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

Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]

[Out]

(2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(13*d^5)

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Rubi [A]  time = 0.0142823, 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} \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(13*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 (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{3/2}+\frac{2 a b (d x)^{7/2}}{d^2}+\frac{b^2 (d x)^{11/2}}{d^4}\right ) \, dx\\ &=\frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0111555, size = 33, normalized size = 0.65 \[ \frac{2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(3/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585

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Maple [A]  time = 0.047, size = 30, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 45\,{b}^{2}{x}^{4}+130\,ab{x}^{2}+117\,{a}^{2} \right ) }{585} \left ( dx \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

2/585*x*(45*b^2*x^4+130*a*b*x^2+117*a^2)*(d*x)^(3/2)

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Maxima [A]  time = 0.954147, size = 55, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (45 \, \left (d x\right )^{\frac{13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac{9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac{5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

2/585*(45*(d*x)^(13/2)*b^2 + 130*(d*x)^(9/2)*a*b*d^2 + 117*(d*x)^(5/2)*a^2*d^4)/d^5

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Fricas [A]  time = 1.21244, size = 86, normalized size = 1.69 \begin{align*} \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} 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^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d*x^6 + 130*a*b*d*x^4 + 117*a^2*d*x^2)*sqrt(d*x)

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Sympy [A]  time = 1.1575, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{2} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a b d^{\frac{3}{2}} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

2*a**2*d**(3/2)*x**(5/2)/5 + 4*a*b*d**(3/2)*x**(9/2)/9 + 2*b**2*d**(3/2)*x**(13/2)/13

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Giac [A]  time = 1.12954, size = 57, normalized size = 1.12 \begin{align*} \frac{2}{13} \, \sqrt{d x} b^{2} d x^{6} + \frac{4}{9} \, \sqrt{d x} a b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{2} d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

2/13*sqrt(d*x)*b^2*d*x^6 + 4/9*sqrt(d*x)*a*b*d*x^4 + 2/5*sqrt(d*x)*a^2*d*x^2