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

Optimal. Leaf size=89 \[ \frac{12 a^2 b^2 (d x)^{5/2}}{5 d^5}+\frac{8 a^3 b \sqrt{d x}}{d^3}-\frac{2 a^4}{3 d (d x)^{3/2}}+\frac{8 a b^3 (d x)^{9/2}}{9 d^7}+\frac{2 b^4 (d x)^{13/2}}{13 d^9} \]

[Out]

(-2*a^4)/(3*d*(d*x)^(3/2)) + (8*a^3*b*Sqrt[d*x])/d^3 + (12*a^2*b^2*(d*x)^(5/2))/(5*d^5) + (8*a*b^3*(d*x)^(9/2)
)/(9*d^7) + (2*b^4*(d*x)^(13/2))/(13*d^9)

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Rubi [A]  time = 0.0418703, 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} \[ \frac{12 a^2 b^2 (d x)^{5/2}}{5 d^5}+\frac{8 a^3 b \sqrt{d x}}{d^3}-\frac{2 a^4}{3 d (d x)^{3/2}}+\frac{8 a b^3 (d x)^{9/2}}{9 d^7}+\frac{2 b^4 (d x)^{13/2}}{13 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0161114, size = 55, normalized size = 0.62 \[ \frac{x \left (1404 a^2 b^2 x^4+4680 a^3 b x^2-390 a^4+520 a b^3 x^6+90 b^4 x^8\right )}{585 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-390*a^4 + 4680*a^3*b*x^2 + 1404*a^2*b^2*x^4 + 520*a*b^3*x^6 + 90*b^4*x^8))/(585*(d*x)^(5/2))

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Maple [A]  time = 0.049, size = 52, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -90\,{b}^{4}{x}^{8}-520\,a{b}^{3}{x}^{6}-1404\,{a}^{2}{b}^{2}{x}^{4}-4680\,{a}^{3}b{x}^{2}+390\,{a}^{4} \right ) x}{585} \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)^2/(d*x)^(5/2),x)

[Out]

-2/585*(-45*b^4*x^8-260*a*b^3*x^6-702*a^2*b^2*x^4-2340*a^3*b*x^2+195*a^4)*x/(d*x)^(5/2)

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Maxima [A]  time = 0.991041, size = 103, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (\frac{195 \, a^{4}}{\left (d x\right )^{\frac{3}{2}}} - \frac{45 \, \left (d x\right )^{\frac{13}{2}} b^{4} + 260 \, \left (d x\right )^{\frac{9}{2}} a b^{3} d^{2} + 702 \, \left (d x\right )^{\frac{5}{2}} a^{2} b^{2} d^{4} + 2340 \, \sqrt{d x} a^{3} b d^{6}}{d^{8}}\right )}}{585 \, 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)^(5/2),x, algorithm="maxima")

[Out]

-2/585*(195*a^4/(d*x)^(3/2) - (45*(d*x)^(13/2)*b^4 + 260*(d*x)^(9/2)*a*b^3*d^2 + 702*(d*x)^(5/2)*a^2*b^2*d^4 +
2340*sqrt(d*x)*a^3*b*d^6)/d^8)/d

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Fricas [A]  time = 1.22907, size = 136, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (45 \, b^{4} x^{8} + 260 \, a b^{3} x^{6} + 702 \, a^{2} b^{2} x^{4} + 2340 \, a^{3} b x^{2} - 195 \, a^{4}\right )} \sqrt{d x}}{585 \, 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)^2/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*b^4*x^8 + 260*a*b^3*x^6 + 702*a^2*b^2*x^4 + 2340*a^3*b*x^2 - 195*a^4)*sqrt(d*x)/(d^3*x^2)

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Sympy [A]  time = 1.69748, size = 88, normalized size = 0.99 \begin{align*} - \frac{2 a^{4}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} + \frac{8 a^{3} b \sqrt{x}}{d^{\frac{5}{2}}} + \frac{12 a^{2} b^{2} x^{\frac{5}{2}}}{5 d^{\frac{5}{2}}} + \frac{8 a b^{3} x^{\frac{9}{2}}}{9 d^{\frac{5}{2}}} + \frac{2 b^{4} x^{\frac{13}{2}}}{13 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)**2/(d*x)**(5/2),x)

[Out]

-2*a**4/(3*d**(5/2)*x**(3/2)) + 8*a**3*b*sqrt(x)/d**(5/2) + 12*a**2*b**2*x**(5/2)/(5*d**(5/2)) + 8*a*b**3*x**(
9/2)/(9*d**(5/2)) + 2*b**4*x**(13/2)/(13*d**(5/2))

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Giac [A]  time = 1.14139, size = 124, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (\frac{195 \, a^{4} d}{\sqrt{d x} x} - \frac{45 \, \sqrt{d x} b^{4} d^{78} x^{6} + 260 \, \sqrt{d x} a b^{3} d^{78} x^{4} + 702 \, \sqrt{d x} a^{2} b^{2} d^{78} x^{2} + 2340 \, \sqrt{d x} a^{3} b d^{78}}{d^{78}}\right )}}{585 \, 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)^2/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/585*(195*a^4*d/(sqrt(d*x)*x) - (45*sqrt(d*x)*b^4*d^78*x^6 + 260*sqrt(d*x)*a*b^3*d^78*x^4 + 702*sqrt(d*x)*a^
2*b^2*d^78*x^2 + 2340*sqrt(d*x)*a^3*b*d^78)/d^78)/d^3