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

Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]

[Out]

(2*a^2*(d*x)^(3/2))/(3*d) + (4*a*b*(d*x)^(7/2))/(7*d^3) + (2*b^2*(d*x)^(11/2))/(11*d^5)

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Rubi [A]  time = 0.0130496, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {14} \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(d*x)^(3/2))/(3*d) + (4*a*b*(d*x)^(7/2))/(7*d^3) + (2*b^2*(d*x)^(11/2))/(11*d^5)

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} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 \sqrt{d x}+\frac{2 a b (d x)^{5/2}}{d^2}+\frac{b^2 (d x)^{9/2}}{d^4}\right ) \, dx\\ &=\frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0086985, size = 33, normalized size = 0.65 \[ \frac{2}{231} x \sqrt{d x} \left (77 a^2+66 a b x^2+21 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[d*x]*(77*a^2 + 66*a*b*x^2 + 21*b^2*x^4))/231

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Maple [A]  time = 0.047, size = 30, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 21\,{b}^{2}{x}^{4}+66\,ab{x}^{2}+77\,{a}^{2} \right ) }{231}\sqrt{dx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*x*(21*b^2*x^4+66*a*b*x^2+77*a^2)*(d*x)^(1/2)

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Maxima [A]  time = 0.974603, size = 55, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (21 \, \left (d x\right )^{\frac{11}{2}} b^{2} + 66 \, \left (d x\right )^{\frac{7}{2}} a b d^{2} + 77 \, \left (d x\right )^{\frac{3}{2}} a^{2} d^{4}\right )}}{231 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*(d*x)^(11/2)*b^2 + 66*(d*x)^(7/2)*a*b*d^2 + 77*(d*x)^(3/2)*a^2*d^4)/d^5

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Fricas [A]  time = 1.29525, size = 73, normalized size = 1.43 \begin{align*} \frac{2}{231} \,{\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*b^2*x^5 + 66*a*b*x^3 + 77*a^2*x)*sqrt(d*x)

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Sympy [A]  time = 0.438113, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{2} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{4 a b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)*(d*x)**(1/2),x)

[Out]

2*a**2*sqrt(d)*x**(3/2)/3 + 4*a*b*sqrt(d)*x**(7/2)/7 + 2*b**2*sqrt(d)*x**(11/2)/11

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Giac [A]  time = 1.11155, size = 61, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (21 \, \sqrt{d x} b^{2} d x^{5} + 66 \, \sqrt{d x} a b d x^{3} + 77 \, \sqrt{d x} a^{2} d x\right )}}{231 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*sqrt(d*x)*b^2*d*x^5 + 66*sqrt(d*x)*a*b*d*x^3 + 77*sqrt(d*x)*a^2*d*x)/d