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]

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

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Rubi [A]  time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, 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 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]]

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.91, size = 18, normalized size = 0.19 \[ \frac {2}{21} \, {\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt {d x} \]

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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giac [A]  time = 0.17, size = 37, normalized size = 0.40 \[ \frac {2}{7} \, \sqrt {d x} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, \sqrt {d x} a x \mathrm {sgn}\left (b x^{2} + a\right ) \]

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]

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

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maple [A]  time = 0.00, size = 39, normalized size = 0.42 \[ \frac {2 \left (3 b \,x^{2}+7 a \right ) \sqrt {d x}\, \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{21 \left (b \,x^{2}+a \right )} \]

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 = 1.40, size = 25, normalized size = 0.27 \[ \frac {2 \, {\left (3 \, \left (d x\right )^{\frac {7}{2}} b + 7 \, \left (d x\right )^{\frac {3}{2}} a d^{2}\right )}}{21 \, d^{3}} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {d\,x}\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 133.05, size = 27, normalized size = 0.29 \[ \frac {2 a \left (d x\right )^{\frac {3}{2}}}{3 d} + \frac {2 b \left (d x\right )^{\frac {7}{2}}}{7 d^{3}} \]

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]

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

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