3.967 \(\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^7} \, dx\)

Optimal. Leaf size=101 \[ \frac{c^2 \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-a d)}{5 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{3 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0693012, antiderivative size = 101, 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} \[ \frac{c^2 \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{5/2} (3 b c-a d)}{5 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{3 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{7/2}}{7 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0267121, size = 91, normalized size = 0.9 \[ \frac{\left (\frac{c x^2}{d}+1\right ) \left (8 c^2 x^4-4 c d x^2+3 d^2\right ) (6 b c-7 a d)}{105 d^3 x^6 \sqrt{c+\frac{d}{x^2}}}-\frac{b \left (c x^2+d\right )}{7 d x^8 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(d + c*x^2))/(7*d*Sqrt[c + d/x^2]*x^8) + ((6*b*c - 7*a*d)*(1 + (c*x^2)/d)*(3*d^2 - 4*c*d*x^2 + 8*c^2*x^4))
/(105*d^3*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 ( 56\,a{c}^{2}d{x}^{6}-48\,b{c}^{3}{x}^{6}-28\,ac{d}^{2}{x}^{4}+24\,b{c}^{2}d{x}^{4}+21\,a{d}^{3}{x}^{2}-18\,bc{d}^{2}{x}^{2}+15\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{105\,{d}^{4}{x}^{8}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)/x^7/(c+d/x^2)^(1/2),x)

[Out]

-1/105*(56*a*c^2*d*x^6-48*b*c^3*x^6-28*a*c*d^2*x^4+24*b*c^2*d*x^4+21*a*d^3*x^2-18*b*c*d^2*x^2+15*b*d^3)*(c*x^2
+d)/((c*x^2+d)/x^2)^(1/2)/d^4/x^8

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Maxima [A]  time = 0.944889, size = 159, normalized size = 1.57 \begin{align*} -\frac{1}{35} \, b{\left (\frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}}}{d^{4}} - \frac{21 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c}{d^{4}} + \frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c^{2}}{d^{4}} - \frac{35 \, \sqrt{c + \frac{d}{x^{2}}} c^{3}}{d^{4}}\right )} - \frac{1}{15} \, a{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{3}} - \frac{10 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{3}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^7/(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

-1/35*b*(5*(c + d/x^2)^(7/2)/d^4 - 21*(c + d/x^2)^(5/2)*c/d^4 + 35*(c + d/x^2)^(3/2)*c^2/d^4 - 35*sqrt(c + d/x
^2)*c^3/d^4) - 1/15*a*(3*(c + d/x^2)^(5/2)/d^3 - 10*(c + d/x^2)^(3/2)*c/d^3 + 15*sqrt(c + d/x^2)*c^2/d^3)

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Fricas [A]  time = 1.34226, size = 192, normalized size = 1.9 \begin{align*} \frac{{\left (8 \,{\left (6 \, b c^{3} - 7 \, a c^{2} d\right )} x^{6} - 4 \,{\left (6 \, b c^{2} d - 7 \, a c d^{2}\right )} x^{4} - 15 \, b d^{3} + 3 \,{\left (6 \, b c d^{2} - 7 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{105 \, d^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^7/(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

1/105*(8*(6*b*c^3 - 7*a*c^2*d)*x^6 - 4*(6*b*c^2*d - 7*a*c*d^2)*x^4 - 15*b*d^3 + 3*(6*b*c*d^2 - 7*a*d^3)*x^2)*s
qrt((c*x^2 + d)/x^2)/(d^4*x^6)

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Sympy [A]  time = 13.6424, size = 270, normalized size = 2.67 \begin{align*} - \frac{\begin{cases} \frac{\frac{a}{3 x^{6}} + \frac{b}{4 x^{8}}}{\sqrt{c}} & \text{for}\: d = 0 \\- \frac{\frac{2 a c \left (\frac{c^{2}}{\sqrt{c + \frac{d}{x^{2}}}} + 2 c \sqrt{c + \frac{d}{x^{2}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3}\right )}{d^{2}} + \frac{2 a \left (- \frac{c^{3}}{\sqrt{c + \frac{d}{x^{2}}}} - 3 c^{2} \sqrt{c + \frac{d}{x^{2}}} + c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{2 b c \left (- \frac{c^{3}}{\sqrt{c + \frac{d}{x^{2}}}} - 3 c^{2} \sqrt{c + \frac{d}{x^{2}}} + c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{3}} + \frac{2 b \left (\frac{c^{4}}{\sqrt{c + \frac{d}{x^{2}}}} + 4 c^{3} \sqrt{c + \frac{d}{x^{2}}} - 2 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} + \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}}}{d} & \text{otherwise} \end{cases}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/x**7/(c+d/x**2)**(1/2),x)

[Out]

-Piecewise(((a/(3*x**6) + b/(4*x**8))/sqrt(c), Eq(d, 0)), (-(2*a*c*(c**2/sqrt(c + d/x**2) + 2*c*sqrt(c + d/x**
2) - (c + d/x**2)**(3/2)/3)/d**2 + 2*a*(-c**3/sqrt(c + d/x**2) - 3*c**2*sqrt(c + d/x**2) + c*(c + d/x**2)**(3/
2) - (c + d/x**2)**(5/2)/5)/d**2 + 2*b*c*(-c**3/sqrt(c + d/x**2) - 3*c**2*sqrt(c + d/x**2) + c*(c + d/x**2)**(
3/2) - (c + d/x**2)**(5/2)/5)/d**3 + 2*b*(c**4/sqrt(c + d/x**2) + 4*c**3*sqrt(c + d/x**2) - 2*c**2*(c + d/x**2
)**(3/2) + 4*c*(c + d/x**2)**(5/2)/5 - (c + d/x**2)**(7/2)/7)/d**3)/d, True))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^7/(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

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