3.1944 \(\int \frac {x^3}{(a+\frac {b}{x^2})^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ \frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}}-\frac {35 b^2}{8 a^4 \sqrt {a+\frac {b}{x^2}}}-\frac {35 b^2}{24 a^3 \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 b x^2}{8 a^2 \left (a+\frac {b}{x^2}\right )^{3/2}}+\frac {x^4}{4 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[Out]

-35/24*b^2/a^3/(a+b/x^2)^(3/2)-7/8*b*x^2/a^2/(a+b/x^2)^(3/2)+1/4*x^4/a/(a+b/x^2)^(3/2)+35/8*b^2*arctanh((a+b/x
^2)^(1/2)/a^(1/2))/a^(9/2)-35/8*b^2/a^4/(a+b/x^2)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {35 x^4 \sqrt {a+\frac {b}{x^2}}}{12 a^3}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {35 b x^2 \sqrt {a+\frac {b}{x^2}}}{8 a^4}-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^2)^(5/2),x]

[Out]

(-35*b*Sqrt[a + b/x^2]*x^2)/(8*a^4) - x^4/(3*a*(a + b/x^2)^(3/2)) - (7*x^4)/(3*a^2*Sqrt[a + b/x^2]) + (35*Sqrt
[a + b/x^2]*x^4)/(12*a^3) + (35*b^2*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(9/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

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

Rubi steps

\begin {align*} \int \frac {x^3}{\left (a+\frac {b}{x^2}\right )^{5/2}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^{5/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )}{6 a}\\ &=-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}-\frac {35 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{6 a^2}\\ &=-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {35 \sqrt {a+\frac {b}{x^2}} x^4}{12 a^3}+\frac {(35 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{8 a^3}\\ &=-\frac {35 b \sqrt {a+\frac {b}{x^2}} x^2}{8 a^4}-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {35 \sqrt {a+\frac {b}{x^2}} x^4}{12 a^3}-\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^2}\right )}{16 a^4}\\ &=-\frac {35 b \sqrt {a+\frac {b}{x^2}} x^2}{8 a^4}-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {35 \sqrt {a+\frac {b}{x^2}} x^4}{12 a^3}-\frac {(35 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^2}}\right )}{8 a^4}\\ &=-\frac {35 b \sqrt {a+\frac {b}{x^2}} x^2}{8 a^4}-\frac {x^4}{3 a \left (a+\frac {b}{x^2}\right )^{3/2}}-\frac {7 x^4}{3 a^2 \sqrt {a+\frac {b}{x^2}}}+\frac {35 \sqrt {a+\frac {b}{x^2}} x^4}{12 a^3}+\frac {35 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8 a^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 112, normalized size = 0.97 \[ \frac {\sqrt {a} \left (6 a^3 x^6-21 a^2 b x^4-140 a b^2 x^2-105 b^3\right )+\frac {105 b^{5/2} \left (a x^2+b\right ) \sqrt {\frac {a x^2}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{x}}{24 a^{9/2} \sqrt {a+\frac {b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^2)^(5/2),x]

[Out]

(Sqrt[a]*(-105*b^3 - 140*a*b^2*x^2 - 21*a^2*b*x^4 + 6*a^3*x^6) + (105*b^(5/2)*(b + a*x^2)*Sqrt[1 + (a*x^2)/b]*
ArcSinh[(Sqrt[a]*x)/Sqrt[b]])/x)/(24*a^(9/2)*Sqrt[a + b/x^2]*(b + a*x^2))

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fricas [A]  time = 1.04, size = 287, normalized size = 2.47 \[ \left [\frac {105 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt {a} \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}} - b\right ) + 2 \, {\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{48 \, {\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}, -\frac {105 \, {\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - {\left (6 \, a^{4} x^{8} - 21 \, a^{3} b x^{6} - 140 \, a^{2} b^{2} x^{4} - 105 \, a b^{3} x^{2}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{24 \, {\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(a)*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + b)/x^2) - b) +
2*(6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2*b^2*x^4 - 105*a*b^3*x^2)*sqrt((a*x^2 + b)/x^2))/(a^7*x^4 + 2*a^6*b*x^2
+ a^5*b^2), -1/24*(105*(a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(-a)*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + b)/x^2)/(a
*x^2 + b)) - (6*a^4*x^8 - 21*a^3*b*x^6 - 140*a^2*b^2*x^4 - 105*a*b^3*x^2)*sqrt((a*x^2 + b)/x^2))/(a^7*x^4 + 2*
a^6*b*x^2 + a^5*b^2)]

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giac [B]  time = 0.37, size = 196, normalized size = 1.69 \[ \frac {1}{8} \, \sqrt {a x^{4} + b x^{2}} {\left (\frac {2 \, x^{2}}{a^{3}} - \frac {11 \, b}{a^{4}}\right )} - \frac {35 \, b^{2} \log \left ({\left | -2 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} - b \right |}\right )}{16 \, a^{\frac {9}{2}}} + \frac {5 \, {\left (21 \, b^{2} \log \left ({\left | b \right |}\right ) + 32 \, b^{2}\right )}}{48 \, a^{\frac {9}{2}}} - \frac {12 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )}^{2} a b^{3} + 21 \, {\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} b^{4} + 10 \, b^{5}}{3 \, {\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="giac")

[Out]

1/8*sqrt(a*x^4 + b*x^2)*(2*x^2/a^3 - 11*b/a^4) - 35/16*b^2*log(abs(-2*(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))*sqrt
(a) - b))/a^(9/2) + 5/48*(21*b^2*log(abs(b)) + 32*b^2)/a^(9/2) - 1/3*(12*(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))^2
*a*b^3 + 21*(sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))*sqrt(a)*b^4 + 10*b^5)/(((sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))*sq
rt(a) + b)^3*a^(9/2))

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maple [A]  time = 0.02, size = 98, normalized size = 0.84 \[ \frac {\left (a \,x^{2}+b \right ) \left (6 a^{\frac {9}{2}} x^{7}-21 a^{\frac {7}{2}} b \,x^{5}-140 a^{\frac {5}{2}} b^{2} x^{3}-105 a^{\frac {3}{2}} b^{3} x +105 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a \,b^{2} \ln \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )\right )}{24 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} a^{\frac {11}{2}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b/x^2)^(5/2),x)

[Out]

1/24*(a*x^2+b)*(6*x^7*a^(9/2)-21*a^(7/2)*x^5*b-140*a^(5/2)*x^3*b^2-105*a^(3/2)*x*b^3+105*ln(a^(1/2)*x+(a*x^2+b
)^(1/2))*(a*x^2+b)^(3/2)*a*b^2)/((a*x^2+b)/x^2)^(5/2)/x^5/a^(11/2)

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maxima [A]  time = 1.90, size = 139, normalized size = 1.20 \[ -\frac {105 \, {\left (a + \frac {b}{x^{2}}\right )}^{3} b^{2} - 175 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} a b^{2} + 56 \, {\left (a + \frac {b}{x^{2}}\right )} a^{2} b^{2} + 8 \, a^{3} b^{2}}{24 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}} a^{4} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{5} + {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{6}\right )}} - \frac {35 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{2}}} + \sqrt {a}}\right )}{16 \, a^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x^2)^(5/2),x, algorithm="maxima")

[Out]

-1/24*(105*(a + b/x^2)^3*b^2 - 175*(a + b/x^2)^2*a*b^2 + 56*(a + b/x^2)*a^2*b^2 + 8*a^3*b^2)/((a + b/x^2)^(7/2
)*a^4 - 2*(a + b/x^2)^(5/2)*a^5 + (a + b/x^2)^(3/2)*a^6) - 35/16*b^2*log((sqrt(a + b/x^2) - sqrt(a))/(sqrt(a +
b/x^2) + sqrt(a)))/a^(9/2)

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mupad [B]  time = 1.92, size = 95, normalized size = 0.82 \[ \frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^2}}}{\sqrt {a}}\right )}{8\,a^{9/2}}-\frac {35\,b^2}{6\,a^3\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}+\frac {x^4}{4\,a\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}-\frac {7\,b\,x^2}{8\,a^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}}-\frac {35\,b^3}{8\,a^4\,x^2\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b/x^2)^(5/2),x)

[Out]

(35*b^2*atanh((a + b/x^2)^(1/2)/a^(1/2)))/(8*a^(9/2)) - (35*b^2)/(6*a^3*(a + b/x^2)^(3/2)) + x^4/(4*a*(a + b/x
^2)^(3/2)) - (7*b*x^2)/(8*a^2*(a + b/x^2)^(3/2)) - (35*b^3)/(8*a^4*x^2*(a + b/x^2)^(3/2))

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sympy [B]  time = 8.92, size = 432, normalized size = 3.72 \[ \frac {6 a^{\frac {89}{2}} b^{75} x^{7}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {21 a^{\frac {87}{2}} b^{76} x^{5}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {140 a^{\frac {85}{2}} b^{77} x^{3}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} - \frac {105 a^{\frac {83}{2}} b^{78} x}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {105 a^{42} b^{\frac {155}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} + \frac {105 a^{41} b^{\frac {157}{2}} \sqrt {\frac {a x^{2}}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} x}{\sqrt {b}} \right )}}{24 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{2} \sqrt {\frac {a x^{2}}{b} + 1} + 24 a^{\frac {91}{2}} b^{\frac {153}{2}} \sqrt {\frac {a x^{2}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b/x**2)**(5/2),x)

[Out]

6*a**(89/2)*b**75*x**7/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/
b + 1)) - 21*a**(87/2)*b**76*x**5/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*s
qrt(a*x**2/b + 1)) - 140*a**(85/2)*b**77*x**3/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*
b**(153/2)*sqrt(a*x**2/b + 1)) - 105*a**(83/2)*b**78*x/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a
**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) + 105*a**42*b**(155/2)*x**2*sqrt(a*x**2/b + 1)*asinh(sqrt(a)*x/sqrt(b)
)/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a**(91/2)*b**(153/2)*sqrt(a*x**2/b + 1)) + 105*a**41*b
**(157/2)*sqrt(a*x**2/b + 1)*asinh(sqrt(a)*x/sqrt(b))/(24*a**(93/2)*b**(151/2)*x**2*sqrt(a*x**2/b + 1) + 24*a*
*(91/2)*b**(153/2)*sqrt(a*x**2/b + 1))

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