3.1.23 \(\int \frac {(2+3 x^2) (5+x^4)^{3/2}}{x} \, dx\) [23]

Optimal. Leaf size=78 \[ \frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right ) \]

[Out]

1/24*(9*x^2+8)*(x^4+5)^(3/2)+225/16*arcsinh(1/5*x^2*5^(1/2))-5*arctanh(1/5*(x^4+5)^(1/2)*5^(1/2))*5^(1/2)+5/16
*(9*x^2+16)*(x^4+5)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1266, 829, 858, 221, 272, 65, 213} \begin {gather*} -5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac {5}{16} \left (9 x^2+16\right ) \sqrt {x^4+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x^2)*(5 + x^4)^(3/2))/x,x]

[Out]

(5*(16 + 9*x^2)*Sqrt[5 + x^4])/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 + (225*ArcSinh[x^2/Sqrt[5]])/16 - 5*Sqrt[
5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]]

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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]]

Rule 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1266

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rubi steps

\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (5+x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 x) \left (5+x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {1}{8} \text {Subst}\left (\int \frac {(40+45 x) \sqrt {5+x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {1}{16} \text {Subst}\left (\int \frac {400+225 x}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )+25 \text {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {25}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+25 \text {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 77, normalized size = 0.99 \begin {gather*} \frac {1}{48} \left (\sqrt {5+x^4} \left (320+225 x^2+16 x^4+18 x^6\right )+675 \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right )+480 \sqrt {5} \tanh ^{-1}\left (\frac {x^2-\sqrt {5+x^4}}{\sqrt {5}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x,x]

[Out]

(Sqrt[5 + x^4]*(320 + 225*x^2 + 16*x^4 + 18*x^6) + 675*ArcTanh[x^2/Sqrt[5 + x^4]] + 480*Sqrt[5]*ArcTanh[(x^2 -
Sqrt[5 + x^4])/Sqrt[5]])/48

________________________________________________________________________________________

Maple [A]
time = 0.26, size = 75, normalized size = 0.96

method	result	size
trager	\(\left (\frac {3}{8} x^{6}+\frac {1}{3} x^{4}+\frac {75}{16} x^{2}+\frac {20}{3}\right ) \sqrt {x^{4}+5}-5 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-5\right )+\sqrt {x^{4}+5}}{x^{2}}\right )-\frac {225 \ln \left (x^{2}-\sqrt {x^{4}+5}\right )}{16}\)	\(70\)
default	\(\frac {3 x^{6} \sqrt {x^{4}+5}}{8}+\frac {75 x^{2} \sqrt {x^{4}+5}}{16}+\frac {225 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}+\frac {x^{4} \sqrt {x^{4}+5}}{3}+\frac {20 \sqrt {x^{4}+5}}{3}-5 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )\)	\(75\)
elliptic	\(\frac {3 x^{6} \sqrt {x^{4}+5}}{8}+\frac {75 x^{2} \sqrt {x^{4}+5}}{16}+\frac {225 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}+\frac {x^{4} \sqrt {x^{4}+5}}{3}+\frac {20 \sqrt {x^{4}+5}}{3}-5 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )\)	\(75\)
meijerg	\(\frac {15 \sqrt {5}\, \left (-\frac {32 \sqrt {\pi }}{9}+\frac {2 \sqrt {\pi }\, \left (\frac {4 x^{4}}{5}+16\right ) \sqrt {1+\frac {x^{4}}{5}}}{9}-\frac {8 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{4}}{5}}}{2}\right )}{3}+\frac {4 \left (\frac {8}{3}-2 \ln \left (2\right )+4 \ln \left (x \right )-\ln \left (5\right )\right ) \sqrt {\pi }}{3}\right )}{8 \sqrt {\pi }}+\frac {\frac {15 \sqrt {\pi }\, x^{2} \sqrt {5}\, \left (\frac {x^{4}}{20}+\frac {5}{8}\right ) \sqrt {1+\frac {x^{4}}{5}}}{2}+\frac {225 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}}{\sqrt {\pi }}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)*(x^4+5)^(3/2)/x,x,method=_RETURNVERBOSE)

[Out]

3/8*x^6*(x^4+5)^(1/2)+75/16*x^2*(x^4+5)^(1/2)+225/16*arcsinh(1/5*x^2*5^(1/2))+1/3*x^4*(x^4+5)^(1/2)+20/3*(x^4+
5)^(1/2)-5*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (62) = 124\).
time = 0.49, size = 138, normalized size = 1.77 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} + \frac {5}{2} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + 5 \, \sqrt {x^{4} + 5} + \frac {75 \, {\left (\frac {3 \, \sqrt {x^{4} + 5}}{x^{2}} - \frac {5 \, {\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}}{16 \, {\left (\frac {2 \, {\left (x^{4} + 5\right )}}{x^{4}} - \frac {{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac {225}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {225}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5)^(3/2)/x,x, algorithm="maxima")

[Out]

1/3*(x^4 + 5)^(3/2) + 5/2*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) + 5*sqrt(x^4 + 5)
+ 75/16*(3*sqrt(x^4 + 5)/x^2 - 5*(x^4 + 5)^(3/2)/x^6)/(2*(x^4 + 5)/x^4 - (x^4 + 5)^2/x^8 - 1) + 225/32*log(sqr
t(x^4 + 5)/x^2 + 1) - 225/32*log(sqrt(x^4 + 5)/x^2 - 1)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 67, normalized size = 0.86 \begin {gather*} \frac {1}{48} \, {\left (18 \, x^{6} + 16 \, x^{4} + 225 \, x^{2} + 320\right )} \sqrt {x^{4} + 5} + 5 \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - \frac {225}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5)^(3/2)/x,x, algorithm="fricas")

[Out]

1/48*(18*x^6 + 16*x^4 + 225*x^2 + 320)*sqrt(x^4 + 5) + 5*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/x^2) - 225/16*
log(-x^2 + sqrt(x^4 + 5))

________________________________________________________________________________________

Sympy [A]
time = 16.34, size = 114, normalized size = 1.46 \begin {gather*} \frac {3 x^{10}}{8 \sqrt {x^{4} + 5}} + \frac {105 x^{6}}{16 \sqrt {x^{4} + 5}} + \frac {375 x^{2}}{16 \sqrt {x^{4} + 5}} + \frac {\left (x^{4} + 5\right )^{\frac {3}{2}}}{3} + 5 \sqrt {x^{4} + 5} + \frac {5 \sqrt {5} \log {\left (x^{4} \right )}}{2} - 5 \sqrt {5} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )} + \frac {225 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)*(x**4+5)**(3/2)/x,x)

[Out]

3*x**10/(8*sqrt(x**4 + 5)) + 105*x**6/(16*sqrt(x**4 + 5)) + 375*x**2/(16*sqrt(x**4 + 5)) + (x**4 + 5)**(3/2)/3
+ 5*sqrt(x**4 + 5) + 5*sqrt(5)*log(x**4)/2 - 5*sqrt(5)*log(sqrt(x**4/5 + 1) + 1) + 225*asinh(sqrt(5)*x**2/5)/
16

________________________________________________________________________________________

Giac [A]
time = 4.03, size = 90, normalized size = 1.15 \begin {gather*} \frac {1}{48} \, \sqrt {x^{4} + 5} {\left ({\left (2 \, {\left (9 \, x^{2} + 8\right )} x^{2} + 225\right )} x^{2} + 320\right )} + 5 \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) - \frac {225}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5)^(3/2)/x,x, algorithm="giac")

[Out]

1/48*sqrt(x^4 + 5)*((2*(9*x^2 + 8)*x^2 + 225)*x^2 + 320) + 5*sqrt(5)*log(-(x^2 + sqrt(5) - sqrt(x^4 + 5))/(x^2
- sqrt(5) - sqrt(x^4 + 5))) - 225/16*log(-x^2 + sqrt(x^4 + 5))

________________________________________________________________________________________

Mupad [B]
time = 0.18, size = 55, normalized size = 0.71 \begin {gather*} \frac {225\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{16}-5\,\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )+\sqrt {x^4+5}\,\left (\frac {3\,x^6}{8}+\frac {x^4}{3}+\frac {75\,x^2}{16}+\frac {20}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 5)^(3/2)*(3*x^2 + 2))/x,x)

[Out]

(225*asinh((5^(1/2)*x^2)/5))/16 - 5*5^(1/2)*atanh((5^(1/2)*(x^4 + 5)^(1/2))/5) + (x^4 + 5)^(1/2)*((75*x^2)/16
+ x^4/3 + (3*x^6)/8 + 20/3)

________________________________________________________________________________________