ) +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x + d)^(3/2)/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

2/15*(5*(x*e + d)^(3/2)*d*sign(b*x + a) + (3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)
*d)*sign(b*x + a))*e^(-1)