3.938 \(\int (a+\frac{b}{x^2}) \sqrt{c+\frac{d}{x^2}} x^8 \, dx\)

Optimal. Leaf size=117 \[ \frac{8 d^2 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}+\frac{x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}-\frac{4 d x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac{a x^9 \left (c+\frac{d}{x^2}\right )^{3/2}}{9 c} \]

[Out]

(8*d^2*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^3)/(315*c^4) - (4*d*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^5)/(105*c^3
) + ((3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^7)/(21*c^2) + (a*(c + d/x^2)^(3/2)*x^9)/(9*c)

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Rubi [A]  time = 0.0616844, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac{8 d^2 x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{315 c^4}+\frac{x^7 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{21 c^2}-\frac{4 d x^5 \left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-2 a d)}{105 c^3}+\frac{a x^9 \left (c+\frac{d}{x^2}\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^8,x]

[Out]

(8*d^2*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^3)/(315*c^4) - (4*d*(3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^5)/(105*c^3
) + ((3*b*c - 2*a*d)*(c + d/x^2)^(3/2)*x^7)/(21*c^2) + (a*(c + d/x^2)^(3/2)*x^9)/(9*c)

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^8 \, dx &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{9 c}+\frac{(9 b c-6 a d) \int \sqrt{c+\frac{d}{x^2}} x^6 \, dx}{9 c}\\ &=\frac{(3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{9 c}-\frac{(4 d (3 b c-2 a d)) \int \sqrt{c+\frac{d}{x^2}} x^4 \, dx}{21 c^2}\\ &=-\frac{4 d (3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^5}{105 c^3}+\frac{(3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{9 c}+\frac{\left (8 d^2 (3 b c-2 a d)\right ) \int \sqrt{c+\frac{d}{x^2}} x^2 \, dx}{105 c^3}\\ &=\frac{8 d^2 (3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^3}{315 c^4}-\frac{4 d (3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^5}{105 c^3}+\frac{(3 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{3/2} x^7}{21 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^9}{9 c}\\ \end{align*}

Mathematica [A]  time = 0.0535571, size = 86, normalized size = 0.74 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (a \left (-30 c^2 d x^4+35 c^3 x^6+24 c d^2 x^2-16 d^3\right )+3 b c \left (15 c^2 x^4-12 c d x^2+8 d^2\right )\right )}{315 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^8,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)*(3*b*c*(8*d^2 - 12*c*d*x^2 + 15*c^2*x^4) + a*(-16*d^3 + 24*c*d^2*x^2 - 30*c^2*d
*x^4 + 35*c^3*x^6)))/(315*c^4)

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Maple [A]  time = 0.006, size = 89, normalized size = 0.8 \begin{align*}{\frac{x \left ( 35\,a{x}^{6}{c}^{3}-30\,a{c}^{2}d{x}^{4}+45\,b{c}^{3}{x}^{4}+24\,ac{d}^{2}{x}^{2}-36\,b{c}^{2}d{x}^{2}-16\,a{d}^{3}+24\,bc{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{315\,{c}^{4}}\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^8*(c+d/x^2)^(1/2),x)

[Out]

1/315*((c*x^2+d)/x^2)^(1/2)*x*(35*a*c^3*x^6-30*a*c^2*d*x^4+45*b*c^3*x^4+24*a*c*d^2*x^2-36*b*c^2*d*x^2-16*a*d^3
+24*b*c*d^2)*(c*x^2+d)/c^4

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Maxima [A]  time = 0.969763, size = 167, normalized size = 1.43 \begin{align*} \frac{{\left (15 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} x^{7} - 42 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d x^{5} + 35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{2} x^{3}\right )} b}{105 \, c^{3}} + \frac{{\left (35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} x^{9} - 135 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} d x^{7} + 189 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d^{2} x^{5} - 105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d^{3} x^{3}\right )} a}{315 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^8*(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

1/105*(15*(c + d/x^2)^(7/2)*x^7 - 42*(c + d/x^2)^(5/2)*d*x^5 + 35*(c + d/x^2)^(3/2)*d^2*x^3)*b/c^3 + 1/315*(35
*(c + d/x^2)^(9/2)*x^9 - 135*(c + d/x^2)^(7/2)*d*x^7 + 189*(c + d/x^2)^(5/2)*d^2*x^5 - 105*(c + d/x^2)^(3/2)*d
^3*x^3)*a/c^4

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Fricas [A]  time = 1.33652, size = 232, normalized size = 1.98 \begin{align*} \frac{{\left (35 \, a c^{4} x^{9} + 5 \,{\left (9 \, b c^{4} + a c^{3} d\right )} x^{7} + 3 \,{\left (3 \, b c^{3} d - 2 \, a c^{2} d^{2}\right )} x^{5} - 4 \,{\left (3 \, b c^{2} d^{2} - 2 \, a c d^{3}\right )} x^{3} + 8 \,{\left (3 \, b c d^{3} - 2 \, a d^{4}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{315 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^8*(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

1/315*(35*a*c^4*x^9 + 5*(9*b*c^4 + a*c^3*d)*x^7 + 3*(3*b*c^3*d - 2*a*c^2*d^2)*x^5 - 4*(3*b*c^2*d^2 - 2*a*c*d^3
)*x^3 + 8*(3*b*c*d^3 - 2*a*d^4)*x)*sqrt((c*x^2 + d)/x^2)/c^4

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Sympy [B]  time = 6.33436, size = 910, normalized size = 7.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**8*(c+d/x**2)**(1/2),x)

[Out]

35*a*c**7*d**(19/2)*x**14*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 +
315*c**4*d**12) + 110*a*c**6*d**(21/2)*x**12*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 9
45*c**5*d**11*x**2 + 315*c**4*d**12) + 114*a*c**5*d**(23/2)*x**10*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945
*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) + 40*a*c**4*d**(25/2)*x**8*sqrt(c*x**2/d + 1)/(315*c*
*7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) - 5*a*c**3*d**(27/2)*x**6*sqrt(c*x*
*2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12) - 30*a*c**2*d**(29
/2)*x**4*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 315*c**4*d**12)
- 40*a*c*d**(31/2)*x**2*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*x**2 + 3
15*c**4*d**12) - 16*a*d**(33/2)*sqrt(c*x**2/d + 1)/(315*c**7*d**9*x**6 + 945*c**6*d**10*x**4 + 945*c**5*d**11*
x**2 + 315*c**4*d**12) + 15*b*c**5*d**(9/2)*x**10*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2
+ 105*c**3*d**6) + 33*b*c**4*d**(11/2)*x**8*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*
c**3*d**6) + 17*b*c**3*d**(13/2)*x**6*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d
**6) + 3*b*c**2*d**(15/2)*x**4*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) +
12*b*c*d**(17/2)*x**2*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 8*b*d**(1
9/2)*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6)

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Giac [A]  time = 1.13264, size = 180, normalized size = 1.54 \begin{align*} \frac{\frac{3 \,{\left (15 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{2}\right )} b \mathrm{sgn}\left (x\right )}{c^{2}} + \frac{{\left (35 \,{\left (c x^{2} + d\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} d + 189 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{3}\right )} a \mathrm{sgn}\left (x\right )}{c^{3}}}{315 \, c} - \frac{8 \,{\left (3 \, b c d^{\frac{7}{2}} - 2 \, a d^{\frac{9}{2}}\right )} \mathrm{sgn}\left (x\right )}{315 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^8*(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

1/315*(3*(15*(c*x^2 + d)^(7/2) - 42*(c*x^2 + d)^(5/2)*d + 35*(c*x^2 + d)^(3/2)*d^2)*b*sgn(x)/c^2 + (35*(c*x^2
+ d)^(9/2) - 135*(c*x^2 + d)^(7/2)*d + 189*(c*x^2 + d)^(5/2)*d^2 - 105*(c*x^2 + d)^(3/2)*d^3)*a*sgn(x)/c^3)/c
- 8/315*(3*b*c*d^(7/2) - 2*a*d^(9/2))*sgn(x)/c^4