3.300 \(\int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{3/2} \, dx\)

Optimal. Leaf size=229 \[ \frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{5 e}-\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 d e}+\frac{3 a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e}-\frac{3 a \sqrt{d} f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 e} \]

[Out]

(3*a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e) + (a*f^2*(d + e*x + f*
Sqrt[a + (e^2*x^2)/f^2])^(3/2))/(2*d*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^
(5/2)/(5*e) - (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2))/(2*d*e*(e*x +
f*Sqrt[a + (e^2*x^2)/f^2])) - (3*a*Sqrt[d]*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a +
(e^2*x^2)/f^2]]/Sqrt[d]])/(2*e)

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Rubi [A]  time = 0.440032, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{5 e}-\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{2 d e}+\frac{3 a f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e}-\frac{3 a \sqrt{d} f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

(3*a*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e) + (a*f^2*(d + e*x + f*
Sqrt[a + (e^2*x^2)/f^2])^(3/2))/(2*d*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^
(5/2)/(5*e) - (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2))/(2*d*e*(e*x +
f*Sqrt[a + (e^2*x^2)/f^2])) - (3*a*Sqrt[d]*f^2*ArcTanh[Sqrt[d + e*x + f*Sqrt[a +
(e^2*x^2)/f^2]]/Sqrt[d]])/(2*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(3/2), x)

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Mathematica [A]  time = 0.222256, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{3/2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(3/2), x)

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Fricas [A]  time = 0.341659, size = 1, normalized size = 0. \[ \left [\frac{15 \, a \sqrt{d} f^{2} \log \left (\frac{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} - 2 \, \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt{d} + 2 \, d}{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) + 2 \,{\left (4 \, e^{2} x^{2} + 12 \, a f^{2} + 9 \, d e x + 2 \, d^{2} +{\left (4 \, e f x - d f\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{20 \, e}, -\frac{15 \, a \sqrt{-d} f^{2} \arctan \left (\frac{\sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{\sqrt{-d}}\right ) -{\left (4 \, e^{2} x^{2} + 12 \, a f^{2} + 9 \, d e x + 2 \, d^{2} +{\left (4 \, e f x - d f\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{10 \, e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/20*(15*a*sqrt(d)*f^2*log((e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) - 2*sqrt(e*x +
f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)*sqrt(d) + 2*d)/(e*x + f*sqrt((e^2*x^2 + a*f^2
)/f^2))) + 2*(4*e^2*x^2 + 12*a*f^2 + 9*d*e*x + 2*d^2 + (4*e*f*x - d*f)*sqrt((e^2
*x^2 + a*f^2)/f^2))*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/e, -1/10*(15*
a*sqrt(-d)*f^2*arctan(sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)/sqrt(-d)) -
(4*e^2*x^2 + 12*a*f^2 + 9*d*e*x + 2*d^2 + (4*e*f*x - d*f)*sqrt((e^2*x^2 + a*f^2)
/f^2))*sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d))/e]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(3/2), x)