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

Optimal. Leaf size=67 \[ \frac{2 (d x)^{3/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 d} \]

[Out]

(2*(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[3/4, -2*p, 7/4, -
((b*x^2)/a)])/(3*d*(1 + (b*x^2)/a)^(2*p))

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Rubi [A]  time = 0.0630344, antiderivative size = 67, 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 \[ \frac{2 (d x)^{3/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[3/4, -2*p, 7/4, -
((b*x^2)/a)])/(3*d*(1 + (b*x^2)/a)^(2*p))

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Rubi in Sympy [A]  time = 17.1153, size = 60, normalized size = 0.9 \[ \frac{2 \left (d x\right )^{\frac{3}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{- 2 p} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d*x)**(3/2)*(1 + b*x**2/a)**(-2*p)*(a**2 + 2*a*b*x**2 + b**2*x**4)**p*hyper((
-2*p, 3/4), (7/4,), -b*x**2/a)/(3*d)

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Mathematica [A]  time = 0.024855, size = 56, normalized size = 0.84 \[ \frac{2}{3} x \sqrt{d x} \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*((a + b*x^2)^2)^p*Hypergeometric2F1[3/4, -2*p, 7/4, -((b*x^2)/a)]
)/(3*(1 + (b*x^2)/a)^(2*p))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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