3.1668 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.117001, antiderivative size = 94, 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 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4934, size = 80, normalized size = 0.85 \[ \frac{2 \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e} + \frac{4 \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e) + 4*sqrt(d + e*x)*(a*e -
b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3*e**2*(a + b*x))

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Mathematica [A]  time = 0.0338062, size = 47, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} (3 a e-2 b d+b e x)}{3 e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 42, normalized size = 0.5 \[{\frac{2\,bex+6\,ae-4\,bd}{3\, \left ( bx+a \right ){e}^{2}}\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.759657, size = 62, normalized size = 0.66 \[ \frac{2 \,{\left (b e^{2} x^{2} - 2 \, b d^{2} + 3 \, a d e -{\left (b d e - 3 \, a e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*e^2*x^2 - 2*b*d^2 + 3*a*d*e - (b*d*e - 3*a*e^2)*x)/(sqrt(e*x + d)*e^2)

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Fricas [A]  time = 0.206145, size = 34, normalized size = 0.36 \[ \frac{2 \,{\left (b e x - 2 \, b d + 3 \, a e\right )} \sqrt{e x + d}}{3 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*e*x - 2*b*d + 3*a*e)*sqrt(e*x + d)/e^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x\right )^{2}}}{\sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211958, size = 70, normalized size = 0.74 \[ \frac{2}{3} \,{\left ({\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 3 \, \sqrt{x e + d} a{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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