3.729 \(\int (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

[Out]

(2*a*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (2*b*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(9*d^3*(a + b*x^2))

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Rubi [A]  time = 0.0288275, 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} \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (2*b*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(9*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 (d x)^{3/2} \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 (d x)^{3/2} \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 (d x)^{3/2}+\frac{b^2 (d x)^{7/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.013489, size = 44, normalized size = 0.47 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (9 a+5 b x^2\right )}{45 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(9*a + 5*b*x^2))/(45*(a + b*x^2))

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Maple [A]  time = 0.046, size = 39, normalized size = 0.4 \begin{align*}{\frac{2\, \left ( 5\,b{x}^{2}+9\,a \right ) x}{45\,b{x}^{2}+45\,a} \left ( dx \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*x*(5*b*x^2+9*a)*(d*x)^(3/2)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [A]  time = 1.00629, size = 30, normalized size = 0.32 \begin{align*} \frac{2}{45} \,{\left (5 \, b d^{\frac{3}{2}} x^{3} + 9 \, a d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*((b*x^2+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2/45*(5*b*d^(3/2)*x^3 + 9*a*d^(3/2)*x)*x^(3/2)

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Fricas [A]  time = 1.32534, size = 54, normalized size = 0.58 \begin{align*} \frac{2}{45} \,{\left (5 \, b d x^{4} + 9 \, a d x^{2}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*((b*x^2+a)^2)^(1/2),x, algorithm="fricas")

[Out]

2/45*(5*b*d*x^4 + 9*a*d*x^2)*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)**(3/2)*((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.2685, size = 55, normalized size = 0.59 \begin{align*} \frac{2}{9} \, \sqrt{d x} b d x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a d x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*((b*x^2+a)^2)^(1/2),x, algorithm="giac")

[Out]

2/9*sqrt(d*x)*b*d*x^4*sgn(b*x^2 + a) + 2/5*sqrt(d*x)*a*d*x^2*sgn(b*x^2 + a)