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

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

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Rubi [A]  time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]))))

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]]

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*}

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Mathematica [A]  time = 0.05, size = 91, normalized size = 0.90 \begin {gather*} \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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(b*(d + c*x^2))/(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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IntegrateAlgebraic [A]  time = 0.07, size = 90, normalized size = 0.89 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-56 a c^2 d x^6+28 a c d^2 x^4-21 a d^3 x^2+48 b c^3 x^6-24 b c^2 d x^4+18 b c d^2 x^2-15 b d^3\right )}{105 d^4 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 86, normalized size = 0.85 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.62, size = 219, normalized size = 2.17 \begin {gather*} \frac {140 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{4} a c + 210 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3} b c^{\frac {3}{2}} + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3} a \sqrt {c} d + 252 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} b c d + 21 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} a d^{2} + 105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} b \sqrt {c} d^{2} + 15 \, b d^{3}}{105 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{7}} \end {gather*}

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]

1/105*(140*(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^4*a*c + 210*(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^3*b*c^(3/2) + 1
05*(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^3*a*sqrt(c)*d + 252*(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^2*b*c*d + 21*(s
qrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^2*a*d^2 + 105*(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))*b*sqrt(c)*d^2 + 15*b*d^3)/
(sqrt(c)*x^2 - sqrt(c*x^4 + d*x^2))^7

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maple [A]  time = 0.05, size = 94, normalized size = 0.93 \begin {gather*} -\frac {\left (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}\right ) \left (c \,x^{2}+d \right )}{105 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, d^{4} x^{8}} \end {gather*}

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 = 0.58, size = 118, normalized size = 1.17 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 4.72, size = 102, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}}\,\left (48\,b\,c^3-56\,a\,c^2\,d\right )}{105\,d^4}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{7\,d\,x^6}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (24\,b\,c^2-28\,a\,c\,d\right )}{105\,d^3\,x^2}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (7\,a\,d-6\,b\,c\right )}{35\,d^2\,x^4} \end {gather*}

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]

((c + d/x^2)^(1/2)*(48*b*c^3 - 56*a*c^2*d))/(105*d^4) - (b*(c + d/x^2)^(1/2))/(7*d*x^6) - ((c + d/x^2)^(1/2)*(
24*b*c^2 - 28*a*c*d))/(105*d^3*x^2) - ((c + d/x^2)^(1/2)*(7*a*d - 6*b*c))/(35*d^2*x^4)

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sympy [A]  time = 18.00, size = 269, normalized size = 2.66 \begin {gather*} \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 {gather*}

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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