3.956 \(\int (a+\frac{b}{x^2}) (c+\frac{d}{x^2})^{3/2} x^6 \, dx\)

Optimal. Leaf size=53 \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]

[Out]

((7*b*c - 2*a*d)*(c + d/x^2)^(5/2)*x^5)/(35*c^2) + (a*(c + d/x^2)^(5/2)*x^7)/(7*c)

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Rubi [A]  time = 0.0253675, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 264} \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x]

[Out]

((7*b*c - 2*a*d)*(c + d/x^2)^(5/2)*x^5)/(35*c^2) + (a*(c + d/x^2)^(5/2)*x^7)/(7*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 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 ) \left (c+\frac{d}{x^2}\right )^{3/2} x^6 \, dx &=\frac{a \left (c+\frac{d}{x^2}\right )^{5/2} x^7}{7 c}+\frac{(7 b c-2 a d) \int \left (c+\frac{d}{x^2}\right )^{3/2} x^4 \, dx}{7 c}\\ &=\frac{(7 b c-2 a d) \left (c+\frac{d}{x^2}\right )^{5/2} x^5}{35 c^2}+\frac{a \left (c+\frac{d}{x^2}\right )^{5/2} x^7}{7 c}\\ \end{align*}

Mathematica [A]  time = 0.0266944, size = 44, normalized size = 0.83 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (5 a c x^2-2 a d+7 b c\right )}{35 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(7*b*c - 2*a*d + 5*a*c*x^2))/(35*c^2)

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Maple [A]  time = 0.004, size = 45, normalized size = 0.9 \begin{align*}{\frac{{x}^{3} \left ( 5\,a{x}^{2}c-2\,ad+7\,bc \right ) \left ( c{x}^{2}+d \right ) }{35\,{c}^{2}} \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^6,x)

[Out]

1/35*((c*x^2+d)/x^2)^(3/2)*x^3*(5*a*c*x^2-2*a*d+7*b*c)*(c*x^2+d)/c^2

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Maxima [A]  time = 0.936792, size = 74, normalized size = 1.4 \begin{align*} \frac{b{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5}}{5 \, c} + \frac{{\left (5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} x^{7} - 7 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d x^{5}\right )} a}{35 \, c^{2}} \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^6,x, algorithm="maxima")

[Out]

1/5*b*(c + d/x^2)^(5/2)*x^5/c + 1/35*(5*(c + d/x^2)^(7/2)*x^7 - 7*(c + d/x^2)^(5/2)*d*x^5)*a/c^2

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Fricas [A]  time = 1.31448, size = 174, normalized size = 3.28 \begin{align*} \frac{{\left (5 \, a c^{3} x^{7} +{\left (7 \, b c^{3} + 8 \, a c^{2} d\right )} x^{5} +{\left (14 \, b c^{2} d + a c d^{2}\right )} x^{3} +{\left (7 \, b c d^{2} - 2 \, a d^{3}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \, c^{2}} \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^6,x, algorithm="fricas")

[Out]

1/35*(5*a*c^3*x^7 + (7*b*c^3 + 8*a*c^2*d)*x^5 + (14*b*c^2*d + a*c*d^2)*x^3 + (7*b*c*d^2 - 2*a*d^3)*x)*sqrt((c*
x^2 + d)/x^2)/c^2

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Sympy [B]  time = 7.04296, size = 498, normalized size = 9.4 \begin{align*} \frac{15 a c^{6} d^{\frac{9}{2}} x^{10} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{33 a c^{5} d^{\frac{11}{2}} x^{8} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{17 a c^{4} d^{\frac{13}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{3 a c^{3} d^{\frac{15}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{12 a c^{2} d^{\frac{17}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{8 a c d^{\frac{19}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{a d^{\frac{3}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{a d^{\frac{5}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c} - \frac{2 a d^{\frac{7}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{2}} + \frac{b c \sqrt{d} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{2 b d^{\frac{3}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{b d^{\frac{5}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c} \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**6,x)

[Out]

15*a*c**6*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*a*c
**5*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*a*c**4*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*a*c**3*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*a*c**2*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*a*c*d**(19/2)*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) + a*d**(3/2)*x**4*sqrt(c*x**2/d + 1)/5 + a*d**(5
/2)*x**2*sqrt(c*x**2/d + 1)/(15*c) - 2*a*d**(7/2)*sqrt(c*x**2/d + 1)/(15*c**2) + b*c*sqrt(d)*x**4*sqrt(c*x**2/
d + 1)/5 + 2*b*d**(3/2)*x**2*sqrt(c*x**2/d + 1)/5 + b*d**(5/2)*sqrt(c*x**2/d + 1)/(5*c)

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Giac [B]  time = 1.0995, size = 203, normalized size = 3.83 \begin{align*} \frac{35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b d \mathrm{sgn}\left (x\right ) + 7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} b \mathrm{sgn}\left (x\right ) + \frac{7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} a d \mathrm{sgn}\left (x\right )}{c} + \frac{{\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 )} a \mathrm{sgn}\left (x\right )}{c}}{105 \, c} - \frac{{\left (7 \, b c d^{\frac{5}{2}} - 2 \, a d^{\frac{7}{2}}\right )} \mathrm{sgn}\left (x\right )}{35 \, c^{2}} \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^6,x, algorithm="giac")

[Out]

1/105*(35*(c*x^2 + d)^(3/2)*b*d*sgn(x) + 7*(3*(c*x^2 + d)^(5/2) - 5*(c*x^2 + d)^(3/2)*d)*b*sgn(x) + 7*(3*(c*x^
2 + d)^(5/2) - 5*(c*x^2 + d)^(3/2)*d)*a*d*sgn(x)/c + (15*(c*x^2 + d)^(7/2) - 42*(c*x^2 + d)^(5/2)*d + 35*(c*x^
2 + d)^(3/2)*d^2)*a*sgn(x)/c)/c - 1/35*(7*b*c*d^(5/2) - 2*a*d^(7/2))*sgn(x)/c^2