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

Optimal. Leaf size=303 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^3}{32 e^5 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{3 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{32 e^5}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right )}{8 e^4}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2}{16 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^4}{8 e} \]

[Out]

(f^2*(2*d*e - b*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^4) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])
^2)/(16*e^3) + (d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^4/(8*e) - (f^2*(2*d*e
- b*f^2)^3*(4*a*e^2 - b^2*f^2))/(32*e^5*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^
2 + e^2*x))/f^2]))) + (3*f^2*(2*d*e - b*f^2)^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])])/(32*e^5)

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Rubi [A]  time = 0.768512, antiderivative size = 303, 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 \[ -\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^3}{32 e^5 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}+\frac{3 f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right )^2 \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{32 e^5}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (2 d e-b f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+e x\right )}{8 e^4}+\frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^2}{16 e^3}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^4}{8 e} \]

Antiderivative was successfully verified.

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

[Out]

(f^2*(2*d*e - b*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^4) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])
^2)/(16*e^3) + (d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^4/(8*e) - (f^2*(2*d*e
- b*f^2)^3*(4*a*e^2 - b^2*f^2))/(32*e^5*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^
2 + e^2*x))/f^2]))) + (3*f^2*(2*d*e - b*f^2)^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])])/(32*e^5)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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))**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.412209, size = 255, normalized size = 0.84 \[ -\frac{3 \left (b^2 f^2-4 a e^2\right ) \left (b f^3-2 d e f\right )^2 \log \left (2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}{32 e^5}+\frac{\sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )} \left (4 b e^2 f^3 \left (-2 a f^2+3 d^2+2 d e x+2 e^2 x^2\right )+8 e^3 f \left (2 a f^2 (2 d+e x)+e x \left (3 d^2+4 d e x+2 e^2 x^2\right )\right )+3 b^3 f^7-2 b^2 e f^5 (6 d+e x)\right )}{16 e^4}+\frac{3}{2} x^2 \left (a e f^2+b d f^2+d^2 e\right )+d x \left (3 a f^2+d^2\right )+e x^3 \left (b f^2+2 d e\right )+e^3 x^4 \]

Antiderivative was successfully verified.

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

[Out]

d*(d^2 + 3*a*f^2)*x + (3*(d^2*e + b*d*f^2 + a*e*f^2)*x^2)/2 + e*(2*d*e + b*f^2)*
x^3 + e^3*x^4 + (Sqrt[a + x*(b + (e^2*x)/f^2)]*(3*b^3*f^7 - 2*b^2*e*f^5*(6*d + e
*x) + 4*b*e^2*f^3*(3*d^2 - 2*a*f^2 + 2*d*e*x + 2*e^2*x^2) + 8*e^3*f*(2*a*f^2*(2*
d + e*x) + e*x*(3*d^2 + 4*d*e*x + 2*e^2*x^2))))/(16*e^4) - (3*(-4*a*e^2 + b^2*f^
2)*(-2*d*e*f + b*f^3)^2*Log[b*f^2 + 2*e*(e*x + f*Sqrt[a + x*(b + (e^2*x)/f^2)])]
)/(32*e^5)

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Maple [B]  time = 0.022, size = 685, normalized size = 2.3 \[ \text{result too large to display} \]

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))^3,x)

[Out]

d^3*x+e^3*x^4-3/8*d^2*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^2-3/2*d/e*b*f^3*(a+b*x+e^2*x^2/f^2)^(1/2)*x+
3/8*d/e^3*b^3*f^5*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)+3/2*f^2*a*e*x^2+3*f^2*a*d*x+1/4*d^4/e+2*x^3*e^2*d+3/2*x^2*
d^2*e+f^3*(a+b*x+e^2*x^2/f^2)^(3/2)*x+2*d/e*(a+b*x+e^2*x^2/f^2)^(3/2)*f^3+3/4*d^
2*f^3/e^2*(a+b*x+e^2*x^2/f^2)^(1/2)*b+3/2*f*d^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)*a+3/8*f^5/e^2*a*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^2-3/2*
d/e*b*f^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)*a-3/4*d/e^3*b^2*f^5*(a+b*x+e^2*x^2/f^2)^(1/2)+3/2*f*d^2*(a+b*x+e^2
*x^2/f^2)^(1/2)*x-1/2*f^5*(a+b*x+e^2*x^2/f^2)^(3/2)/e^2*b+3/16*f^7/e^4*b^3*(a+b*
x+e^2*x^2/f^2)^(1/2)+f^2*x^3*b*e+3/2*f^2*x^2*b*d+3/8*f^5/e^2*b^2*(a+b*x+e^2*x^2/
f^2)^(1/2)*x-3/32*f^7/e^4*b^4*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)

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320528, size = 466, normalized size = 1.54 \[ \frac{32 \, e^{8} x^{4} + 32 \,{\left (b e^{6} f^{2} + 2 \, d e^{7}\right )} x^{3} + 48 \,{\left (d^{2} e^{6} +{\left (b d e^{5} + a e^{6}\right )} f^{2}\right )} x^{2} + 32 \,{\left (3 \, a d e^{5} f^{2} + d^{3} e^{5}\right )} x + 3 \,{\left (b^{4} f^{8} - 16 \, a d^{2} e^{4} f^{2} - 4 \,{\left (b^{3} d e + a b^{2} e^{2}\right )} f^{6} + 4 \,{\left (b^{2} d^{2} e^{2} + 4 \, a b d e^{3}\right )} f^{4}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \,{\left (3 \, b^{3} e f^{7} + 16 \, e^{7} f x^{3} - 4 \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} f^{5} + 4 \,{\left (3 \, b d^{2} e^{3} + 8 \, a d e^{4}\right )} f^{3} + 8 \,{\left (b e^{5} f^{3} + 4 \, d e^{6} f\right )} x^{2} - 2 \,{\left (b^{2} e^{3} f^{5} - 12 \, d^{2} e^{5} f - 4 \,{\left (b d e^{4} + 2 \, a e^{5}\right )} f^{3}\right )} x\right )} \sqrt{\frac{b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{32 \, e^{5}} \]

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)^3,x, algorithm="fricas")

[Out]

1/32*(32*e^8*x^4 + 32*(b*e^6*f^2 + 2*d*e^7)*x^3 + 48*(d^2*e^6 + (b*d*e^5 + a*e^6
)*f^2)*x^2 + 32*(3*a*d*e^5*f^2 + d^3*e^5)*x + 3*(b^4*f^8 - 16*a*d^2*e^4*f^2 - 4*
(b^3*d*e + a*b^2*e^2)*f^6 + 4*(b^2*d^2*e^2 + 4*a*b*d*e^3)*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^3*e*f^7 + 16*e^7*f*x
^3 - 4*(3*b^2*d*e^2 + 2*a*b*e^3)*f^5 + 4*(3*b*d^2*e^3 + 8*a*d*e^4)*f^3 + 8*(b*e^
5*f^3 + 4*d*e^6*f)*x^2 - 2*(b^2*e^3*f^5 - 12*d^2*e^5*f - 4*(b*d*e^4 + 2*a*e^5)*f
^3)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))/e^5

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

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))**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.283567, size = 504, normalized size = 1.66 \[ b f^{2} x^{3} e + \frac{3}{2} \, b d f^{2} x^{2} + \frac{3}{2} \, a f^{2} x^{2} e + 3 \, a d f^{2} x + x^{4} e^{3} + 2 \, d x^{3} e^{2} + \frac{3}{2} \, d^{2} x^{2} e + d^{3} x + \frac{3}{32} \,{\left (b^{4} f^{7}{\left | f \right |} - 4 \, b^{3} d f^{5}{\left | f \right |} e - 4 \, a b^{2} f^{5}{\left | f \right |} e^{2} + 4 \, b^{2} d^{2} f^{3}{\left | f \right |} e^{2} + 16 \, a b d f^{3}{\left | f \right |} e^{3} - 16 \, a d^{2} f{\left | f \right |} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | -b f^{2} - 2 \,{\left (x e - \sqrt{b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right ) + \frac{1}{16} \, \sqrt{b f^{2} x + a f^{2} + x^{2} e^{2}}{\left (2 \,{\left (4 \,{\left (\frac{2 \, x{\left | f \right |} e^{2}}{f} + \frac{{\left (b f^{6}{\left | f \right |} e^{6} + 4 \, d f^{4}{\left | f \right |} e^{7}\right )} e^{\left (-6\right )}}{f^{5}}\right )} x - \frac{{\left (b^{2} f^{8}{\left | f \right |} e^{4} - 4 \, b d f^{6}{\left | f \right |} e^{5} - 8 \, a f^{6}{\left | f \right |} e^{6} - 12 \, d^{2} f^{4}{\left | f \right |} e^{6}\right )} e^{\left (-6\right )}}{f^{5}}\right )} x + \frac{{\left (3 \, b^{3} f^{10}{\left | f \right |} e^{2} - 12 \, b^{2} d f^{8}{\left | f \right |} e^{3} - 8 \, a b f^{8}{\left | f \right |} e^{4} + 12 \, b d^{2} f^{6}{\left | f \right |} e^{4} + 32 \, a d f^{6}{\left | f \right |} e^{5}\right )} e^{\left (-6\right )}}{f^{5}}\right )} \]

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)^3,x, algorithm="giac")

[Out]

b*f^2*x^3*e + 3/2*b*d*f^2*x^2 + 3/2*a*f^2*x^2*e + 3*a*d*f^2*x + x^4*e^3 + 2*d*x^
3*e^2 + 3/2*d^2*x^2*e + d^3*x + 3/32*(b^4*f^7*abs(f) - 4*b^3*d*f^5*abs(f)*e - 4*
a*b^2*f^5*abs(f)*e^2 + 4*b^2*d^2*f^3*abs(f)*e^2 + 16*a*b*d*f^3*abs(f)*e^3 - 16*a
*d^2*f*abs(f)*e^4)*e^(-5)*ln(abs(-b*f^2 - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^
2))*e)) + 1/16*sqrt(b*f^2*x + a*f^2 + x^2*e^2)*(2*(4*(2*x*abs(f)*e^2/f + (b*f^6*
abs(f)*e^6 + 4*d*f^4*abs(f)*e^7)*e^(-6)/f^5)*x - (b^2*f^8*abs(f)*e^4 - 4*b*d*f^6
*abs(f)*e^5 - 8*a*f^6*abs(f)*e^6 - 12*d^2*f^4*abs(f)*e^6)*e^(-6)/f^5)*x + (3*b^3
*f^10*abs(f)*e^2 - 12*b^2*d*f^8*abs(f)*e^3 - 8*a*b*f^8*abs(f)*e^4 + 12*b*d^2*f^6
*abs(f)*e^4 + 32*a*d*f^6*abs(f)*e^5)*e^(-6)/f^5)