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

Optimal. Leaf size=93 \[ \frac{2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

[Out]

(2*a*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (2*b*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(7*d^3*(a + b*x^2))

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Rubi [A]  time = 0.0290742, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1112, 14} \[ \frac{2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(2*a*(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d*(a + b*x^2)) + (2*b*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(7*d^3*(a + b*x^2))

Rule 1112

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \sqrt{d x} \left (a b+b^2 x^2\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a b \sqrt{d x}+\frac{b^2 (d x)^{5/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{2 a (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac{2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0132072, size = 44, normalized size = 0.47 \[ \frac{2 \sqrt{d x} \sqrt{\left (a+b x^2\right )^2} \left (7 a x+3 b x^3\right )}{21 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(7*a*x + 3*b*x^3))/(21*(a + b*x^2))

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Maple [A]  time = 0.043, size = 39, normalized size = 0.4 \begin{align*}{\frac{2\, \left ( 3\,b{x}^{2}+7\,a \right ) x}{21\,b{x}^{2}+21\,a}\sqrt{dx}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*x*(3*b*x^2+7*a)*(d*x)^(1/2)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [A]  time = 1.03298, size = 30, normalized size = 0.32 \begin{align*} \frac{2}{21} \,{\left (3 \, b \sqrt{d} x^{3} + 7 \, a \sqrt{d} x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b*sqrt(d)*x^3 + 7*a*sqrt(d)*x)*sqrt(x)

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Fricas [A]  time = 1.25723, size = 46, normalized size = 0.49 \begin{align*} \frac{2}{21} \,{\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b*x^3 + 7*a*x)*sqrt(d*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(1/2)*((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.19223, size = 59, normalized size = 0.63 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{d x} b d x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 7 \, \sqrt{d x} a d x \mathrm{sgn}\left (b x^{2} + a\right )\right )}}{21 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*sqrt(d*x)*b*d*x^3*sgn(b*x^2 + a) + 7*sqrt(d*x)*a*d*x*sgn(b*x^2 + a))/d