3.299 \(\int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{5/2} \, dx\)

Optimal. Leaf size=268 \[ -\frac{5 a d^{3/2} f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e}-\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{2 d e}+\frac{5 a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{6 e}+\frac{5 a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e} \]

[Out]

(5*a*d*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e) + (5*a*f^2*(d + e*x
+ f*Sqrt[a + (e^2*x^2)/f^2])^(3/2))/(6*e) + (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^
2)/f^2])^(5/2))/(2*d*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(7/2)/(7*e) - (a
*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(7/2))/(2*d*e*(e*x + f*Sqrt[a + (e^2*
x^2)/f^2])) - (5*a*d^(3/2)*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 = 0.477121, antiderivative size = 268, 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 \[ -\frac{5 a d^{3/2} f^2 \tanh ^{-1}\left (\frac{\sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{d}}\right )}{2 e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e}-\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{2 d e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{2 d e}+\frac{5 a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{6 e}+\frac{5 a d f^2 \sqrt{f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x}}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2),x]

[Out]

(5*a*d*f^2*Sqrt[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*e) + (5*a*f^2*(d + e*x
+ f*Sqrt[a + (e^2*x^2)/f^2])^(3/2))/(6*e) + (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^
2)/f^2])^(5/2))/(2*d*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(7/2)/(7*e) - (a
*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(7/2))/(2*d*e*(e*x + f*Sqrt[a + (e^2*
x^2)/f^2])) - (5*a*d^(3/2)*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. \[ \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{5}{2}}\, dx \]

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))**(5/2),x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(5/2), x)

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Mathematica [A]  time = 0.631118, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2),x]

[Out]

Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(5/2), x]

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{{\frac{5}{2}}}\, dx \]

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))^(5/2),x)

[Out]

int((d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(5/2), x)

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Fricas [A]  time = 0.348758, size = 1, normalized size = 0. \[ \left [\frac{105 \, a d^{\frac{3}{2}} f^{2} \log \left (\frac{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} - 2 \, \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d} \sqrt{d} + 2 \, d}{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}\right ) + 2 \,{\left (24 \, e^{3} x^{3} + 36 \, d e^{2} x^{2} + 116 \, a d f^{2} + 6 \, d^{3} +{\left (32 \, a e f^{2} + 39 \, d^{2} e\right )} x +{\left (24 \, e^{2} f x^{2} + 20 \, a f^{3} + 36 \, d e f x - 3 \, d^{2} f\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{84 \, e}, -\frac{105 \, a \sqrt{-d} d f^{2} \arctan \left (\frac{\sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{\sqrt{-d}}\right ) -{\left (24 \, e^{3} x^{3} + 36 \, d e^{2} x^{2} + 116 \, a d f^{2} + 6 \, d^{3} +{\left (32 \, a e f^{2} + 39 \, d^{2} e\right )} x +{\left (24 \, e^{2} f x^{2} + 20 \, a f^{3} + 36 \, d e f x - 3 \, d^{2} f\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right )} \sqrt{e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}} + d}}{42 \, e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(5/2),x, algorithm="fricas")

[Out]

[1/84*(105*a*d^(3/2)*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*(24*e^3*x^3 + 36*d*e^2*x^2 + 116*a*d*f^2 + 6*d^3 + (32*a*e*f^2 + 3
9*d^2*e)*x + (24*e^2*f*x^2 + 20*a*f^3 + 36*d*e*f*x - 3*d^2*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, -1/42*(105*a*sqrt(-
d)*d*f^2*arctan(sqrt(e*x + f*sqrt((e^2*x^2 + a*f^2)/f^2) + d)/sqrt(-d)) - (24*e^
3*x^3 + 36*d*e^2*x^2 + 116*a*d*f^2 + 6*d^3 + (32*a*e*f^2 + 39*d^2*e)*x + (24*e^2
*f*x^2 + 20*a*f^3 + 36*d*e*f*x - 3*d^2*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. \[ \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{\frac{5}{2}}\, dx \]

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))**(5/2),x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(5/2),x, algorithm="giac")

[Out]

integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^(5/2), x)