3.811 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^p}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=65 \[ \frac{2 \sqrt{d x} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{1}{4},-2 p;\frac{5}{4};-\frac{b x^2}{a}\right )}{d} \]

[Out]

(2*Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[1/4, -2*p, 5/4, -((
b*x^2)/a)])/(d*(1 + (b*x^2)/a)^(2*p))

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Rubi [A]  time = 0.0622319, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 \sqrt{d x} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{1}{4},-2 p;\frac{5}{4};-\frac{b x^2}{a}\right )}{d} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/Sqrt[d*x],x]

[Out]

(2*Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[1/4, -2*p, 5/4, -((
b*x^2)/a)])/(d*(1 + (b*x^2)/a)^(2*p))

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Rubi in Sympy [A]  time = 17.3414, size = 58, normalized size = 0.89 \[ \frac{2 \sqrt{d x} \left (1 + \frac{b x^{2}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**p/(d*x)**(1/2),x)

[Out]

2*sqrt(d*x)*(1 + b*x**2/a)**(-2*p)*(a**2 + 2*a*b*x**2 + b**2*x**4)**p*hyper((-2*
p, 1/4), (5/4,), -b*x**2/a)/d

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Mathematica [A]  time = 0.0237895, size = 54, normalized size = 0.83 \[ \frac{2 x \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{1}{4},-2 p;\frac{5}{4};-\frac{b x^2}{a}\right )}{\sqrt{d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/Sqrt[d*x],x]

[Out]

(2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[1/4, -2*p, 5/4, -((b*x^2)/a)])/(Sqrt[d*
x]*(1 + (b*x^2)/a)^(2*p))

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \[ \int{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}{\frac{1}{\sqrt{dx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(1/2),x)

[Out]

int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x),x, algorithm="maxima")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{\sqrt{d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**p/(d*x)**(1/2),x)

[Out]

Integral(((a + b*x**2)**2)**p/sqrt(d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x),x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/sqrt(d*x), x)