3.9.10 \(\int \sqrt {d x} (a^2+2 a b x^2+b^2 x^4)^p \, dx\) [810]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 67, 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 = {1127, 371} \begin {gather*} \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} \end {gather*}

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))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1127

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 \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, 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 \sqrt {d x} \left (1+\frac {b x^2}{a}\right )^{2 p} \, dx\\ &=\frac {2 (d x)^{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 \, _2F_1\left (\frac {3}{4},-2 p;\frac {7}{4};-\frac {b x^2}{a}\right )}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 56, normalized size = 0.84 \begin {gather*} \frac {2}{3} x \sqrt {d x} \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \, _2F_1\left (\frac {3}{4},-2 p;\frac {7}{4};-\frac {b x^2}{a}\right ) \end {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, 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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, 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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{p}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {d\,x}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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