3.2072 \(\int \frac{(d+e x)^{2/3}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=566 \[ \frac{3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt{a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right ) \sqrt{\frac{\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \sqrt [3]{c d^2-a e^2}+c^{2/3} d^{4/3} \left (\frac{e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 c d e \sqrt{d (a e+c d x)} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt{-\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}}}+\frac{3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(3*(a*e + c*d*x)*(d + e*x)^(2/3))/(2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2]) + (3^(3/4)*(c*d^2 - a*e^2)^(2/3)*Sqrt[a*d*e + c*d^2*x]*(d + e*x)^(2/3)*((c
*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))*Sqrt[((c*d^2 - a*e^2)
^(2/3) + c^(1/3)*d^(2/3)*(c*d^2 - a*e^2)^(1/3)*(1 + (e*x)/d)^(1/3) + c^(2/3)*d^(
4/3)*(1 + (e*x)/d)^(2/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)
*(1 + (e*x)/d)^(1/3))^2]*EllipticF[ArcCos[((c*d^2 - a*e^2)^(1/3) - (1 - Sqrt[3])
*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(
1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))], (2 + Sqrt[3])/4])/(4*c*d*e*Sqrt[d*(a*e + c*d
*x)]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]*Sqrt[-((c^(1/3)*d^(2/3)*(1 + (e
*x)/d)^(1/3)*((c*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3)))/((c*
d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))^2)])

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Rubi [A]  time = 1.68469, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128 \[ \frac{3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt{a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right ) \sqrt{\frac{\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \sqrt [3]{c d^2-a e^2}+c^{2/3} d^{4/3} \left (\frac{e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{4 c d e \sqrt{d (a e+c d x)} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt{-\frac{\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt{3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac{e x}{d}+1}\right )^2}}}+\frac{3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a*e + c*d*x)*(d + e*x)^(2/3))/(2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2]) + (3^(3/4)*(c*d^2 - a*e^2)^(2/3)*Sqrt[a*d*e + c*d^2*x]*(d + e*x)^(2/3)*((c
*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))*Sqrt[((c*d^2 - a*e^2)
^(2/3) + c^(1/3)*d^(2/3)*(c*d^2 - a*e^2)^(1/3)*(1 + (e*x)/d)^(1/3) + c^(2/3)*d^(
4/3)*(1 + (e*x)/d)^(2/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)
*(1 + (e*x)/d)^(1/3))^2]*EllipticF[ArcCos[((c*d^2 - a*e^2)^(1/3) - (1 - Sqrt[3])
*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))/((c*d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(
1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))], (2 + Sqrt[3])/4])/(4*c*d*e*Sqrt[d*(a*e + c*d
*x)]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]*Sqrt[-((c^(1/3)*d^(2/3)*(1 + (e
*x)/d)^(1/3)*((c*d^2 - a*e^2)^(1/3) - c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3)))/((c*
d^2 - a*e^2)^(1/3) - (1 + Sqrt[3])*c^(1/3)*d^(2/3)*(1 + (e*x)/d)^(1/3))^2)])

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Rubi in Sympy [A]  time = 70.2318, size = 468, normalized size = 0.83 \[ \frac{3 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 c d \sqrt [3]{d + e x}} - \frac{3^{\frac{3}{4}} \sqrt{\frac{c^{\frac{2}{3}} d^{\frac{2}{3}} \left (d + e x\right )^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{d + e x} \sqrt [3]{a e^{2} - c d^{2}} + \left (a e^{2} - c d^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{c} \sqrt [3]{d} \left (1 + \sqrt{3}\right ) \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}\right )^{2}}} \left (a e^{2} - c d^{2}\right )^{\frac{2}{3}} \left (\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{c} \sqrt [3]{d} \left (- \sqrt{3} + 1\right ) \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}}{\sqrt [3]{c} \sqrt [3]{d} \left (1 + \sqrt{3}\right ) \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{4 c d e \sqrt{\frac{\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{d + e x} \left (\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}\right )}{\left (\sqrt [3]{c} \sqrt [3]{d} \left (1 + \sqrt{3}\right ) \sqrt [3]{d + e x} + \sqrt [3]{a e^{2} - c d^{2}}\right )^{2}}} \sqrt [3]{d + e x} \sqrt{a e + c d x} \sqrt{a e - \frac{c d^{2}}{e} + \frac{c d \left (d + e x\right )}{e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*c*d*(d + e*x)**(1/3)) - 3**(
3/4)*sqrt((c**(2/3)*d**(2/3)*(d + e*x)**(2/3) - c**(1/3)*d**(1/3)*(d + e*x)**(1/
3)*(a*e**2 - c*d**2)**(1/3) + (a*e**2 - c*d**2)**(2/3))/(c**(1/3)*d**(1/3)*(1 +
sqrt(3))*(d + e*x)**(1/3) + (a*e**2 - c*d**2)**(1/3))**2)*(a*e**2 - c*d**2)**(2/
3)*(c**(1/3)*d**(1/3)*(d + e*x)**(1/3) + (a*e**2 - c*d**2)**(1/3))*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))*elliptic_f(acos((c**(1/3)*d**(1/3)*(-sqrt(3) +
1)*(d + e*x)**(1/3) + (a*e**2 - c*d**2)**(1/3))/(c**(1/3)*d**(1/3)*(1 + sqrt(3)
)*(d + e*x)**(1/3) + (a*e**2 - c*d**2)**(1/3))), sqrt(3)/4 + 1/2)/(4*c*d*e*sqrt(
c**(1/3)*d**(1/3)*(d + e*x)**(1/3)*(c**(1/3)*d**(1/3)*(d + e*x)**(1/3) + (a*e**2
- c*d**2)**(1/3))/(c**(1/3)*d**(1/3)*(1 + sqrt(3))*(d + e*x)**(1/3) + (a*e**2 -
c*d**2)**(1/3))**2)*(d + e*x)**(1/3)*sqrt(a*e + c*d*x)*sqrt(a*e - c*d**2/e + c*
d*(d + e*x)/e))

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Mathematica [C]  time = 0.282566, size = 124, normalized size = 0.22 \[ \frac{3 (d+e x)^{2/3} \left (\left (c d^2-a e^2\right ) \sqrt{\frac{e (a e+c d x)}{a e^2-c d^2}} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{c d (d+e x)}{c d^2-a e^2}\right )+e (a e+c d x)\right )}{2 c d e \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(d + e*x)^(2/3)*(e*(a*e + c*d*x) + (c*d^2 - a*e^2)*Sqrt[(e*(a*e + c*d*x))/(-(
c*d^2) + a*e^2)]*Hypergeometric2F1[1/6, 1/2, 7/6, (c*d*(d + e*x))/(c*d^2 - a*e^2
)]))/(2*c*d*e*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{1 \left ( ex+d \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt{aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^(2/3)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x), x)

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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((e*x+d)**(2/3)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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