3.39 \(\int \frac{1}{\left (b x+c x^2\right )^{7/3}} \, dx\)

Optimal. Leaf size=838 \[ \frac{15 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 \sqrt [3]{2} c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{5 \sqrt [6]{2} 3^{3/4} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}+\frac{15 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{\sqrt [3]{2} c \left (c x^2+b x\right )^{7/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{15 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac{c x (b+c x)}{b^2}} \left (c x^2+b x\right )^{7/3}}+\frac{3 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac{c x (b+c x)}{b^2}\right )^{4/3} \left (c x^2+b x\right )^{7/3}} \]

[Out]

(3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(4*c*(-((c*x*(b + c*x))/b^2))^(
4/3)*(b*x + c*x^2)^(7/3)) + (15*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(2
*c*(-((c*x*(b + c*x))/b^2))^(1/3)*(b*x + c*x^2)^(7/3)) + (15*(b + 2*c*x)*(-((c*(
b*x + c*x^2))/b^2))^(7/3))/(2^(1/3)*c*(b*x + c*x^2)^(7/3)*(1 - Sqrt[3] - 2^(2/3)
*(-((c*x*(b + c*x))/b^2))^(1/3))) + (15*3^(1/4)*Sqrt[2 + Sqrt[3]]*b^2*(-((c*(b*x
+ c*x^2))/b^2))^(7/3)*(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]*EllipticE[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]])/(2*2^(1/3)*c*(b + 2*c*x)*(b*x + c*x
^2)^(7/3)*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)]) - (5*2^(1/6)*3^(3/4)*b^2*(-((c*(b*x +
c*x^2))/b^2))^(7/3)*(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 + Sq
rt[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]])/(c*(b + 2*c*x)*(b*x + c*x^2)^(7/3)*Sqr
t[-((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 = 1.99409, antiderivative size = 838, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{15 \sqrt [4]{3} \sqrt{2+\sqrt{3}} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 \sqrt [3]{2} c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{5 \sqrt [6]{2} 3^{3/4} b^2 \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{c (b+2 c x) \left (c x^2+b x\right )^{7/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}+\frac{15 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{\sqrt [3]{2} c \left (c x^2+b x\right )^{7/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{15 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{2 c \sqrt [3]{-\frac{c x (b+c x)}{b^2}} \left (c x^2+b x\right )^{7/3}}+\frac{3 (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{7/3}}{4 c \left (-\frac{c x (b+c x)}{b^2}\right )^{4/3} \left (c x^2+b x\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(b*x + c*x^2)^(-7/3),x]

[Out]

(3*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(4*c*(-((c*x*(b + c*x))/b^2))^(
4/3)*(b*x + c*x^2)^(7/3)) + (15*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2))^(7/3))/(2
*c*(-((c*x*(b + c*x))/b^2))^(1/3)*(b*x + c*x^2)^(7/3)) + (15*(b + 2*c*x)*(-((c*(
b*x + c*x^2))/b^2))^(7/3))/(2^(1/3)*c*(b*x + c*x^2)^(7/3)*(1 - Sqrt[3] - 2^(2/3)
*(-((c*x*(b + c*x))/b^2))^(1/3))) + (15*3^(1/4)*Sqrt[2 + Sqrt[3]]*b^2*(-((c*(b*x
+ c*x^2))/b^2))^(7/3)*(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]*EllipticE[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]])/(2*2^(1/3)*c*(b + 2*c*x)*(b*x + c*x
^2)^(7/3)*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)]) - (5*2^(1/6)*3^(3/4)*b^2*(-((c*(b*x +
c*x^2))/b^2))^(7/3)*(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 + Sq
rt[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]])/(c*(b + 2*c*x)*(b*x + c*x^2)^(7/3)*Sqr
t[-((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 = 82.6812, size = 728, normalized size = 0.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(7/3),x)

[Out]

15*2**(2/3)*3**(1/4)*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)**(7/3)*sqrt(sqrt(3) + 2)*(-(1 - (-b - 2*c*x)**2/b**2)
**(1/3) + 1)*elliptic_e(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))/(4*c*sqrt((
(1 - (-b - 2*c*x)**2/b**2)**(1/3) - 1)/(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqr
t(3) + 1)**2)*(b + 2*c*x)*(b*x + c*x**2)**(7/3)) - 5*2**(1/6)*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)**(7
/3)*(-(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))/(c*sqrt(((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)*(b + 2*c*x)*(b*x + c*x**2)**(7/3)) +
15*2**(2/3)*(c*(-b*x - c*x**2)/b**2)**(7/3)*(b + 2*c*x)/(2*c*(b*x + c*x**2)**(7
/3)*(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1)) + 15*2**(2/3)*(c*(-b*x -
c*x**2)/b**2)**(7/3)*(b + 2*c*x)/(2*c*(1 - (-b - 2*c*x)**2/b**2)**(1/3)*(b*x +
c*x**2)**(7/3)) + 3*2**(2/3)*(c*(-b*x - c*x**2)/b**2)**(7/3)*(b + 2*c*x)/(c*(1 -
(-b - 2*c*x)**2/b**2)**(4/3)*(b*x + c*x**2)**(7/3))

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Mathematica [C]  time = 0.0986271, size = 90, normalized size = 0.11 \[ \frac{-3 b^3+24 b^2 c x-30 c^2 x^2 (b+c x) \sqrt [3]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{c x}{b}\right )+90 b c^2 x^2+60 c^3 x^3}{4 b^4 (x (b+c x))^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-7/3),x]

[Out]

(-3*b^3 + 24*b^2*c*x + 90*b*c^2*x^2 + 60*c^3*x^3 - 30*c^2*x^2*(b + c*x)*(1 + (c*
x)/b)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((c*x)/b)])/(4*b^4*(x*(b + c*x))^(
4/3))

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Maple [F]  time = 0.144, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(7/3),x)

[Out]

int(1/(c*x^2+b*x)^(7/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-7/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(-7/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-7/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**(7/3),x)

[Out]

Integral((b*x + c*x**2)**(-7/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-7/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(-7/3), x)