3.20.44 \(\int \frac {x^3}{(a+\frac {b}{x^2})^{5/2}} \, dx\) [1944]

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

[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 = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 44, 53, 65, 214} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^2)^(5/2),x]

[Out]

(-35*b^2)/(24*a^3*(a + b/x^2)^(3/2)) - (35*b^2)/(8*a^4*Sqrt[a + b/x^2]) - (7*b*x^2)/(8*a^2*(a + b/x^2)^(3/2))
+ x^4/(4*a*(a + b/x^2)^(3/2)) + (35*b^2*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(8*a^(9/2))

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 214

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

Rule 272

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} \text {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 \text {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 \text {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) \text {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 ) \text {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) \text {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 = 0.13, size = 109, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a} x \left (-105 b^3-140 a b^2 x^2-21 a^2 b x^4+6 a^3 x^6\right )-105 b^2 \left (b+a x^2\right )^{3/2} \log \left (-\sqrt {a} x+\sqrt {b+a x^2}\right )}{24 a^{9/2} \sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^2)^(5/2),x]

[Out]

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

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Maple [A]
time = 0.12, size = 98, normalized size = 0.84

method	result	size
default	\(\frac {\left (a \,x^{2}+b \right ) \left (6 x^{7} a^{\frac {9}{2}}-21 a^{\frac {7}{2}} b \,x^{5}-140 a^{\frac {5}{2}} b^{2} x^{3}-105 a^{\frac {3}{2}} b^{3} x +105 \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right ) \left (a \,x^{2}+b \right )^{\frac {3}{2}} a \,b^{2}\right )}{24 \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} x^{5} a^{\frac {11}{2}}}\)	\(98\)
risch	\(\frac {\left (2 a \,x^{2}-11 b \right ) \left (a \,x^{2}+b \right )}{8 a^{4} \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}+\frac {\left (\frac {35 b^{2} \ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+b}\right )}{8 a^{\frac {9}{2}}}+\frac {b^{3} \sqrt {a \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{12 a^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}}-\frac {5 b^{2} \sqrt {a \left (x -\frac {\sqrt {-a b}}{a}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{a}\right )}}{3 a^{5} \left (x -\frac {\sqrt {-a b}}{a}\right )}-\frac {5 b^{2} \sqrt {a \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{3 a^{5} \left (x +\frac {\sqrt {-a b}}{a}\right )}-\frac {b^{3} \sqrt {a \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )}}{12 a^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{a}\right )^{2}}\right ) \sqrt {a \,x^{2}+b}}{\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}\)	\(350\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(a*x^2+b)*(6*x^7*a^(9/2)-21*a^(7/2)*b*x^5-140*a^(5/2)*b^2*x^3-105*a^(3/2)*b^3*x+105*ln(x*a^(1/2)+(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 = 0.52, size = 139, normalized size = 1.20 \begin {gather*} -\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}}} \end {gather*}

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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Fricas [A]
time = 0.39, size = 287, normalized size = 2.47 \begin {gather*} \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 ] \end {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (107) = 214\).
time = 6.42, size = 432, normalized size = 3.72 \begin {gather*} \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}} \end {gather*}

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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Giac [A]
time = 1.38, size = 114, normalized size = 0.98 \begin {gather*} \frac {{\left ({\left (3 \, x^{2} {\left (\frac {2 \, x^{2}}{a \mathrm {sgn}\left (x\right )} - \frac {7 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} - \frac {140 \, b^{2}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} x^{2} - \frac {105 \, b^{3}}{a^{4} \mathrm {sgn}\left (x\right )}\right )} x}{24 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}}} + \frac {35 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{\frac {9}{2}}} - \frac {35 \, b^{2} \log \left ({\left | -\sqrt {a} x + \sqrt {a x^{2} + b} \right |}\right )}{8 \, a^{\frac {9}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((3*x^2*(2*x^2/(a*sgn(x)) - 7*b/(a^2*sgn(x))) - 140*b^2/(a^3*sgn(x)))*x^2 - 105*b^3/(a^4*sgn(x)))*x/(a*x^
2 + b)^(3/2) + 35/16*b^2*log(abs(b))*sgn(x)/a^(9/2) - 35/8*b^2*log(abs(-sqrt(a)*x + sqrt(a*x^2 + b)))/(a^(9/2)
*sgn(x))

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Mupad [B]
time = 1.92, size = 95, normalized size = 0.82 \begin {gather*} \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}} \end {gather*}

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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