3.45 \(\int \frac{1}{\left (b x+c x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=83 \[ \frac{4 \sqrt{2} \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{b \sqrt [4]{b x+c x^2}}-\frac{4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}} \]

[Out]

(-4*(b + 2*c*x))/(b^2*(b*x + c*x^2)^(1/4)) + (4*Sqrt[2]*(-((c*(b*x + c*x^2))/b^2
))^(1/4)*EllipticE[ArcSin[1 + (2*c*x)/b]/2, 2])/(b*(b*x + c*x^2)^(1/4))

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Rubi [A]  time = 0.0732425, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{4 \sqrt{2} \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{b \sqrt [4]{b x+c x^2}}-\frac{4 (b+2 c x)}{b^2 \sqrt [4]{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-5/4),x]

[Out]

(-4*(b + 2*c*x))/(b^2*(b*x + c*x^2)^(1/4)) + (4*Sqrt[2]*(-((c*(b*x + c*x^2))/b^2
))^(1/4)*EllipticE[ArcSin[1 + (2*c*x)/b]/2, 2])/(b*(b*x + c*x^2)^(1/4))

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Rubi in Sympy [A]  time = 13.5915, size = 75, normalized size = 0.9 \[ \frac{4 \sqrt{2} \sqrt [4]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} E\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{b \sqrt [4]{b x + c x^{2}}} - \frac{4 \left (b + 2 c x\right )}{b^{2} \sqrt [4]{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(5/4),x)

[Out]

4*sqrt(2)*(c*(-b*x - c*x**2)/b**2)**(1/4)*elliptic_e(asin(1 + 2*c*x/b)/2, 2)/(b*
(b*x + c*x**2)**(1/4)) - 4*(b + 2*c*x)/(b**2*(b*x + c*x**2)**(1/4))

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Mathematica [C]  time = 0.053344, size = 59, normalized size = 0.71 \[ -\frac{4 \left (-4 c x \sqrt [4]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )+3 b+6 c x\right )}{3 b^2 \sqrt [4]{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(3*b + 6*c*x - 4*c*x*(1 + (c*x)/b)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -(
(c*x)/b)]))/(3*b^2*(x*(b + c*x))^(1/4))

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Maple [F]  time = 0.09, 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(1/(c*x^2+b*x)^(5/4),x)

[Out]

int(1/(c*x^2+b*x)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{4}}}\,{d x} \]

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. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{4}}}, x\right ) \]

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. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{4}}}\,{d x} \]

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)