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

Optimal. Leaf size=91 \[ \frac{12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b (d x)^{7/2}}{7 d^3}+\frac{2 a^4 (d x)^{3/2}}{3 d}+\frac{8 a b^3 (d x)^{15/2}}{15 d^7}+\frac{2 b^4 (d x)^{19/2}}{19 d^9} \]

[Out]

(2*a^4*(d*x)^(3/2))/(3*d) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (12*a^2*b^2*(d*x)^(11/2))/(11*d^5) + (8*a*b^3*(d*x
)^(15/2))/(15*d^7) + (2*b^4*(d*x)^(19/2))/(19*d^9)

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Rubi [A]  time = 0.040864, 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} \[ \frac{12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b (d x)^{7/2}}{7 d^3}+\frac{2 a^4 (d x)^{3/2}}{3 d}+\frac{8 a b^3 (d x)^{15/2}}{15 d^7}+\frac{2 b^4 (d x)^{19/2}}{19 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0139412, size = 55, normalized size = 0.6 \[ \frac{2 x \sqrt{d x} \left (11970 a^2 b^2 x^4+12540 a^3 b x^2+7315 a^4+5852 a b^3 x^6+1155 b^4 x^8\right )}{21945} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[d*x]*(7315*a^4 + 12540*a^3*b*x^2 + 11970*a^2*b^2*x^4 + 5852*a*b^3*x^6 + 1155*b^4*x^8))/21945

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Maple [A]  time = 0.049, size = 52, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 1155\,{b}^{4}{x}^{8}+5852\,a{b}^{3}{x}^{6}+11970\,{a}^{2}{b}^{2}{x}^{4}+12540\,{a}^{3}b{x}^{2}+7315\,{a}^{4} \right ) }{21945}\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)^2*(d*x)^(1/2),x)

[Out]

2/21945*x*(1155*b^4*x^8+5852*a*b^3*x^6+11970*a^2*b^2*x^4+12540*a^3*b*x^2+7315*a^4)*(d*x)^(1/2)

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Maxima [A]  time = 0.983754, size = 99, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (1155 \, \left (d x\right )^{\frac{19}{2}} b^{4} + 5852 \, \left (d x\right )^{\frac{15}{2}} a b^{3} d^{2} + 11970 \, \left (d x\right )^{\frac{11}{2}} a^{2} b^{2} d^{4} + 12540 \, \left (d x\right )^{\frac{7}{2}} a^{3} b d^{6} + 7315 \, \left (d x\right )^{\frac{3}{2}} a^{4} d^{8}\right )}}{21945 \, d^{9}} \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)^2*(d*x)^(1/2),x, algorithm="maxima")

[Out]

2/21945*(1155*(d*x)^(19/2)*b^4 + 5852*(d*x)^(15/2)*a*b^3*d^2 + 11970*(d*x)^(11/2)*a^2*b^2*d^4 + 12540*(d*x)^(7
/2)*a^3*b*d^6 + 7315*(d*x)^(3/2)*a^4*d^8)/d^9

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Fricas [A]  time = 1.30301, size = 138, normalized size = 1.52 \begin{align*} \frac{2}{21945} \,{\left (1155 \, b^{4} x^{9} + 5852 \, a b^{3} x^{7} + 11970 \, a^{2} b^{2} x^{5} + 12540 \, a^{3} b x^{3} + 7315 \, a^{4} 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)^2*(d*x)^(1/2),x, algorithm="fricas")

[Out]

2/21945*(1155*b^4*x^9 + 5852*a*b^3*x^7 + 11970*a^2*b^2*x^5 + 12540*a^3*b*x^3 + 7315*a^4*x)*sqrt(d*x)

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Sympy [A]  time = 1.24744, size = 90, normalized size = 0.99 \begin{align*} \frac{2 a^{4} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{8 a^{3} b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{12 a^{2} b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} + \frac{8 a b^{3} \sqrt{d} x^{\frac{15}{2}}}{15} + \frac{2 b^{4} \sqrt{d} x^{\frac{19}{2}}}{19} \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)**2*(d*x)**(1/2),x)

[Out]

2*a**4*sqrt(d)*x**(3/2)/3 + 8*a**3*b*sqrt(d)*x**(7/2)/7 + 12*a**2*b**2*sqrt(d)*x**(11/2)/11 + 8*a*b**3*sqrt(d)
*x**(15/2)/15 + 2*b**4*sqrt(d)*x**(19/2)/19

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Giac [A]  time = 1.11738, size = 107, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (1155 \, \sqrt{d x} b^{4} d x^{9} + 5852 \, \sqrt{d x} a b^{3} d x^{7} + 11970 \, \sqrt{d x} a^{2} b^{2} d x^{5} + 12540 \, \sqrt{d x} a^{3} b d x^{3} + 7315 \, \sqrt{d x} a^{4} d x\right )}}{21945 \, 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)^2*(d*x)^(1/2),x, algorithm="giac")

[Out]

2/21945*(1155*sqrt(d*x)*b^4*d*x^9 + 5852*sqrt(d*x)*a*b^3*d*x^7 + 11970*sqrt(d*x)*a^2*b^2*d*x^5 + 12540*sqrt(d*
x)*a^3*b*d*x^3 + 7315*sqrt(d*x)*a^4*d*x)/d