3.306 \(\int \sqrt{a x+b \sqrt{c+\frac{a^2 x^2}{b^2}}} \, dx\)

Optimal. Leaf size=69 \[ \frac{\left (b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x\right )^{3/2}}{3 a}-\frac{b^2 c}{a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}} \]

[Out]

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

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Rubi [A]  time = 0.108216, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x\right )^{3/2}}{3 a}-\frac{b^2 c}{a \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} + c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.133149, size = 57, normalized size = 0.83 \[ \frac{2 \left (2 a x-b \sqrt{\frac{a^2 x^2}{b^2}+c}\right ) \sqrt{b \sqrt{\frac{a^2 x^2}{b^2}+c}+a x}}{3 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int \sqrt{ax+b\sqrt{c+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((a*x+b*(c+a^2*x^2/b^2)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + \sqrt{\frac{a^{2} x^{2}}{b^{2}} + c} b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x + sqrt(a^2*x^2/b^2 + c)*b), x)

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Fricas [A]  time = 0.331644, size = 130, normalized size = 1.88 \[ \frac{2 \,{\left (a^{2} x^{2} + a b x \sqrt{\frac{a^{2} x^{2} + b^{2} c}{b^{2}}} - b^{2} c\right )} \sqrt{a x + b \sqrt{\frac{a^{2} x^{2} + b^{2} c}{b^{2}}}}}{3 \,{\left (a^{2} x + a b \sqrt{\frac{a^{2} x^{2} + b^{2} c}{b^{2}}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} + c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x + \sqrt{\frac{a^{2} x^{2}}{b^{2}} + c} b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x + sqrt(a^2*x^2/b^2 + c)*b), x)