3.968 \(\int \frac{(a+\frac{b}{x^2}) x^4}{\sqrt{c+\frac{d}{x^2}}} \, dx\)

Optimal. Leaf size=82 \[ \frac{x^3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^2}-\frac{2 d x \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^3}+\frac{a x^5 \sqrt{c+\frac{d}{x^2}}}{5 c} \]

[Out]

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

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Rubi [A]  time = 0.0333523, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 191} \[ \frac{x^3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^2}-\frac{2 d x \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{15 c^3}+\frac{a x^5 \sqrt{c+\frac{d}{x^2}}}{5 c} \]

Antiderivative was successfully verified.

[In]

Int[((a + b/x^2)*x^4)/Sqrt[c + d/x^2],x]

[Out]

(-2*d*(5*b*c - 4*a*d)*Sqrt[c + d/x^2]*x)/(15*c^3) + ((5*b*c - 4*a*d)*Sqrt[c + d/x^2]*x^3)/(15*c^2) + (a*Sqrt[c
+ d/x^2]*x^5)/(5*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 191

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

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) x^4}{\sqrt{c+\frac{d}{x^2}}} \, dx &=\frac{a \sqrt{c+\frac{d}{x^2}} x^5}{5 c}+\frac{(5 b c-4 a d) \int \frac{x^2}{\sqrt{c+\frac{d}{x^2}}} \, dx}{5 c}\\ &=\frac{(5 b c-4 a d) \sqrt{c+\frac{d}{x^2}} x^3}{15 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^5}{5 c}-\frac{(2 d (5 b c-4 a d)) \int \frac{1}{\sqrt{c+\frac{d}{x^2}}} \, dx}{15 c^2}\\ &=-\frac{2 d (5 b c-4 a d) \sqrt{c+\frac{d}{x^2}} x}{15 c^3}+\frac{(5 b c-4 a d) \sqrt{c+\frac{d}{x^2}} x^3}{15 c^2}+\frac{a \sqrt{c+\frac{d}{x^2}} x^5}{5 c}\\ \end{align*}

Mathematica [A]  time = 0.0440407, size = 56, normalized size = 0.68 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (a \left (3 c^2 x^4-4 c d x^2+8 d^2\right )+5 b c \left (c x^2-2 d\right )\right )}{15 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b/x^2)*x^4)/Sqrt[c + d/x^2],x]

[Out]

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

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Maple [A]  time = 0.007, size = 67, normalized size = 0.8 \begin{align*}{\frac{ \left ( 3\,a{x}^{4}{c}^{2}-4\,acd{x}^{2}+5\,b{c}^{2}{x}^{2}+8\,a{d}^{2}-10\,bcd \right ) \left ( c{x}^{2}+d \right ) }{15\,x{c}^{3}}{\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^4/(c+d/x^2)^(1/2),x)

[Out]

1/15/x*(3*a*c^2*x^4-4*a*c*d*x^2+5*b*c^2*x^2+8*a*d^2-10*b*c*d)*(c*x^2+d)/((c*x^2+d)/x^2)^(1/2)/c^3

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Maxima [A]  time = 0.949943, size = 115, normalized size = 1.4 \begin{align*} \frac{{\left ({\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3} - 3 \, \sqrt{c + \frac{d}{x^{2}}} d x\right )} b}{3 \, c^{2}} + \frac{{\left (3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 10 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d x^{3} + 15 \, \sqrt{c + \frac{d}{x^{2}}} d^{2} x\right )} a}{15 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4/(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35639, size = 132, normalized size = 1.61 \begin{align*} \frac{{\left (3 \, a c^{2} x^{5} +{\left (5 \, b c^{2} - 4 \, a c d\right )} x^{3} - 2 \,{\left (5 \, b c d - 4 \, a d^{2}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4/(c+d/x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.24329, size = 338, normalized size = 4.12 \begin{align*} \frac{3 a c^{4} d^{\frac{9}{2}} x^{8} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{2} + 15 c^{3} d^{6}} + \frac{2 a c^{3} d^{\frac{11}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{2} + 15 c^{3} d^{6}} + \frac{3 a c^{2} d^{\frac{13}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{2} + 15 c^{3} d^{6}} + \frac{12 a c d^{\frac{15}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{2} + 15 c^{3} d^{6}} + \frac{8 a d^{\frac{17}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{5} d^{4} x^{4} + 30 c^{4} d^{5} x^{2} + 15 c^{3} d^{6}} + \frac{b \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} - \frac{2 b d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**4/(c+d/x**2)**(1/2),x)

[Out]

3*a*c**4*d**(9/2)*x**8*sqrt(c*x**2/d + 1)/(15*c**5*d**4*x**4 + 30*c**4*d**5*x**2 + 15*c**3*d**6) + 2*a*c**3*d*
*(11/2)*x**6*sqrt(c*x**2/d + 1)/(15*c**5*d**4*x**4 + 30*c**4*d**5*x**2 + 15*c**3*d**6) + 3*a*c**2*d**(13/2)*x*
*4*sqrt(c*x**2/d + 1)/(15*c**5*d**4*x**4 + 30*c**4*d**5*x**2 + 15*c**3*d**6) + 12*a*c*d**(15/2)*x**2*sqrt(c*x*
*2/d + 1)/(15*c**5*d**4*x**4 + 30*c**4*d**5*x**2 + 15*c**3*d**6) + 8*a*d**(17/2)*sqrt(c*x**2/d + 1)/(15*c**5*d
**4*x**4 + 30*c**4*d**5*x**2 + 15*c**3*d**6) + b*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/(3*c) - 2*b*d**(3/2)*sqrt(c*x
**2/d + 1)/(3*c**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x^{2}}\right )} x^{4}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4/(c+d/x^2)^(1/2),x, algorithm="giac")

[Out]

integrate((a + b/x^2)*x^4/sqrt(c + d/x^2), x)