3.665 \(\int (d x)^{5/2} (a^2+2 a b x^2+b^2 x^4) \, dx\)

Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{7/2}}{7 d}+\frac{4 a b (d x)^{11/2}}{11 d^3}+\frac{2 b^2 (d x)^{15/2}}{15 d^5} \]

[Out]

(2*a^2*(d*x)^(7/2))/(7*d) + (4*a*b*(d*x)^(11/2))/(11*d^3) + (2*b^2*(d*x)^(15/2))/(15*d^5)

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Rubi [A]  time = 0.0136195, 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} \[ \frac{2 a^2 (d x)^{7/2}}{7 d}+\frac{4 a b (d x)^{11/2}}{11 d^3}+\frac{2 b^2 (d x)^{15/2}}{15 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*a^2*(d*x)^(7/2))/(7*d) + (4*a*b*(d*x)^(11/2))/(11*d^3) + (2*b^2*(d*x)^(15/2))/(15*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)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{5/2}+\frac{2 a b (d x)^{9/2}}{d^2}+\frac{b^2 (d x)^{13/2}}{d^4}\right ) \, dx\\ &=\frac{2 a^2 (d x)^{7/2}}{7 d}+\frac{4 a b (d x)^{11/2}}{11 d^3}+\frac{2 b^2 (d x)^{15/2}}{15 d^5}\\ \end{align*}

Mathematica [A]  time = 0.014221, size = 33, normalized size = 0.65 \[ \frac{2 x (d x)^{5/2} \left (165 a^2+210 a b x^2+77 b^2 x^4\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*x*(d*x)^(5/2)*(165*a^2 + 210*a*b*x^2 + 77*b^2*x^4))/1155

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Maple [A]  time = 0.046, size = 30, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 77\,{b}^{2}{x}^{4}+210\,ab{x}^{2}+165\,{a}^{2} \right ) }{1155} \left ( dx \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

2/1155*x*(77*b^2*x^4+210*a*b*x^2+165*a^2)*(d*x)^(5/2)

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Maxima [A]  time = 0.954908, size = 55, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (77 \, \left (d x\right )^{\frac{15}{2}} b^{2} + 210 \, \left (d x\right )^{\frac{11}{2}} a b d^{2} + 165 \, \left (d x\right )^{\frac{7}{2}} a^{2} d^{4}\right )}}{1155 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

2/1155*(77*(d*x)^(15/2)*b^2 + 210*(d*x)^(11/2)*a*b*d^2 + 165*(d*x)^(7/2)*a^2*d^4)/d^5

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Fricas [A]  time = 1.2434, size = 96, normalized size = 1.88 \begin{align*} \frac{2}{1155} \,{\left (77 \, b^{2} d^{2} x^{7} + 210 \, a b d^{2} x^{5} + 165 \, a^{2} d^{2} x^{3}\right )} \sqrt{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

2/1155*(77*b^2*d^2*x^7 + 210*a*b*d^2*x^5 + 165*a^2*d^2*x^3)*sqrt(d*x)

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Sympy [A]  time = 2.58678, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{2} d^{\frac{5}{2}} x^{\frac{7}{2}}}{7} + \frac{4 a b d^{\frac{5}{2}} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} d^{\frac{5}{2}} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

2*a**2*d**(5/2)*x**(7/2)/7 + 4*a*b*d**(5/2)*x**(11/2)/11 + 2*b**2*d**(5/2)*x**(15/2)/15

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Giac [A]  time = 1.11216, size = 65, normalized size = 1.27 \begin{align*} \frac{2}{15} \, \sqrt{d x} b^{2} d^{2} x^{7} + \frac{4}{11} \, \sqrt{d x} a b d^{2} x^{5} + \frac{2}{7} \, \sqrt{d x} a^{2} d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

2/15*sqrt(d*x)*b^2*d^2*x^7 + 4/11*sqrt(d*x)*a*b*d^2*x^5 + 2/7*sqrt(d*x)*a^2*d^2*x^3