3.678 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^2}{(d x)^{7/2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{4 a^2 b^2 (d x)^{3/2}}{d^5}-\frac{8 a^3 b}{d^3 \sqrt{d x}}-\frac{2 a^4}{5 d (d x)^{5/2}}+\frac{8 a b^3 (d x)^{7/2}}{7 d^7}+\frac{2 b^4 (d x)^{11/2}}{11 d^9} \]

[Out]

(-2*a^4)/(5*d*(d*x)^(5/2)) - (8*a^3*b)/(d^3*Sqrt[d*x]) + (4*a^2*b^2*(d*x)^(3/2))/d^5 + (8*a*b^3*(d*x)^(7/2))/(
7*d^7) + (2*b^4*(d*x)^(11/2))/(11*d^9)

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Rubi [A]  time = 0.0415425, antiderivative size = 87, 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} \[ \frac{4 a^2 b^2 (d x)^{3/2}}{d^5}-\frac{8 a^3 b}{d^3 \sqrt{d x}}-\frac{2 a^4}{5 d (d x)^{5/2}}+\frac{8 a b^3 (d x)^{7/2}}{7 d^7}+\frac{2 b^4 (d x)^{11/2}}{11 d^9} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0193988, size = 60, normalized size = 0.69 \[ \frac{2 \sqrt{d x} \left (770 a^2 b^2 x^4-1540 a^3 b x^2-77 a^4+220 a b^3 x^6+35 b^4 x^8\right )}{385 d^4 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(7/2),x]

[Out]

(2*Sqrt[d*x]*(-77*a^4 - 1540*a^3*b*x^2 + 770*a^2*b^2*x^4 + 220*a*b^3*x^6 + 35*b^4*x^8))/(385*d^4*x^3)

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Maple [A]  time = 0.047, size = 52, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -70\,{b}^{4}{x}^{8}-440\,a{b}^{3}{x}^{6}-1540\,{a}^{2}{b}^{2}{x}^{4}+3080\,{a}^{3}b{x}^{2}+154\,{a}^{4} \right ) x}{385} \left ( dx \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

-2/385*(-35*b^4*x^8-220*a*b^3*x^6-770*a^2*b^2*x^4+1540*a^3*b*x^2+77*a^4)*x/(d*x)^(7/2)

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Maxima [A]  time = 0.983806, size = 111, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (\frac{77 \,{\left (20 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )}}{\left (d x\right )^{\frac{5}{2}} d^{2}} - \frac{5 \,{\left (7 \, \left (d x\right )^{\frac{11}{2}} b^{4} + 44 \, \left (d x\right )^{\frac{7}{2}} a b^{3} d^{2} + 154 \, \left (d x\right )^{\frac{3}{2}} a^{2} b^{2} d^{4}\right )}}{d^{8}}\right )}}{385 \, 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)^(7/2),x, algorithm="maxima")

[Out]

-2/385*(77*(20*a^3*b*d^2*x^2 + a^4*d^2)/((d*x)^(5/2)*d^2) - 5*(7*(d*x)^(11/2)*b^4 + 44*(d*x)^(7/2)*a*b^3*d^2 +
154*(d*x)^(3/2)*a^2*b^2*d^4)/d^8)/d

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Fricas [A]  time = 1.21486, size = 135, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (35 \, b^{4} x^{8} + 220 \, a b^{3} x^{6} + 770 \, a^{2} b^{2} x^{4} - 1540 \, a^{3} b x^{2} - 77 \, a^{4}\right )} \sqrt{d x}}{385 \, d^{4} x^{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)^(7/2),x, algorithm="fricas")

[Out]

2/385*(35*b^4*x^8 + 220*a*b^3*x^6 + 770*a^2*b^2*x^4 - 1540*a^3*b*x^2 - 77*a^4)*sqrt(d*x)/(d^4*x^3)

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Sympy [A]  time = 2.30402, size = 87, normalized size = 1. \begin{align*} - \frac{2 a^{4}}{5 d^{\frac{7}{2}} x^{\frac{5}{2}}} - \frac{8 a^{3} b}{d^{\frac{7}{2}} \sqrt{x}} + \frac{4 a^{2} b^{2} x^{\frac{3}{2}}}{d^{\frac{7}{2}}} + \frac{8 a b^{3} x^{\frac{7}{2}}}{7 d^{\frac{7}{2}}} + \frac{2 b^{4} x^{\frac{11}{2}}}{11 d^{\frac{7}{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)**(7/2),x)

[Out]

-2*a**4/(5*d**(7/2)*x**(5/2)) - 8*a**3*b/(d**(7/2)*sqrt(x)) + 4*a**2*b**2*x**(3/2)/d**(7/2) + 8*a*b**3*x**(7/2
)/(7*d**(7/2)) + 2*b**4*x**(11/2)/(11*d**(7/2))

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Giac [A]  time = 1.11246, size = 128, normalized size = 1.47 \begin{align*} -\frac{2 \,{\left (\frac{77 \,{\left (20 \, a^{3} b d^{3} x^{2} + a^{4} d^{3}\right )}}{\sqrt{d x} d^{2} x^{2}} - \frac{5 \,{\left (7 \, \sqrt{d x} b^{4} d^{55} x^{5} + 44 \, \sqrt{d x} a b^{3} d^{55} x^{3} + 154 \, \sqrt{d x} a^{2} b^{2} d^{55} x\right )}}{d^{55}}\right )}}{385 \, d^{4}} \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)^(7/2),x, algorithm="giac")

[Out]

-2/385*(77*(20*a^3*b*d^3*x^2 + a^4*d^3)/(sqrt(d*x)*d^2*x^2) - 5*(7*sqrt(d*x)*b^4*d^55*x^5 + 44*sqrt(d*x)*a*b^3
*d^55*x^3 + 154*sqrt(d*x)*a^2*b^2*d^55*x)/d^55)/d^4