3.40 \(\int \left (b x+c x^2\right )^{5/4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{5 b^2 (b+2 c x) \sqrt [4]{b x+c x^2}}{84 c^2}+\frac{5 b^5 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{84 \sqrt{2} c^3 \left (b x+c x^2\right )^{3/4}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/4}}{7 c} \]

[Out]

(-5*b^2*(b + 2*c*x)*(b*x + c*x^2)^(1/4))/(84*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(
5/4))/(7*c) + (5*b^5*(-((c*(b*x + c*x^2))/b^2))^(3/4)*EllipticF[ArcSin[1 + (2*c*
x)/b]/2, 2])/(84*Sqrt[2]*c^3*(b*x + c*x^2)^(3/4))

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Rubi [A]  time = 0.110834, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{5 b^2 (b+2 c x) \sqrt [4]{b x+c x^2}}{84 c^2}+\frac{5 b^5 \left (-\frac{c \left (b x+c x^2\right )}{b^2}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{84 \sqrt{2} c^3 \left (b x+c x^2\right )^{3/4}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/4}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/4),x]

[Out]

(-5*b^2*(b + 2*c*x)*(b*x + c*x^2)^(1/4))/(84*c^2) + ((b + 2*c*x)*(b*x + c*x^2)^(
5/4))/(7*c) + (5*b^5*(-((c*(b*x + c*x^2))/b^2))^(3/4)*EllipticF[ArcSin[1 + (2*c*
x)/b]/2, 2])/(84*Sqrt[2]*c^3*(b*x + c*x^2)^(3/4))

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Rubi in Sympy [A]  time = 17.2335, size = 109, normalized size = 0.92 \[ \frac{5 \sqrt{2} b^{5} \left (\frac{c \left (- b x - c x^{2}\right )}{b^{2}}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{168 c^{3} \left (b x + c x^{2}\right )^{\frac{3}{4}}} - \frac{5 b^{2} \left (b + 2 c x\right ) \sqrt [4]{b x + c x^{2}}}{84 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{4}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/4),x)

[Out]

5*sqrt(2)*b**5*(c*(-b*x - c*x**2)/b**2)**(3/4)*elliptic_f(asin(1 + 2*c*x/b)/2, 2
)/(168*c**3*(b*x + c*x**2)**(3/4)) - 5*b**2*(b + 2*c*x)*(b*x + c*x**2)**(1/4)/(8
4*c**2) + (b + 2*c*x)*(b*x + c*x**2)**(5/4)/(7*c)

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Mathematica [C]  time = 0.070116, size = 94, normalized size = 0.79 \[ \frac{x \left (5 b^4 \left (\frac{c x}{b}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{c x}{b}\right )-5 b^4-3 b^3 c x+38 b^2 c^2 x^2+60 b c^3 x^3+24 c^4 x^4\right )}{84 c^2 (x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/4),x]

[Out]

(x*(-5*b^4 - 3*b^3*c*x + 38*b^2*c^2*x^2 + 60*b*c^3*x^3 + 24*c^4*x^4 + 5*b^4*(1 +
(c*x)/b)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((c*x)/b)]))/(84*c^2*(x*(b + c
*x))^(3/4))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/4),x)

[Out]

int((c*x^2+b*x)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{5}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(5/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{5}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(5/4),x)

[Out]

Integral((b*x + c*x**2)**(5/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x\right )}^{\frac{5}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/4), x)