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

Optimal. Leaf size=237 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2}{16 e^4 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{8 e^4}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right )}{8 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e} \]

[Out]

(f^2*(4*a*e^2 - b^2*f^2)*(e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]))/(8*e^3) + (d +
e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^3/(6*e) - (f^2*(2*d*e - b*f^2)^2*(4*a*e^
2 - b^2*f^2))/(16*e^4*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2])))
+ (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*Log[b*f^2 + 2*e*(e*x + f*Sqrt[a + (x
*(b*f^2 + e^2*x))/f^2])])/(8*e^4)

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Rubi [A]  time = 0.604273, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2}{16 e^4 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{8 e^4}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right )}{8 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

(f^2*(4*a*e^2 - b^2*f^2)*(e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]))/(8*e^3) + (d +
e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^3/(6*e) - (f^2*(2*d*e - b*f^2)^2*(4*a*e^
2 - b^2*f^2))/(16*e^4*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2])))
+ (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*Log[b*f^2 + 2*e*(e*x + f*Sqrt[a + (x
*(b*f^2 + e^2*x))/f^2])])/(8*e^4)

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

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))**2,x)

[Out]

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

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Mathematica [A]  time = 0.293019, size = 179, normalized size = 0.76 \[ \frac{f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right ) \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}{8 e^4}+\frac{\sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )} \left (4 e^2 f \left (2 a f^2+e x (3 d+2 e x)\right )-3 b^2 f^5+2 b e f^3 (3 d+e x)\right )}{12 e^3}+x \left (a f^2+d^2\right )+\frac{1}{2} x^2 \left (b f^2+2 d e\right )+\frac{2 e^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

(d^2 + a*f^2)*x + ((2*d*e + b*f^2)*x^2)/2 + (2*e^2*x^3)/3 + (Sqrt[a + x*(b + (e^
2*x)/f^2)]*(-3*b^2*f^5 + 2*b*e*f^3*(3*d + e*x) + 4*e^2*f*(2*a*f^2 + e*x*(3*d + 2
*e*x))))/(12*e^3) + (f^2*(-2*d*e + b*f^2)*(-4*a*e^2 + b^2*f^2)*Log[b*f^2 + 2*e*(
e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])])/(8*e^4)

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Maple [A]  time = 0.01, size = 409, normalized size = 1.7 \[{f}^{2}xa+{\frac{{x}^{2}b{f}^{2}}{2}}+{\frac{2\,{x}^{3}{e}^{2}}{3}}+{\frac{d{f}^{3}b}{2\,{e}^{2}}\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}+fd\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}x+{adf\ln \left ({1 \left ({\frac{b}{2}}+{\frac{{e}^{2}x}{{f}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}-{\frac{d{f}^{3}{b}^{2}}{4\,{e}^{2}}\ln \left ({1 \left ({\frac{b}{2}}+{\frac{{e}^{2}x}{{f}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+{\frac{2\,{f}^{3}}{3\,e} \left ( a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}{f}^{5}}{4\,{e}^{3}}\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}-{\frac{b{f}^{3}x}{2\,e}\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}-{\frac{b{f}^{3}a}{2\,e}\ln \left ({1 \left ({\frac{b}{2}}+{\frac{{e}^{2}x}{{f}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+{\frac{{b}^{3}{f}^{5}}{8\,{e}^{3}}\ln \left ({1 \left ({\frac{b}{2}}+{\frac{{e}^{2}x}{{f}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+{x}^{2}de+x{d}^{2}+{\frac{{d}^{3}}{3\,e}} \]

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))^2,x)

[Out]

f^2*x*a+1/2*x^2*b*f^2+2/3*x^3*e^2+1/2*d*f^3/e^2*(a+b*x+e^2*x^2/f^2)^(1/2)*b+f*d*
(a+b*x+e^2*x^2/f^2)^(1/2)*x+f*d*ln((1/2*b+e^2*x/f^2)/(1/f^2*e^2)^(1/2)+(a+b*x+e^
2*x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*a-1/4*d*f^3/e^2*ln((1/2*b+e^2*x/f^2)/(1/f^2*
e^2)^(1/2)+(a+b*x+e^2*x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*b^2+2/3*(a+b*x+e^2*x^2/f
^2)^(3/2)*f^3/e-1/4*b^2*f^5/e^3*(a+b*x+e^2*x^2/f^2)^(1/2)-1/2*b*f^3/e*(a+b*x+e^2
*x^2/f^2)^(1/2)*x-1/2*b*f^3/e*ln((1/2*b+e^2*x/f^2)/(1/f^2*e^2)^(1/2)+(a+b*x+e^2*
x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*a+1/8*b^3*f^5/e^3*ln((1/2*b+e^2*x/f^2)/(1/f^2*
e^2)^(1/2)+(a+b*x+e^2*x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)+x^2*d*e+x*d^2+1/3*d^3/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301302, size = 296, normalized size = 1.25 \[ \frac{16 \, e^{6} x^{3} + 12 \,{\left (b e^{4} f^{2} + 2 \, d e^{5}\right )} x^{2} + 24 \,{\left (a e^{4} f^{2} + d^{2} e^{4}\right )} x - 3 \,{\left (b^{3} f^{6} + 8 \, a d e^{3} f^{2} - 2 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} f^{4}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) - 2 \,{\left (3 \, b^{2} e f^{5} - 8 \, e^{5} f x^{2} - 2 \,{\left (3 \, b d e^{2} + 4 \, a e^{3}\right )} f^{3} - 2 \,{\left (b e^{3} f^{3} + 6 \, d e^{4} f\right )} x\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{24 \, e^{4}} \]

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)^2,x, algorithm="fricas")

[Out]

1/24*(16*e^6*x^3 + 12*(b*e^4*f^2 + 2*d*e^5)*x^2 + 24*(a*e^4*f^2 + d^2*e^4)*x - 3
*(b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4)*log(-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*(3*b^2*e*f^5 - 8*e^5*f*x^2 - 2*
(3*b*d*e^2 + 4*a*e^3)*f^3 - 2*(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))/e^4

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

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))**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27989, size = 302, normalized size = 1.27 \[ \frac{1}{2} \, b f^{2} x^{2} + a f^{2} x + \frac{2}{3} \, x^{3} e^{2} + d x^{2} e + d^{2} x - \frac{1}{8} \,{\left (b^{3} f^{5}{\left | f \right |} - 2 \, b^{2} d f^{3}{\left | f \right |} e - 4 \, a b f^{3}{\left | f \right |} e^{2} + 8 \, a d f{\left | f \right |} e^{3}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -b f^{2} - 2 \,{\left (x e - \sqrt{b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right ) + \frac{1}{12} \, \sqrt{b f^{2} x + a f^{2} + x^{2} e^{2}}{\left (2 \,{\left (\frac{4 \, x{\left | f \right |} e}{f} + \frac{{\left (b f^{3}{\left | f \right |} e^{3} + 6 \, d f{\left | f \right |} e^{4}\right )} e^{\left (-4\right )}}{f^{2}}\right )} x - \frac{{\left (3 \, b^{2} f^{5}{\left | f \right |} e - 6 \, b d f^{3}{\left | f \right |} e^{2} - 8 \, a f^{3}{\left | f \right |} e^{3}\right )} e^{\left (-4\right )}}{f^{2}}\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/2*b*f^2*x^2 + a*f^2*x + 2/3*x^3*e^2 + d*x^2*e + d^2*x - 1/8*(b^3*f^5*abs(f) -
2*b^2*d*f^3*abs(f)*e - 4*a*b*f^3*abs(f)*e^2 + 8*a*d*f*abs(f)*e^3)*e^(-4)*ln(abs(
-b*f^2 - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*e)) + 1/12*sqrt(b*f^2*x + a*f
^2 + x^2*e^2)*(2*(4*x*abs(f)*e/f + (b*f^3*abs(f)*e^3 + 6*d*f*abs(f)*e^4)*e^(-4)/
f^2)*x - (3*b^2*f^5*abs(f)*e - 6*b*d*f^3*abs(f)*e^2 - 8*a*f^3*abs(f)*e^3)*e^(-4)
/f^2)