3.684 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=127 \[ \frac{30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac{40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac{6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac{12 a^5 b \sqrt{d x}}{d^3}-\frac{2 a^6}{3 d (d x)^{3/2}}+\frac{12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac{2 b^6 (d x)^{21/2}}{21 d^{13}} \]

[Out]

(-2*a^6)/(3*d*(d*x)^(3/2)) + (12*a^5*b*Sqrt[d*x])/d^3 + (6*a^4*b^2*(d*x)^(5/2))/d^5 + (40*a^3*b^3*(d*x)^(9/2))
/(9*d^7) + (30*a^2*b^4*(d*x)^(13/2))/(13*d^9) + (12*a*b^5*(d*x)^(17/2))/(17*d^11) + (2*b^6*(d*x)^(21/2))/(21*d
^13)

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Rubi [A]  time = 0.0621556, antiderivative size = 127, 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} \[ \frac{30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac{40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac{6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac{12 a^5 b \sqrt{d x}}{d^3}-\frac{2 a^6}{3 d (d x)^{3/2}}+\frac{12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac{2 b^6 (d x)^{21/2}}{21 d^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2),x]

[Out]

(-2*a^6)/(3*d*(d*x)^(3/2)) + (12*a^5*b*Sqrt[d*x])/d^3 + (6*a^4*b^2*(d*x)^(5/2))/d^5 + (40*a^3*b^3*(d*x)^(9/2))
/(9*d^7) + (30*a^2*b^4*(d*x)^(13/2))/(13*d^9) + (12*a*b^5*(d*x)^(17/2))/(17*d^11) + (2*b^6*(d*x)^(21/2))/(21*d
^13)

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 )^3}{(d x)^{5/2}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{(d x)^{5/2}} \, dx}{b^6}\\ &=\frac{\int \left (\frac{a^6 b^6}{(d x)^{5/2}}+\frac{6 a^5 b^7}{d^2 \sqrt{d x}}+\frac{15 a^4 b^8 (d x)^{3/2}}{d^4}+\frac{20 a^3 b^9 (d x)^{7/2}}{d^6}+\frac{15 a^2 b^{10} (d x)^{11/2}}{d^8}+\frac{6 a b^{11} (d x)^{15/2}}{d^{10}}+\frac{b^{12} (d x)^{19/2}}{d^{12}}\right ) \, dx}{b^6}\\ &=-\frac{2 a^6}{3 d (d x)^{3/2}}+\frac{12 a^5 b \sqrt{d x}}{d^3}+\frac{6 a^4 b^2 (d x)^{5/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{9/2}}{9 d^7}+\frac{30 a^2 b^4 (d x)^{13/2}}{13 d^9}+\frac{12 a b^5 (d x)^{17/2}}{17 d^{11}}+\frac{2 b^6 (d x)^{21/2}}{21 d^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0232515, size = 77, normalized size = 0.61 \[ \frac{2 x \left (16065 a^2 b^4 x^8+30940 a^3 b^3 x^6+41769 a^4 b^2 x^4+83538 a^5 b x^2-4641 a^6+4914 a b^5 x^{10}+663 b^6 x^{12}\right )}{13923 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/(d*x)^(5/2),x]

[Out]

(2*x*(-4641*a^6 + 83538*a^5*b*x^2 + 41769*a^4*b^2*x^4 + 30940*a^3*b^3*x^6 + 16065*a^2*b^4*x^8 + 4914*a*b^5*x^1
0 + 663*b^6*x^12))/(13923*(d*x)^(5/2))

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Maple [A]  time = 0.049, size = 74, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -1326\,{b}^{6}{x}^{12}-9828\,a{b}^{5}{x}^{10}-32130\,{a}^{2}{b}^{4}{x}^{8}-61880\,{a}^{3}{b}^{3}{x}^{6}-83538\,{a}^{4}{b}^{2}{x}^{4}-167076\,{a}^{5}b{x}^{2}+9282\,{a}^{6} \right ) x}{13923} \left ( dx \right ) ^{-{\frac{5}{2}}}} \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)^3/(d*x)^(5/2),x)

[Out]

-2/13923*(-663*b^6*x^12-4914*a*b^5*x^10-16065*a^2*b^4*x^8-30940*a^3*b^3*x^6-41769*a^4*b^2*x^4-83538*a^5*b*x^2+
4641*a^6)*x/(d*x)^(5/2)

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Maxima [A]  time = 1.01702, size = 146, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (\frac{4641 \, a^{6}}{\left (d x\right )^{\frac{3}{2}}} - \frac{663 \, \left (d x\right )^{\frac{21}{2}} b^{6} + 4914 \, \left (d x\right )^{\frac{17}{2}} a b^{5} d^{2} + 16065 \, \left (d x\right )^{\frac{13}{2}} a^{2} b^{4} d^{4} + 30940 \, \left (d x\right )^{\frac{9}{2}} a^{3} b^{3} d^{6} + 41769 \, \left (d x\right )^{\frac{5}{2}} a^{4} b^{2} d^{8} + 83538 \, \sqrt{d x} a^{5} b d^{10}}{d^{12}}\right )}}{13923 \, 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)^3/(d*x)^(5/2),x, algorithm="maxima")

[Out]

-2/13923*(4641*a^6/(d*x)^(3/2) - (663*(d*x)^(21/2)*b^6 + 4914*(d*x)^(17/2)*a*b^5*d^2 + 16065*(d*x)^(13/2)*a^2*
b^4*d^4 + 30940*(d*x)^(9/2)*a^3*b^3*d^6 + 41769*(d*x)^(5/2)*a^4*b^2*d^8 + 83538*sqrt(d*x)*a^5*b*d^10)/d^12)/d

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Fricas [A]  time = 1.30171, size = 204, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (663 \, b^{6} x^{12} + 4914 \, a b^{5} x^{10} + 16065 \, a^{2} b^{4} x^{8} + 30940 \, a^{3} b^{3} x^{6} + 41769 \, a^{4} b^{2} x^{4} + 83538 \, a^{5} b x^{2} - 4641 \, a^{6}\right )} \sqrt{d x}}{13923 \, d^{3} x^{2}} \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)^3/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/13923*(663*b^6*x^12 + 4914*a*b^5*x^10 + 16065*a^2*b^4*x^8 + 30940*a^3*b^3*x^6 + 41769*a^4*b^2*x^4 + 83538*a^
5*b*x^2 - 4641*a^6)*sqrt(d*x)/(d^3*x^2)

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Sympy [A]  time = 3.33922, size = 128, normalized size = 1.01 \begin{align*} - \frac{2 a^{6}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} + \frac{12 a^{5} b \sqrt{x}}{d^{\frac{5}{2}}} + \frac{6 a^{4} b^{2} x^{\frac{5}{2}}}{d^{\frac{5}{2}}} + \frac{40 a^{3} b^{3} x^{\frac{9}{2}}}{9 d^{\frac{5}{2}}} + \frac{30 a^{2} b^{4} x^{\frac{13}{2}}}{13 d^{\frac{5}{2}}} + \frac{12 a b^{5} x^{\frac{17}{2}}}{17 d^{\frac{5}{2}}} + \frac{2 b^{6} x^{\frac{21}{2}}}{21 d^{\frac{5}{2}}} \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)**3/(d*x)**(5/2),x)

[Out]

-2*a**6/(3*d**(5/2)*x**(3/2)) + 12*a**5*b*sqrt(x)/d**(5/2) + 6*a**4*b**2*x**(5/2)/d**(5/2) + 40*a**3*b**3*x**(
9/2)/(9*d**(5/2)) + 30*a**2*b**4*x**(13/2)/(13*d**(5/2)) + 12*a*b**5*x**(17/2)/(17*d**(5/2)) + 2*b**6*x**(21/2
)/(21*d**(5/2))

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Giac [A]  time = 1.12439, size = 176, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (\frac{4641 \, a^{6} d}{\sqrt{d x} x} - \frac{663 \, \sqrt{d x} b^{6} d^{210} x^{10} + 4914 \, \sqrt{d x} a b^{5} d^{210} x^{8} + 16065 \, \sqrt{d x} a^{2} b^{4} d^{210} x^{6} + 30940 \, \sqrt{d x} a^{3} b^{3} d^{210} x^{4} + 41769 \, \sqrt{d x} a^{4} b^{2} d^{210} x^{2} + 83538 \, \sqrt{d x} a^{5} b d^{210}}{d^{210}}\right )}}{13923 \, d^{3}} \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)^3/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/13923*(4641*a^6*d/(sqrt(d*x)*x) - (663*sqrt(d*x)*b^6*d^210*x^10 + 4914*sqrt(d*x)*a*b^5*d^210*x^8 + 16065*sq
rt(d*x)*a^2*b^4*d^210*x^6 + 30940*sqrt(d*x)*a^3*b^3*d^210*x^4 + 41769*sqrt(d*x)*a^4*b^2*d^210*x^2 + 83538*sqrt
(d*x)*a^5*b*d^210)/d^210)/d^3