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

Optimal. Leaf size=91 \[ \frac{4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac{8 a^3 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 (d x)^{7/2}}{7 d}+\frac{8 a b^3 (d x)^{19/2}}{19 d^7}+\frac{2 b^4 (d x)^{23/2}}{23 d^9} \]

[Out]

(2*a^4*(d*x)^(7/2))/(7*d) + (8*a^3*b*(d*x)^(11/2))/(11*d^3) + (4*a^2*b^2*(d*x)^(15/2))/(5*d^5) + (8*a*b^3*(d*x
)^(19/2))/(19*d^7) + (2*b^4*(d*x)^(23/2))/(23*d^9)

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Rubi [A]  time = 0.0448823, 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} \[ \frac{4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac{8 a^3 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 (d x)^{7/2}}{7 d}+\frac{8 a b^3 (d x)^{19/2}}{19 d^7}+\frac{2 b^4 (d x)^{23/2}}{23 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^4*(d*x)^(7/2))/(7*d) + (8*a^3*b*(d*x)^(11/2))/(11*d^3) + (4*a^2*b^2*(d*x)^(15/2))/(5*d^5) + (8*a*b^3*(d*x
)^(19/2))/(19*d^7) + (2*b^4*(d*x)^(23/2))/(23*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)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int (d x)^{5/2} \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4 (d x)^{5/2}+\frac{4 a^3 b^5 (d x)^{9/2}}{d^2}+\frac{6 a^2 b^6 (d x)^{13/2}}{d^4}+\frac{4 a b^7 (d x)^{17/2}}{d^6}+\frac{b^8 (d x)^{21/2}}{d^8}\right ) \, dx}{b^4}\\ &=\frac{2 a^4 (d x)^{7/2}}{7 d}+\frac{8 a^3 b (d x)^{11/2}}{11 d^3}+\frac{4 a^2 b^2 (d x)^{15/2}}{5 d^5}+\frac{8 a b^3 (d x)^{19/2}}{19 d^7}+\frac{2 b^4 (d x)^{23/2}}{23 d^9}\\ \end{align*}

Mathematica [A]  time = 0.0213344, size = 55, normalized size = 0.6 \[ \frac{2 x (d x)^{5/2} \left (67298 a^2 b^2 x^4+61180 a^3 b x^2+24035 a^4+35420 a b^3 x^6+7315 b^4 x^8\right )}{168245} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(d*x)^(5/2)*(24035*a^4 + 61180*a^3*b*x^2 + 67298*a^2*b^2*x^4 + 35420*a*b^3*x^6 + 7315*b^4*x^8))/168245

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Maple [A]  time = 0.048, size = 52, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 7315\,{b}^{4}{x}^{8}+35420\,a{b}^{3}{x}^{6}+67298\,{a}^{2}{b}^{2}{x}^{4}+61180\,{a}^{3}b{x}^{2}+24035\,{a}^{4} \right ) }{168245} \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)^2,x)

[Out]

2/168245*x*(7315*b^4*x^8+35420*a*b^3*x^6+67298*a^2*b^2*x^4+61180*a^3*b*x^2+24035*a^4)*(d*x)^(5/2)

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Maxima [A]  time = 1.08328, size = 99, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (7315 \, \left (d x\right )^{\frac{23}{2}} b^{4} + 35420 \, \left (d x\right )^{\frac{19}{2}} a b^{3} d^{2} + 67298 \, \left (d x\right )^{\frac{15}{2}} a^{2} b^{2} d^{4} + 61180 \, \left (d x\right )^{\frac{11}{2}} a^{3} b d^{6} + 24035 \, \left (d x\right )^{\frac{7}{2}} a^{4} d^{8}\right )}}{168245 \, d^{9}} \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)^2,x, algorithm="maxima")

[Out]

2/168245*(7315*(d*x)^(23/2)*b^4 + 35420*(d*x)^(19/2)*a*b^3*d^2 + 67298*(d*x)^(15/2)*a^2*b^2*d^4 + 61180*(d*x)^
(11/2)*a^3*b*d^6 + 24035*(d*x)^(7/2)*a^4*d^8)/d^9

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Fricas [A]  time = 1.22778, size = 173, normalized size = 1.9 \begin{align*} \frac{2}{168245} \,{\left (7315 \, b^{4} d^{2} x^{11} + 35420 \, a b^{3} d^{2} x^{9} + 67298 \, a^{2} b^{2} d^{2} x^{7} + 61180 \, a^{3} b d^{2} x^{5} + 24035 \, a^{4} 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)^2,x, algorithm="fricas")

[Out]

2/168245*(7315*b^4*d^2*x^11 + 35420*a*b^3*d^2*x^9 + 67298*a^2*b^2*d^2*x^7 + 61180*a^3*b*d^2*x^5 + 24035*a^4*d^
2*x^3)*sqrt(d*x)

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Sympy [A]  time = 5.3947, size = 90, normalized size = 0.99 \begin{align*} \frac{2 a^{4} d^{\frac{5}{2}} x^{\frac{7}{2}}}{7} + \frac{8 a^{3} b d^{\frac{5}{2}} x^{\frac{11}{2}}}{11} + \frac{4 a^{2} b^{2} d^{\frac{5}{2}} x^{\frac{15}{2}}}{5} + \frac{8 a b^{3} d^{\frac{5}{2}} x^{\frac{19}{2}}}{19} + \frac{2 b^{4} d^{\frac{5}{2}} x^{\frac{23}{2}}}{23} \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)**2,x)

[Out]

2*a**4*d**(5/2)*x**(7/2)/7 + 8*a**3*b*d**(5/2)*x**(11/2)/11 + 4*a**2*b**2*d**(5/2)*x**(15/2)/5 + 8*a*b**3*d**(
5/2)*x**(19/2)/19 + 2*b**4*d**(5/2)*x**(23/2)/23

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Giac [A]  time = 1.11827, size = 116, normalized size = 1.27 \begin{align*} \frac{2}{23} \, \sqrt{d x} b^{4} d^{2} x^{11} + \frac{8}{19} \, \sqrt{d x} a b^{3} d^{2} x^{9} + \frac{4}{5} \, \sqrt{d x} a^{2} b^{2} d^{2} x^{7} + \frac{8}{11} \, \sqrt{d x} a^{3} b d^{2} x^{5} + \frac{2}{7} \, \sqrt{d x} a^{4} 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)^2,x, algorithm="giac")

[Out]

2/23*sqrt(d*x)*b^4*d^2*x^11 + 8/19*sqrt(d*x)*a*b^3*d^2*x^9 + 4/5*sqrt(d*x)*a^2*b^2*d^2*x^7 + 8/11*sqrt(d*x)*a^
3*b*d^2*x^5 + 2/7*sqrt(d*x)*a^4*d^2*x^3