3.34 \(\int \frac{1}{\left (b x+c x^2\right )^{8/3}} \, dx\)

Optimal. Leaf size=448 \[ \frac{21 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3}}{5 c \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{8/3}}+\frac{3 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3}}{5 c \left (-\frac{c x (b+c x)}{b^2}\right )^{5/3} \left (b x+c x^2\right )^{8/3}}+\frac{14 \sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 c (b+2 c x) \left (b x+c x^2\right )^{8/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

[Out]

(3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(8/3))/(5*c*(-((c*x*(b + c*x))/b^2))^(
5/3)*(b*x + c*x^2)^(8/3)) + (21*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(8/3))/(5
*c*(-((c*x*(b + c*x))/b^2))^(2/3)*(b*x + c*x^2)^(8/3)) + (14*2^(1/3)*3^(3/4)*Sqr
t[2 - Sqrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(8/3)*(1 - 2^(2/3)*(-((c*x*(b + c*
x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-
((c*x*(b + c*x))/b^2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1
/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/
(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(5*c*(
b + 2*c*x)*(b*x + c*x^2)^(8/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3
))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)])

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Rubi [A]  time = 1.06324, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{21 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3}}{5 c \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} \left (b x+c x^2\right )^{8/3}}+\frac{3 (b+2 c x) \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3}}{5 c \left (-\frac{c x (b+c x)}{b^2}\right )^{5/3} \left (b x+c x^2\right )^{8/3}}+\frac{14 \sqrt [3]{2} 3^{3/4} \sqrt{2-\sqrt{3}} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{8/3} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 c (b+2 c x) \left (b x+c x^2\right )^{8/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-8/3),x]

[Out]

(3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(8/3))/(5*c*(-((c*x*(b + c*x))/b^2))^(
5/3)*(b*x + c*x^2)^(8/3)) + (21*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(8/3))/(5
*c*(-((c*x*(b + c*x))/b^2))^(2/3)*(b*x + c*x^2)^(8/3)) + (14*2^(1/3)*3^(3/4)*Sqr
t[2 - Sqrt[3]]*b^2*(-((c*(b*x + c*x^2))/b^2))^(8/3)*(1 - 2^(2/3)*(-((c*x*(b + c*
x))/b^2))^(1/3))*Sqrt[(1 + 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3) + 2*2^(1/3)*(-
((c*x*(b + c*x))/b^2))^(2/3))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1
/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))/
(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))], -7 + 4*Sqrt[3]])/(5*c*(
b + 2*c*x)*(b*x + c*x^2)^(8/3)*Sqrt[-((1 - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3
))/(1 - Sqrt[3] - 2^(2/3)*(-((c*x*(b + c*x))/b^2))^(1/3))^2)])

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Rubi in Sympy [A]  time = 41.4834, size = 400, normalized size = 0.89 \[ \frac{14 \sqrt [3]{2} \cdot 3^{\frac{3}{4}} b^{2} \sqrt{\frac{\left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} + \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{8}{3}} \sqrt{- \sqrt{3} + 2} \left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} + 1 + \sqrt{3}}{- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{5 c \sqrt{\frac{\sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - 1}{\left (- \sqrt [3]{1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}} - \sqrt{3} + 1\right )^{2}}} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{8}{3}}} + \frac{42 \sqrt [3]{2} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{8}{3}} \left (b + 2 c x\right )}{5 c \left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{2}{3}} \left (b x + c x^{2}\right )^{\frac{8}{3}}} + \frac{24 \sqrt [3]{2} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{8}{3}} \left (b + 2 c x\right )}{5 c \left (1 - \frac{\left (- b - 2 c x\right )^{2}}{b^{2}}\right )^{\frac{5}{3}} \left (b x + c x^{2}\right )^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(8/3),x)

[Out]

14*2**(1/3)*3**(3/4)*b**2*sqrt(((1 - (-b - 2*c*x)**2/b**2)**(2/3) + (1 - (-b - 2
*c*x)**2/b**2)**(1/3) + 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1)**2
)*(c*(-b*x - c*x**2)/b**2)**(8/3)*sqrt(-sqrt(3) + 2)*(-(1 - (-b - 2*c*x)**2/b**2
)**(1/3) + 1)*elliptic_f(asin((-(1 - (-b - 2*c*x)**2/b**2)**(1/3) + 1 + sqrt(3))
/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(5*c*sqrt(
((1 - (-b - 2*c*x)**2/b**2)**(1/3) - 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sq
rt(3) + 1)**2)*(b + 2*c*x)*(b*x + c*x**2)**(8/3)) + 42*2**(1/3)*(c*(-b*x - c*x**
2)/b**2)**(8/3)*(b + 2*c*x)/(5*c*(1 - (-b - 2*c*x)**2/b**2)**(2/3)*(b*x + c*x**2
)**(8/3)) + 24*2**(1/3)*(c*(-b*x - c*x**2)/b**2)**(8/3)*(b + 2*c*x)/(5*c*(1 - (-
b - 2*c*x)**2/b**2)**(5/3)*(b*x + c*x**2)**(8/3))

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Mathematica [C]  time = 0.0972201, size = 90, normalized size = 0.2 \[ \frac{-3 b^3+15 b^2 c x+42 c^2 x^2 (b+c x) \left (\frac{c x}{b}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{c x}{b}\right )+63 b c^2 x^2+42 c^3 x^3}{5 b^4 (x (b+c x))^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-8/3),x]

[Out]

(-3*b^3 + 15*b^2*c*x + 63*b*c^2*x^2 + 42*c^3*x^3 + 42*c^2*x^2*(b + c*x)*(1 + (c*
x)/b)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((c*x)/b)])/(5*b^4*(x*(b + c*x))^(
5/3))

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Maple [F]  time = 0.142, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{8}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(8/3),x)

[Out]

int(1/(c*x^2+b*x)^(8/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-8/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(-8/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )}{\left (c x^{2} + b x\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-8/3),x, algorithm="fricas")

[Out]

integral(1/((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*(c*x^2 + b*x)^(2/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{8}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**(8/3),x)

[Out]

Integral((b*x + c*x**2)**(-8/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-8/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(-8/3), x)