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

Optimal. Leaf size=91 \[ \frac{12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac{8 a^3 b (d x)^{9/2}}{9 d^3}+\frac{2 a^4 (d x)^{5/2}}{5 d}+\frac{8 a b^3 (d x)^{17/2}}{17 d^7}+\frac{2 b^4 (d x)^{21/2}}{21 d^9} \]

[Out]

(2*a^4*(d*x)^(5/2))/(5*d) + (8*a^3*b*(d*x)^(9/2))/(9*d^3) + (12*a^2*b^2*(d*x)^(13/2))/(13*d^5) + (8*a*b^3*(d*x
)^(17/2))/(17*d^7) + (2*b^4*(d*x)^(21/2))/(21*d^9)

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Rubi [A]  time = 0.0439847, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {28, 270} \[ \frac{12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac{8 a^3 b (d x)^{9/2}}{9 d^3}+\frac{2 a^4 (d x)^{5/2}}{5 d}+\frac{8 a b^3 (d x)^{17/2}}{17 d^7}+\frac{2 b^4 (d x)^{21/2}}{21 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^4*(d*x)^(5/2))/(5*d) + (8*a^3*b*(d*x)^(9/2))/(9*d^3) + (12*a^2*b^2*(d*x)^(13/2))/(13*d^5) + (8*a*b^3*(d*x
)^(17/2))/(17*d^7) + (2*b^4*(d*x)^(21/2))/(21*d^9)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int (d x)^{3/2} \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4 (d x)^{3/2}+\frac{4 a^3 b^5 (d x)^{7/2}}{d^2}+\frac{6 a^2 b^6 (d x)^{11/2}}{d^4}+\frac{4 a b^7 (d x)^{15/2}}{d^6}+\frac{b^8 (d x)^{19/2}}{d^8}\right ) \, dx}{b^4}\\ &=\frac{2 a^4 (d x)^{5/2}}{5 d}+\frac{8 a^3 b (d x)^{9/2}}{9 d^3}+\frac{12 a^2 b^2 (d x)^{13/2}}{13 d^5}+\frac{8 a b^3 (d x)^{17/2}}{17 d^7}+\frac{2 b^4 (d x)^{21/2}}{21 d^9}\\ \end{align*}

Mathematica [A]  time = 0.0178492, size = 55, normalized size = 0.6 \[ \frac{2 x (d x)^{3/2} \left (32130 a^2 b^2 x^4+30940 a^3 b x^2+13923 a^4+16380 a b^3 x^6+3315 b^4 x^8\right )}{69615} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(3/2)*(13923*a^4 + 30940*a^3*b*x^2 + 32130*a^2*b^2*x^4 + 16380*a*b^3*x^6 + 3315*b^4*x^8))/69615

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Maple [A]  time = 0.049, size = 52, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 3315\,{b}^{4}{x}^{8}+16380\,a{b}^{3}{x}^{6}+32130\,{a}^{2}{b}^{2}{x}^{4}+30940\,{a}^{3}b{x}^{2}+13923\,{a}^{4} \right ) }{69615} \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)^2,x)

[Out]

2/69615*x*(3315*b^4*x^8+16380*a*b^3*x^6+32130*a^2*b^2*x^4+30940*a^3*b*x^2+13923*a^4)*(d*x)^(3/2)

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Maxima [A]  time = 1.00658, size = 99, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (3315 \, \left (d x\right )^{\frac{21}{2}} b^{4} + 16380 \, \left (d x\right )^{\frac{17}{2}} a b^{3} d^{2} + 32130 \, \left (d x\right )^{\frac{13}{2}} a^{2} b^{2} d^{4} + 30940 \, \left (d x\right )^{\frac{9}{2}} a^{3} b d^{6} + 13923 \, \left (d x\right )^{\frac{5}{2}} a^{4} d^{8}\right )}}{69615 \, d^{9}} \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)^2,x, algorithm="maxima")

[Out]

2/69615*(3315*(d*x)^(21/2)*b^4 + 16380*(d*x)^(17/2)*a*b^3*d^2 + 32130*(d*x)^(13/2)*a^2*b^2*d^4 + 30940*(d*x)^(
9/2)*a^3*b*d^6 + 13923*(d*x)^(5/2)*a^4*d^8)/d^9

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Fricas [A]  time = 1.32095, size = 158, normalized size = 1.74 \begin{align*} \frac{2}{69615} \,{\left (3315 \, b^{4} d x^{10} + 16380 \, a b^{3} d x^{8} + 32130 \, a^{2} b^{2} d x^{6} + 30940 \, a^{3} b d x^{4} + 13923 \, a^{4} 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)^2,x, algorithm="fricas")

[Out]

2/69615*(3315*b^4*d*x^10 + 16380*a*b^3*d*x^8 + 32130*a^2*b^2*d*x^6 + 30940*a^3*b*d*x^4 + 13923*a^4*d*x^2)*sqrt
(d*x)

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Sympy [A]  time = 2.77763, size = 90, normalized size = 0.99 \begin{align*} \frac{2 a^{4} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{8 a^{3} b d^{\frac{3}{2}} x^{\frac{9}{2}}}{9} + \frac{12 a^{2} b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} + \frac{8 a b^{3} d^{\frac{3}{2}} x^{\frac{17}{2}}}{17} + \frac{2 b^{4} d^{\frac{3}{2}} x^{\frac{21}{2}}}{21} \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)**2,x)

[Out]

2*a**4*d**(3/2)*x**(5/2)/5 + 8*a**3*b*d**(3/2)*x**(9/2)/9 + 12*a**2*b**2*d**(3/2)*x**(13/2)/13 + 8*a*b**3*d**(
3/2)*x**(17/2)/17 + 2*b**4*d**(3/2)*x**(21/2)/21

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Giac [A]  time = 1.11493, size = 103, normalized size = 1.13 \begin{align*} \frac{2}{21} \, \sqrt{d x} b^{4} d x^{10} + \frac{8}{17} \, \sqrt{d x} a b^{3} d x^{8} + \frac{12}{13} \, \sqrt{d x} a^{2} b^{2} d x^{6} + \frac{8}{9} \, \sqrt{d x} a^{3} b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{4} 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)^2,x, algorithm="giac")

[Out]

2/21*sqrt(d*x)*b^4*d*x^10 + 8/17*sqrt(d*x)*a*b^3*d*x^8 + 12/13*sqrt(d*x)*a^2*b^2*d*x^6 + 8/9*sqrt(d*x)*a^3*b*d
*x^4 + 2/5*sqrt(d*x)*a^4*d*x^2