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

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

[Out]

(-35*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))

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Rubi [A]  time = 0.0594738, antiderivative size = 116, normalized size of antiderivative = 1., 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 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]]

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]

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*}

Mathematica [A]  time = 0.183139, size = 112, normalized size = 0.97 \[ \frac{\sqrt{a} \left (-21 a^2 b x^4+6 a^3 x^6-140 a b^2 x^2-105 b^3\right )+\frac{105 b^{5/2} \left (a x^2+b\right ) \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{x}}{24 a^{9/2} \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

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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Maple [A]  time = 0.013, size = 98, normalized size = 0.8 \begin{align*}{\frac{a{x}^{2}+b}{24\,{x}^{5}} \left ( 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 \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}a{b}^{2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+1/x^2*b)^(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(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89258, size = 628, normalized size = 5.41 \begin{align*} \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{align*}

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]  time = 7.39183, size = 432, normalized size = 3.72 \begin{align*} \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{align*}

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 = 1.29005, size = 193, normalized size = 1.66 \begin{align*} -\frac{1}{24} \, b^{2}{\left (\frac{8 \,{\left (a + \frac{9 \,{\left (a x^{2} + b\right )}}{x^{2}}\right )} x^{2}}{{\left (a x^{2} + b\right )} a^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}}} + \frac{105 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{3 \,{\left (13 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{11 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}\right )}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{4}}\right )} \end{align*}

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*b^2*(8*(a + 9*(a*x^2 + b)/x^2)*x^2/((a*x^2 + b)*a^4*sqrt((a*x^2 + b)/x^2)) + 105*arctan(sqrt((a*x^2 + b)
/x^2)/sqrt(-a))/(sqrt(-a)*a^4) - 3*(13*a*sqrt((a*x^2 + b)/x^2) - 11*(a*x^2 + b)*sqrt((a*x^2 + b)/x^2)/x^2)/((a
- (a*x^2 + b)/x^2)^2*a^4))