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

Optimal. Leaf size=436 \[ -\frac{f^2 \left (4 a e-\frac{b^2 f^2}{e}\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{2 \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{4 e^2 \left (2 d e-b f^2\right )}-\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{16 \sqrt{2} e^{9/2}}+\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{16 e^4}+\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{24 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e} \]

[Out]

(5*f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*
x^2)/f^2]])/(16*e^4) + (5*f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e
^2*x^2)/f^2])^(3/2))/(24*e^3) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b
*x + (e^2*x^2)/f^2])^(5/2))/(4*e^2*(2*d*e - b*f^2)) + (d + e*x + f*Sqrt[a + b*x
+ (e^2*x^2)/f^2])^(7/2)/(7*e) - (f^2*(4*a*e - (b^2*f^2)/e)*(d + e*x + f*Sqrt[a +
b*x + (e^2*x^2)/f^2])^(7/2))/(2*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a +
(x*(b*f^2 + e^2*x))/f^2]))) - (5*f^2*(2*d*e - b*f^2)^(3/2)*(4*a*e^2 - b^2*f^2)*A
rcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/Sqrt[2*
d*e - b*f^2]])/(16*Sqrt[2]*e^(9/2))

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Rubi [A]  time = 1.41367, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ -\frac{f^2 \left (4 a e-\frac{b^2 f^2}{e}\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{2 \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{5/2}}{4 e^2 \left (2 d e-b f^2\right )}-\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{\sqrt{2 d e-b f^2}}\right )}{16 \sqrt{2} e^{9/2}}+\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \sqrt{f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x}}{16 e^4}+\frac{5 f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{24 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{7/2}}{7 e} \]

Antiderivative was successfully verified.

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

[Out]

(5*f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*
x^2)/f^2]])/(16*e^4) + (5*f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e
^2*x^2)/f^2])^(3/2))/(24*e^3) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b
*x + (e^2*x^2)/f^2])^(5/2))/(4*e^2*(2*d*e - b*f^2)) + (d + e*x + f*Sqrt[a + b*x
+ (e^2*x^2)/f^2])^(7/2)/(7*e) - (f^2*(4*a*e - (b^2*f^2)/e)*(d + e*x + f*Sqrt[a +
b*x + (e^2*x^2)/f^2])^(7/2))/(2*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a +
(x*(b*f^2 + e^2*x))/f^2]))) - (5*f^2*(2*d*e - b*f^2)^(3/2)*(4*a*e^2 - b^2*f^2)*A
rcTanh[(Sqrt[2]*Sqrt[e]*Sqrt[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/Sqrt[2*
d*e - b*f^2]])/(16*Sqrt[2]*e^(9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x+f*(a+b*x+e**2*x**2/f**2)**(1/2))**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.797402, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^{5/2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.527733, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/672*(105*sqrt(1/2)*(b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4)*sq
rt(-(b*f^2 - 2*d*e)/e)*log(-(b*f^2 - 2*e^2*x + 4*sqrt(1/2)*sqrt(e*x + f*sqrt((b*
f^2*x + e^2*x^2 + a*f^2)/f^2) + d)*e*sqrt(-(b*f^2 - 2*d*e)/e) - 2*e*f*sqrt((b*f^
2*x + e^2*x^2 + a*f^2)/f^2) - 4*d*e)/(b*f^2 + 2*e^2*x + 2*e*f*sqrt((b*f^2*x + e^
2*x^2 + a*f^2)/f^2))) + 2*(105*b^3*f^6 + 192*e^6*x^3 + 48*d^3*e^3 - 56*(5*b^2*d*
e + 6*a*b*e^2)*f^4 + 4*(21*b*d^2*e^2 + 232*a*d*e^3)*f^2 + 144*(b*e^4*f^2 + 2*d*e
^5)*x^2 + 2*(7*b^2*e^2*f^4 + 156*d^2*e^4 - 4*(3*b*d*e^3 - 32*a*e^4)*f^2)*x - 2*(
35*b^2*e*f^5 - 96*e^5*f*x^2 + 12*d^2*e^3*f - 4*(21*b*d*e^2 + 20*a*e^3)*f^3 - 24*
(b*e^3*f^3 + 6*d*e^4*f)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*s
qrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d))/e^4, -1/336*(105*sqrt(1/2)*(b^3*f^6 +
8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4)*sqrt((b*f^2 - 2*d*e)/e)*arctan(2*s
qrt(1/2)*sqrt(e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2) + d)/sqrt((b*f^2 - 2
*d*e)/e)) - (105*b^3*f^6 + 192*e^6*x^3 + 48*d^3*e^3 - 56*(5*b^2*d*e + 6*a*b*e^2)
*f^4 + 4*(21*b*d^2*e^2 + 232*a*d*e^3)*f^2 + 144*(b*e^4*f^2 + 2*d*e^5)*x^2 + 2*(7
*b^2*e^2*f^4 + 156*d^2*e^4 - 4*(3*b*d*e^3 - 32*a*e^4)*f^2)*x - 2*(35*b^2*e*f^5 -
96*e^5*f*x^2 + 12*d^2*e^3*f - 4*(21*b*d*e^2 + 20*a*e^3)*f^3 - 24*(b*e^3*f^3 + 6
*d*e^4*f)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))*sqrt(e*x + f*sqrt((b*f^2*x +
e^2*x^2 + a*f^2)/f^2) + d))/e^4]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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