3.47 \(\int \frac{1}{\left (b x+c x^2\right )^{13/4}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac{112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac{4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac{448 \sqrt{2} c^2 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{15 b^5 \sqrt [4]{b x+c x^2}} \]

[Out]

(-4*(b + 2*c*x))/(9*b^2*(b*x + c*x^2)^(9/4)) + (112*c*(b + 2*c*x))/(45*b^4*(b*x
+ c*x^2)^(5/4)) - (448*c^2*(b + 2*c*x))/(15*b^6*(b*x + c*x^2)^(1/4)) + (448*Sqrt
[2]*c^2*(-((c*(b*x + c*x^2))/b^2))^(1/4)*EllipticE[ArcSin[1 + (2*c*x)/b]/2, 2])/
(15*b^5*(b*x + c*x^2)^(1/4))

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Rubi [A]  time = 0.129558, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{448 c^2 (b+2 c x)}{15 b^6 \sqrt [4]{b x+c x^2}}+\frac{112 c (b+2 c x)}{45 b^4 \left (b x+c x^2\right )^{5/4}}-\frac{4 (b+2 c x)}{9 b^2 \left (b x+c x^2\right )^{9/4}}+\frac{448 \sqrt{2} c^2 \sqrt [4]{-\frac{c \left (b x+c x^2\right )}{b^2}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{2 c x}{b}+1\right )\right |2\right )}{15 b^5 \sqrt [4]{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-13/4),x]

[Out]

(-4*(b + 2*c*x))/(9*b^2*(b*x + c*x^2)^(9/4)) + (112*c*(b + 2*c*x))/(45*b^4*(b*x
+ c*x^2)^(5/4)) - (448*c^2*(b + 2*c*x))/(15*b^6*(b*x + c*x^2)^(1/4)) + (448*Sqrt
[2]*c^2*(-((c*(b*x + c*x^2))/b^2))^(1/4)*EllipticE[ArcSin[1 + (2*c*x)/b]/2, 2])/
(15*b^5*(b*x + c*x^2)^(1/4))

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Rubi in Sympy [A]  time = 19.6136, size = 139, normalized size = 0.95 \[ - \frac{4 \left (b + 2 c x\right )}{9 b^{2} \left (b x + c x^{2}\right )^{\frac{9}{4}}} + \frac{112 c \left (b + 2 c x\right )}{45 b^{4} \left (b x + c x^{2}\right )^{\frac{5}{4}}} + \frac{448 \sqrt{2} c^{2} \sqrt [4]{\frac{c \left (- b x - c x^{2}\right )}{b^{2}}} E\left (\frac{\operatorname{asin}{\left (1 + \frac{2 c x}{b} \right )}}{2}\middle | 2\right )}{15 b^{5} \sqrt [4]{b x + c x^{2}}} - \frac{448 c^{2} \left (b + 2 c x\right )}{15 b^{6} \sqrt [4]{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x**2+b*x)**(13/4),x)

[Out]

-4*(b + 2*c*x)/(9*b**2*(b*x + c*x**2)**(9/4)) + 112*c*(b + 2*c*x)/(45*b**4*(b*x
+ c*x**2)**(5/4)) + 448*sqrt(2)*c**2*(c*(-b*x - c*x**2)/b**2)**(1/4)*elliptic_e(
asin(1 + 2*c*x/b)/2, 2)/(15*b**5*(b*x + c*x**2)**(1/4)) - 448*c**2*(b + 2*c*x)/(
15*b**6*(b*x + c*x**2)**(1/4))

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Mathematica [C]  time = 0.144171, size = 114, normalized size = 0.78 \[ -\frac{4 \left (5 b^5-18 b^4 c x+252 b^3 c^2 x^2+1288 b^2 c^3 x^3+1680 b c^4 x^4-448 c^3 x^3 (b+c x)^2 \sqrt [4]{\frac{c x}{b}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{c x}{b}\right )+672 c^5 x^5\right )}{45 b^6 (x (b+c x))^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-13/4),x]

[Out]

(-4*(5*b^5 - 18*b^4*c*x + 252*b^3*c^2*x^2 + 1288*b^2*c^3*x^3 + 1680*b*c^4*x^4 +
672*c^5*x^5 - 448*c^3*x^3*(b + c*x)^2*(1 + (c*x)/b)^(1/4)*Hypergeometric2F1[1/4,
3/4, 7/4, -((c*x)/b)]))/(45*b^6*(x*(b + c*x))^(9/4))

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx \right ) ^{-{\frac{13}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^(13/4),x)

[Out]

int(1/(c*x^2+b*x)^(13/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-13/4),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(-13/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}\right )}{\left (c x^{2} + b x\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-13/4),x, algorithm="fricas")

[Out]

integral(1/((c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3)*(c*x^2 + b*x)^(1/4))
, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x + c x^{2}\right )^{\frac{13}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**(13/4),x)

[Out]

Integral((b*x + c*x**2)**(-13/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(-13/4),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(-13/4), x)