3.35 \(\int \left (b x+c x^2\right )^{5/3} \, dx\)

Optimal. Leaf size=842 \[ -\frac{15 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (c x^2+b x\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) b^2}{364 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}+\frac{5\ 3^{3/4} \left (c x^2+b x\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) b^2}{91\ 2^{5/6} c (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{15 (b+2 c x) \left (c x^2+b x\right )^{5/3}}{182 \sqrt [3]{2} c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{5/3} (b+2 c x) \left (c x^2+b x\right )^{5/3}}{26 c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3}}+\frac{15 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (c x^2+b x\right )^{5/3}}{364 c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3}} \]

[Out]

(15*(-((c*x*(b + c*x))/b^2))^(2/3)*(b + 2*c*x)*(b*x + c*x^2)^(5/3))/(364*c*(-((c
*(b*x + c*x^2))/b^2))^(5/3)) + (3*(-((c*x*(b + c*x))/b^2))^(5/3)*(b + 2*c*x)*(b*
x + c*x^2)^(5/3))/(26*c*(-((c*(b*x + c*x^2))/b^2))^(5/3)) - (15*(b + 2*c*x)*(b*x
+ c*x^2)^(5/3))/(182*2^(1/3)*c*(-((c*(b*x + c*x^2))/b^2))^(5/3)*(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*(b*
x + c*x^2)^(5/3)*(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]])/(364*2^(1/3)*c*(b + 2*c*x)*(-((c*(b*x + c
*x^2))/b^2))^(5/3)*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*3^(3/4)*b^2*(b*x + c*x^2
)^(5/3)*(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]])/(91*2^(5/6)*c*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2
))^(5/3)*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 = 2.02013, antiderivative size = 842, 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 \[ -\frac{15 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (c x^2+b x\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) b^2}{364 \sqrt [3]{2} c (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}+\frac{5\ 3^{3/4} \left (c x^2+b x\right )^{5/3} \left (1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}\right ) \sqrt{\frac{2 \sqrt [3]{2} \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3}+2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+1}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}+\sqrt{3}+1}{-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right ) b^2}{91\ 2^{5/6} c (b+2 c x) \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \sqrt{-\frac{1-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}}{\left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )^2}}}-\frac{15 (b+2 c x) \left (c x^2+b x\right )^{5/3}}{182 \sqrt [3]{2} c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3} \left (-2^{2/3} \sqrt [3]{-\frac{c x (b+c x)}{b^2}}-\sqrt{3}+1\right )}+\frac{3 \left (-\frac{c x (b+c x)}{b^2}\right )^{5/3} (b+2 c x) \left (c x^2+b x\right )^{5/3}}{26 c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3}}+\frac{15 \left (-\frac{c x (b+c x)}{b^2}\right )^{2/3} (b+2 c x) \left (c x^2+b x\right )^{5/3}}{364 c \left (-\frac{c \left (c x^2+b x\right )}{b^2}\right )^{5/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(b*x + c*x^2)^(5/3),x]

[Out]

(15*(-((c*x*(b + c*x))/b^2))^(2/3)*(b + 2*c*x)*(b*x + c*x^2)^(5/3))/(364*c*(-((c
*(b*x + c*x^2))/b^2))^(5/3)) + (3*(-((c*x*(b + c*x))/b^2))^(5/3)*(b + 2*c*x)*(b*
x + c*x^2)^(5/3))/(26*c*(-((c*(b*x + c*x^2))/b^2))^(5/3)) - (15*(b + 2*c*x)*(b*x
+ c*x^2)^(5/3))/(182*2^(1/3)*c*(-((c*(b*x + c*x^2))/b^2))^(5/3)*(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*(b*
x + c*x^2)^(5/3)*(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]])/(364*2^(1/3)*c*(b + 2*c*x)*(-((c*(b*x + c
*x^2))/b^2))^(5/3)*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*3^(3/4)*b^2*(b*x + c*x^2
)^(5/3)*(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]])/(91*2^(5/6)*c*(b + 2*c*x)*(-((c*(b*x + c*x^2))/b^2
))^(5/3)*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 = 82.2921, size = 731, normalized size = 0.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/3),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)*sqrt(sqrt(3) + 2)*(b*x + c*x**2)**(5/3)*(-(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))/(728*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)*(c*(-b*x - c*x**2)/b**2)**(5/3)*(b + 2*c*x)) + 5*2**(1/6)*3**(3/4)*b**2*s
qrt(((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)*(b*x + c*x**2)**(5/3)*(-(
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))/(182*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)*(c*(-b*x - c*x**2)/b**2)**(5/3)*(b + 2*
c*x)) + 3*2**(2/3)*(1 - (-b - 2*c*x)**2/b**2)**(5/3)*(b + 2*c*x)*(b*x + c*x**2)*
*(5/3)/(416*c*(c*(-b*x - c*x**2)/b**2)**(5/3)) + 15*2**(2/3)*(1 - (-b - 2*c*x)**
2/b**2)**(2/3)*(b + 2*c*x)*(b*x + c*x**2)**(5/3)/(1456*c*(c*(-b*x - c*x**2)/b**2
)**(5/3)) - 15*2**(2/3)*(b + 2*c*x)*(b*x + c*x**2)**(5/3)/(364*c*(c*(-b*x - c*x*
*2)/b**2)**(5/3)*(-(1 - (-b - 2*c*x)**2/b**2)**(1/3) - sqrt(3) + 1))

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Mathematica [C]  time = 0.0699761, size = 94, normalized size = 0.11 \[ \frac{3 x \left (5 b^4 \sqrt [3]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{c x}{b}\right )-5 b^4-b^3 c x+46 b^2 c^2 x^2+70 b c^3 x^3+28 c^4 x^4\right )}{364 c^2 \sqrt [3]{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/3),x]

[Out]

(3*x*(-5*b^4 - b^3*c*x + 46*b^2*c^2*x^2 + 70*b*c^3*x^3 + 28*c^4*x^4 + 5*b^4*(1 +
(c*x)/b)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((c*x)/b)]))/(364*c^2*(x*(b +
c*x))^(1/3))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/3),x)

[Out]

int((c*x^2+b*x)^(5/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{5}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/3),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(5/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{5}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(5/3),x)

[Out]

Integral((b*x + c*x**2)**(5/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/3), x)