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

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

[Out]

8/5*a^3*b*(d*x)^(5/2)/d^3+4/3*a^2*b^2*(d*x)^(9/2)/d^5+8/13*a*b^3*(d*x)^(13/2)/d^7+2/17*b^4*(d*x)^(17/2)/d^9+2*
a^4*(d*x)^(1/2)/d

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Rubi [A]  time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 55, normalized size = 0.62 \[ \frac {2 \left (3315 a^4 x+2652 a^3 b x^3+2210 a^2 b^2 x^5+1020 a b^3 x^7+195 b^4 x^9\right )}{3315 \sqrt {d x}} \]

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])

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fricas [A]  time = 0.82, size = 53, normalized size = 0.60 \[ \frac {2 \, {\left (195 \, b^{4} x^{8} + 1020 \, a b^{3} x^{6} + 2210 \, a^{2} b^{2} x^{4} + 2652 \, a^{3} b x^{2} + 3315 \, a^{4}\right )} \sqrt {d x}}{3315 \, d} \]

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

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giac [A]  time = 0.26, size = 73, normalized size = 0.82 \[ \frac {2 \, {\left (195 \, \sqrt {d x} b^{4} x^{8} + 1020 \, \sqrt {d x} a b^{3} x^{6} + 2210 \, \sqrt {d x} a^{2} b^{2} x^{4} + 2652 \, \sqrt {d x} a^{3} b x^{2} + 3315 \, \sqrt {d x} a^{4}\right )}}{3315 \, d} \]

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

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maple [A]  time = 0.01, size = 52, normalized size = 0.58 \[ \frac {2 \left (195 b^{4} x^{8}+1020 a \,b^{3} x^{6}+2210 a^{2} b^{2} x^{4}+2652 a^{3} b \,x^{2}+3315 a^{4}\right ) x}{3315 \sqrt {d x}} \]

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)

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maxima [A]  time = 1.36, size = 90, normalized size = 1.01 \[ \frac {2 \, {\left (9945 \, \sqrt {d x} a^{4} + \frac {585 \, \left (d x\right )^{\frac {17}{2}} b^{4}}{d^{8}} + \frac {3060 \, \left (d x\right )^{\frac {13}{2}} a b^{3}}{d^{6}} + \frac {4420 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2}}{d^{4}} + 442 \, {\left (\frac {5 \, \left (d x\right )^{\frac {9}{2}} b^{2}}{d^{4}} + \frac {18 \, \left (d x\right )^{\frac {5}{2}} a b}{d^{2}}\right )} a^{2}\right )}}{9945 \, d} \]

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

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mupad [B]  time = 0.03, size = 71, normalized size = 0.80 \[ \frac {2\,a^4\,\sqrt {d\,x}}{d}+\frac {2\,b^4\,{\left (d\,x\right )}^{17/2}}{17\,d^9}+\frac {4\,a^2\,b^2\,{\left (d\,x\right )}^{9/2}}{3\,d^5}+\frac {8\,a^3\,b\,{\left (d\,x\right )}^{5/2}}{5\,d^3}+\frac {8\,a\,b^3\,{\left (d\,x\right )}^{13/2}}{13\,d^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^2/(d*x)^(1/2),x)

[Out]

(2*a^4*(d*x)^(1/2))/d + (2*b^4*(d*x)^(17/2))/(17*d^9) + (4*a^2*b^2*(d*x)^(9/2))/(3*d^5) + (8*a^3*b*(d*x)^(5/2)
)/(5*d^3) + (8*a*b^3*(d*x)^(13/2))/(13*d^7)

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sympy [A]  time = 1.35, size = 88, normalized size = 0.99 \[ \frac {2 a^{4} \sqrt {x}}{\sqrt {d}} + \frac {8 a^{3} b x^{\frac {5}{2}}}{5 \sqrt {d}} + \frac {4 a^{2} b^{2} x^{\frac {9}{2}}}{3 \sqrt {d}} + \frac {8 a b^{3} x^{\frac {13}{2}}}{13 \sqrt {d}} + \frac {2 b^{4} x^{\frac {17}{2}}}{17 \sqrt {d}} \]

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))

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