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

Optimal. Leaf size=89 \[ \frac{4 a^2 b^2 (d x)^{9/2}}{3 d^5}+\frac{8 a^3 b (d x)^{5/2}}{5 d^3}+\frac{2 a^4 \sqrt{d x}}{d}+\frac{8 a b^3 (d x)^{13/2}}{13 d^7}+\frac{2 b^4 (d x)^{17/2}}{17 d^9} \]

[Out]

(2*a^4*Sqrt[d*x])/d + (8*a^3*b*(d*x)^(5/2))/(5*d^3) + (4*a^2*b^2*(d*x)^(9/2))/(3*d^5) + (8*a*b^3*(d*x)^(13/2))
/(13*d^7) + (2*b^4*(d*x)^(17/2))/(17*d^9)

________________________________________________________________________________________

Rubi [A]  time = 0.0413977, antiderivative size = 89, 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} \[ \frac{4 a^2 b^2 (d x)^{9/2}}{3 d^5}+\frac{8 a^3 b (d x)^{5/2}}{5 d^3}+\frac{2 a^4 \sqrt{d x}}{d}+\frac{8 a b^3 (d x)^{13/2}}{13 d^7}+\frac{2 b^4 (d x)^{17/2}}{17 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0153181, size = 55, normalized size = 0.62 \[ \frac{2 \left (2210 a^2 b^2 x^5+2652 a^3 b x^3+3315 a^4 x+1020 a b^3 x^7+195 b^4 x^9\right )}{3315 \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3315*a^4*x + 2652*a^3*b*x^3 + 2210*a^2*b^2*x^5 + 1020*a*b^3*x^7 + 195*b^4*x^9))/(3315*Sqrt[d*x])

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 390\,{b}^{4}{x}^{8}+2040\,a{b}^{3}{x}^{6}+4420\,{a}^{2}{b}^{2}{x}^{4}+5304\,{a}^{3}b{x}^{2}+6630\,{a}^{4} \right ) x}{3315}{\frac{1}{\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]

2/3315*(195*b^4*x^8+1020*a*b^3*x^6+2210*a^2*b^2*x^4+2652*a^3*b*x^2+3315*a^4)*x/(d*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 0.978911, size = 122, normalized size = 1.37 \begin{align*} \frac{2 \,{\left (9945 \, \sqrt{d x} a^{4} + \frac{585 \, \left (d x\right )^{\frac{17}{2}} b^{4}}{d^{8}} + \frac{3060 \, \left (d x\right )^{\frac{13}{2}} a b^{3}}{d^{6}} + \frac{4420 \, \left (d x\right )^{\frac{9}{2}} a^{2} b^{2}}{d^{4}} + 442 \,{\left (\frac{5 \, \left (d x\right )^{\frac{9}{2}} b^{2}}{d^{4}} + \frac{18 \, \left (d x\right )^{\frac{5}{2}} a b}{d^{2}}\right )} a^{2}\right )}}{9945 \, 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="maxima")

[Out]

2/9945*(9945*sqrt(d*x)*a^4 + 585*(d*x)^(17/2)*b^4/d^8 + 3060*(d*x)^(13/2)*a*b^3/d^6 + 4420*(d*x)^(9/2)*a^2*b^2
/d^4 + 442*(5*(d*x)^(9/2)*b^2/d^4 + 18*(d*x)^(5/2)*a*b/d^2)*a^2)/d

________________________________________________________________________________________

Fricas [A]  time = 1.23957, size = 132, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (195 \, b^{4} x^{8} + 1020 \, a b^{3} x^{6} + 2210 \, a^{2} b^{2} x^{4} + 2652 \, a^{3} b x^{2} + 3315 \, a^{4}\right )} \sqrt{d x}}{3315 \, 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="fricas")

[Out]

2/3315*(195*b^4*x^8 + 1020*a*b^3*x^6 + 2210*a^2*b^2*x^4 + 2652*a^3*b*x^2 + 3315*a^4)*sqrt(d*x)/d

________________________________________________________________________________________

Sympy [A]  time = 1.70209, size = 88, normalized size = 0.99 \begin{align*} \frac{2 a^{4} \sqrt{x}}{\sqrt{d}} + \frac{8 a^{3} b x^{\frac{5}{2}}}{5 \sqrt{d}} + \frac{4 a^{2} b^{2} x^{\frac{9}{2}}}{3 \sqrt{d}} + \frac{8 a b^{3} x^{\frac{13}{2}}}{13 \sqrt{d}} + \frac{2 b^{4} x^{\frac{17}{2}}}{17 \sqrt{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)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12595, size = 99, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (195 \, \sqrt{d x} b^{4} x^{8} + 1020 \, \sqrt{d x} a b^{3} x^{6} + 2210 \, \sqrt{d x} a^{2} b^{2} x^{4} + 2652 \, \sqrt{d x} a^{3} b x^{2} + 3315 \, \sqrt{d x} a^{4}\right )}}{3315 \, 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]

2/3315*(195*sqrt(d*x)*b^4*x^8 + 1020*sqrt(d*x)*a*b^3*x^6 + 2210*sqrt(d*x)*a^2*b^2*x^4 + 2652*sqrt(d*x)*a^3*b*x
^2 + 3315*sqrt(d*x)*a^4)/d