3.978 \(\int \frac{a+\frac{b}{x^2}}{(c+\frac{d}{x^2})^{3/2} x^7} \, dx\)

Optimal. Leaf size=100 \[ -\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^4} \]

[Out]

-((c^2*(b*c - a*d))/(d^4*Sqrt[c + d/x^2])) - (c*(3*b*c - 2*a*d)*Sqrt[c + d/x^2])/d^4 + ((3*b*c - a*d)*(c + d/x
^2)^(3/2))/(3*d^4) - (b*(c + d/x^2)^(5/2))/(5*d^4)

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Rubi [A]  time = 0.0710595, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {446, 77} \[ -\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x]

[Out]

-((c^2*(b*c - a*d))/(d^4*Sqrt[c + d/x^2])) - (c*(3*b*c - 2*a*d)*Sqrt[c + d/x^2])/d^4 + ((3*b*c - a*d)*(c + d/x
^2)^(3/2))/(3*d^4) - (b*(c + d/x^2)^(5/2))/(5*d^4)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^7} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (a+b x)}{(c+d x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{c^2 (b c-a d)}{d^3 (c+d x)^{3/2}}+\frac{c (3 b c-2 a d)}{d^3 \sqrt{c+d x}}+\frac{(-3 b c+a d) \sqrt{c+d x}}{d^3}+\frac{b (c+d x)^{3/2}}{d^3}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}-\frac{c (3 b c-2 a d) \sqrt{c+\frac{d}{x^2}}}{d^4}+\frac{(3 b c-a d) \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0256347, size = 81, normalized size = 0.81 \[ \frac{-5 a d x^2 \left (-8 c^2 x^4-4 c d x^2+d^2\right )-3 b \left (8 c^2 d x^4+16 c^3 x^6-2 c d^2 x^2+d^3\right )}{15 d^4 x^6 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x]

[Out]

(-5*a*d*x^2*(d^2 - 4*c*d*x^2 - 8*c^2*x^4) - 3*b*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4 + 16*c^3*x^6))/(15*d^4*Sqrt[c
+ d/x^2]*x^6)

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Maple [A]  time = 0.007, size = 94, normalized size = 0.9 \begin{align*}{\frac{ \left ( 40\,a{c}^{2}d{x}^{6}-48\,b{c}^{3}{x}^{6}+20\,ac{d}^{2}{x}^{4}-24\,b{c}^{2}d{x}^{4}-5\,a{d}^{3}{x}^{2}+6\,bc{d}^{2}{x}^{2}-3\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{4}{x}^{8}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x)

[Out]

1/15*(40*a*c^2*d*x^6-48*b*c^3*x^6+20*a*c*d^2*x^4-24*b*c^2*d*x^4-5*a*d^3*x^2+6*b*c*d^2*x^2-3*b*d^3)*(c*x^2+d)/(
(c*x^2+d)/x^2)^(3/2)/d^4/x^8

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Maxima [A]  time = 0.949499, size = 157, normalized size = 1.57 \begin{align*} -\frac{1}{5} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{4}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{4}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{4}} + \frac{5 \, c^{3}}{\sqrt{c + \frac{d}{x^{2}}} d^{4}}\right )} - \frac{1}{3} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{3}} - \frac{6 \, \sqrt{c + \frac{d}{x^{2}}} c}{d^{3}} - \frac{3 \, c^{2}}{\sqrt{c + \frac{d}{x^{2}}} d^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

-1/5*b*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*sqrt(c + d/x^2)*c^2/d^4 + 5*c^3/(sqrt(c + d/x^2
)*d^4)) - 1/3*a*((c + d/x^2)^(3/2)/d^3 - 6*sqrt(c + d/x^2)*c/d^3 - 3*c^2/(sqrt(c + d/x^2)*d^3))

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Fricas [A]  time = 1.61303, size = 204, normalized size = 2.04 \begin{align*} -\frac{{\left (8 \,{\left (6 \, b c^{3} - 5 \, a c^{2} d\right )} x^{6} + 4 \,{\left (6 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{4} + 3 \, b d^{3} -{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \,{\left (c d^{4} x^{6} + d^{5} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

-1/15*(8*(6*b*c^3 - 5*a*c^2*d)*x^6 + 4*(6*b*c^2*d - 5*a*c*d^2)*x^4 + 3*b*d^3 - (6*b*c*d^2 - 5*a*d^3)*x^2)*sqrt
((c*x^2 + d)/x^2)/(c*d^4*x^6 + d^5*x^4)

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Sympy [A]  time = 10.0171, size = 90, normalized size = 0.9 \begin{align*} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{c^{2} \left (a d - b c\right )}{d^{4} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 3 b c\right )}{3 d^{4}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (- 2 a c d + 3 b c^{2}\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**7,x)

[Out]

-b*(c + d/x**2)**(5/2)/(5*d**4) + c**2*(a*d - b*c)/(d**4*sqrt(c + d/x**2)) - (c + d/x**2)**(3/2)*(a*d - 3*b*c)
/(3*d**4) - sqrt(c + d/x**2)*(-2*a*c*d + 3*b*c**2)/d**4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^7), x)