3.662 \(\int \frac{x^2}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx\)

Optimal. Leaf size=398 \[ \frac{d^2 (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}+\frac{\tanh ^{-1}\left (\frac{b+2 c (d+e x)^3}{2 \sqrt{c} \sqrt{a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt{c} e^3} \]

[Out]

(d^2*(d + e*x)*Sqrt[1 + (2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c
*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*(d + e
*x)^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])])/(e^
3*Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6]) - (d*(d + e*x)^2*Sqrt[1 + (2*c*(d + e
*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c
])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c]), (-2
*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])])/(e^3*Sqrt[a + b*(d + e*x)^3 + c*(d + e
*x)^6]) + ArcTanh[(b + 2*c*(d + e*x)^3)/(2*Sqrt[c]*Sqrt[a + b*(d + e*x)^3 + c*(d
+ e*x)^6])]/(3*Sqrt[c]*e^3)

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Rubi [A]  time = 1.6095, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{d^2 (d+e x) \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}-\frac{d (d+e x)^2 \sqrt{\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c (d+e x)^3}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{2 c (d+e x)^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c (d+e x)^3}{b+\sqrt{b^2-4 a c}}\right )}{e^3 \sqrt{a+b (d+e x)^3+c (d+e x)^6}}+\frac{\tanh ^{-1}\left (\frac{b+2 c (d+e x)^3}{2 \sqrt{c} \sqrt{a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt{c} e^3} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^2/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]

[Out]

(d^2*(d + e*x)*Sqrt[1 + (2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c
*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*(d + e
*x)^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])])/(e^
3*Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6]) - (d*(d + e*x)^2*Sqrt[1 + (2*c*(d + e
*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c
])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c]), (-2
*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])])/(e^3*Sqrt[a + b*(d + e*x)^3 + c*(d + e
*x)^6]) + ArcTanh[(b + 2*c*(d + e*x)^3)/(2*Sqrt[c]*Sqrt[a + b*(d + e*x)^3 + c*(d
+ e*x)^6])]/(3*Sqrt[c]*e^3)

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Rubi in Sympy [A]  time = 101.353, size = 354, normalized size = 0.89 \[ \frac{\operatorname{atanh}{\left (\frac{b + 2 c \left (d + e x\right )^{3}}{2 \sqrt{c} \sqrt{a + b \left (d + e x\right )^{3} + c \left (d + e x\right )^{6}}} \right )}}{3 \sqrt{c} e^{3}} + \frac{d^{2} \left (d + e x\right ) \sqrt{a + b \left (d + e x\right )^{3} + c \left (d + e x\right )^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{4}{3},- \frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a e^{3} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} - \frac{d \left (d + e x\right )^{2} \sqrt{a + b \left (d + e x\right )^{3} + c \left (d + e x\right )^{6}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},\frac{1}{2},\frac{5}{3},- \frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a e^{3} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c \left (d + e x\right )^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(a+b*(e*x+d)**3+c*(e*x+d)**6)**(1/2),x)

[Out]

atanh((b + 2*c*(d + e*x)**3)/(2*sqrt(c)*sqrt(a + b*(d + e*x)**3 + c*(d + e*x)**6
)))/(3*sqrt(c)*e**3) + d**2*(d + e*x)*sqrt(a + b*(d + e*x)**3 + c*(d + e*x)**6)*
appellf1(1/3, 1/2, 1/2, 4/3, -2*c*(d + e*x)**3/(b - sqrt(-4*a*c + b**2)), -2*c*(
d + e*x)**3/(b + sqrt(-4*a*c + b**2)))/(a*e**3*sqrt(2*c*(d + e*x)**3/(b - sqrt(-
4*a*c + b**2)) + 1)*sqrt(2*c*(d + e*x)**3/(b + sqrt(-4*a*c + b**2)) + 1)) - d*(d
+ e*x)**2*sqrt(a + b*(d + e*x)**3 + c*(d + e*x)**6)*appellf1(2/3, 1/2, 1/2, 5/3
, -2*c*(d + e*x)**3/(b - sqrt(-4*a*c + b**2)), -2*c*(d + e*x)**3/(b + sqrt(-4*a*
c + b**2)))/(a*e**3*sqrt(2*c*(d + e*x)**3/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*
c*(d + e*x)**3/(b + sqrt(-4*a*c + b**2)) + 1))

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Mathematica [A]  time = 2.21127, size = 0, normalized size = 0. \[ \int \frac{x^2}{\sqrt{a+b (d+e x)^3+c (d+e x)^6}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^2/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6],x]

[Out]

Integrate[x^2/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6], x]

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int{{x}^{2}{\frac{1}{\sqrt{a+b \left ( ex+d \right ) ^{3}+c \left ( ex+d \right ) ^{6}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2),x)

[Out]

int(x^2/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{{\left (e x + d\right )}^{6} c +{\left (e x + d\right )}^{3} b + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{c e^{6} x^{6} + 6 \, c d e^{5} x^{5} + 15 \, c d^{2} e^{4} x^{4} + c d^{6} +{\left (20 \, c d^{3} + b\right )} e^{3} x^{3} + 3 \,{\left (5 \, c d^{4} + b d\right )} e^{2} x^{2} + b d^{3} + 3 \,{\left (2 \, c d^{5} + b d^{2}\right )} e x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(c*e^6*x^6 + 6*c*d*e^5*x^5 + 15*c*d^2*e^4*x^4 + c*d^6 + (20*c*d
^3 + b)*e^3*x^3 + 3*(5*c*d^4 + b*d)*e^2*x^2 + b*d^3 + 3*(2*c*d^5 + b*d^2)*e*x +
a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(a+b*(e*x+d)**3+c*(e*x+d)**6)**(1/2),x)

[Out]

Integral(x**2/sqrt(a + b*d**3 + 3*b*d**2*e*x + 3*b*d*e**2*x**2 + b*e**3*x**3 + c
*d**6 + 6*c*d**5*e*x + 15*c*d**4*e**2*x**2 + 20*c*d**3*e**3*x**3 + 15*c*d**2*e**
4*x**4 + 6*c*d*e**5*x**5 + c*e**6*x**6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{{\left (e x + d\right )}^{6} c +{\left (e x + d\right )}^{3} b + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a),x, algorithm="giac")

[Out]

integrate(x^2/sqrt((e*x + d)^6*c + (e*x + d)^3*b + a), x)