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

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

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Rubi [A]  time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1113, 364} \[ \frac {2 \sqrt {d x} \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {1}{4},-2 p;\frac {5}{4};-\frac {b x^2}{a}\right )}{d} \]

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))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1113

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^2 +
c*x^4)^FracPart[p])/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{
a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{\sqrt {d x}} \, dx &=\left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^{2 p}}{\sqrt {d x}} \, dx\\ &=\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 \, _2F_1\left (\frac {1}{4},-2 p;\frac {5}{4};-\frac {b x^2}{a}\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 0.01, 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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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\sqrt {d x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{\sqrt {d x}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\sqrt {d x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(1/2),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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{\sqrt {d x}}\, dx \]

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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