3.1.22 \(\int x (2+3 x^2) (5+x^4)^{3/2} \, dx\) [22]

Optimal. Leaf size=60 \[ \frac {15}{8} x^2 \sqrt {5+x^4}+\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]

[Out]

1/4*x^2*(x^4+5)^(3/2)+3/10*(x^4+5)^(5/2)+75/8*arcsinh(1/5*x^2*5^(1/2))+15/8*x^2*(x^4+5)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1262, 655, 201, 221} \begin {gather*} \frac {3}{10} \left (x^4+5\right )^{5/2}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac {15}{8} x^2 \sqrt {x^4+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(2 + 3*x^2)*(5 + x^4)^(3/2),x]

[Out]

(15*x^2*Sqrt[5 + x^4])/8 + (x^2*(5 + x^4)^(3/2))/4 + (3*(5 + x^4)^(5/2))/10 + (75*ArcSinh[x^2/Sqrt[5]])/8

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 1262

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

Rubi steps

\begin {align*} \int x \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int (2+3 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {3}{10} \left (5+x^4\right )^{5/2}+\text {Subst}\left (\int \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {15}{4} \text {Subst}\left (\int \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {15}{8} x^2 \sqrt {5+x^4}+\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {75}{8} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {15}{8} x^2 \sqrt {5+x^4}+\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 54, normalized size = 0.90 \begin {gather*} \frac {1}{40} \sqrt {5+x^4} \left (300+125 x^2+120 x^4+10 x^6+12 x^8\right )+\frac {75}{8} \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[5 + x^4]*(300 + 125*x^2 + 120*x^4 + 10*x^6 + 12*x^8))/40 + (75*ArcTanh[x^2/Sqrt[5 + x^4]])/8

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Maple [A]
time = 0.13, size = 46, normalized size = 0.77

method	result	size
risch	\(\frac {\left (12 x^{8}+10 x^{6}+120 x^{4}+125 x^{2}+300\right ) \sqrt {x^{4}+5}}{40}+\frac {75 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{8}\)	\(44\)
default	\(\frac {3 \left (x^{4}+5\right )^{\frac {5}{2}}}{10}+\frac {x^{6} \sqrt {x^{4}+5}}{4}+\frac {25 x^{2} \sqrt {x^{4}+5}}{8}+\frac {75 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{8}\)	\(46\)
trager	\(\left (\frac {3}{10} x^{8}+\frac {1}{4} x^{6}+3 x^{4}+\frac {25}{8} x^{2}+\frac {15}{2}\right ) \sqrt {x^{4}+5}-\frac {75 \ln \left (x^{2}-\sqrt {x^{4}+5}\right )}{8}\)	\(48\)
elliptic	\(\frac {3 x^{8} \sqrt {x^{4}+5}}{10}+3 x^{4} \sqrt {x^{4}+5}+\frac {15 \sqrt {x^{4}+5}}{2}+\frac {x^{6} \sqrt {x^{4}+5}}{4}+\frac {25 x^{2} \sqrt {x^{4}+5}}{8}+\frac {75 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{8}\)	\(70\)
meijerg	\(\frac {225 \sqrt {5}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {2}{25} x^{8}+\frac {4}{5} x^{4}+2\right ) \sqrt {1+\frac {x^{4}}{5}}}{15}\right )}{16 \sqrt {\pi }}+\frac {5 \sqrt {\pi }\, x^{2} \sqrt {5}\, \left (\frac {x^{4}}{20}+\frac {5}{8}\right ) \sqrt {1+\frac {x^{4}}{5}}+\frac {75 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{8}}{\sqrt {\pi }}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(x^4+5)^(5/2)+1/4*x^6*(x^4+5)^(1/2)+25/8*x^2*(x^4+5)^(1/2)+75/8*arcsinh(1/5*x^2*5^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (45) = 90\).
time = 0.50, size = 95, normalized size = 1.58 \begin {gather*} \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} + \frac {25 \, {\left (\frac {3 \, \sqrt {x^{4} + 5}}{x^{2}} - \frac {5 \, {\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}}{8 \, {\left (\frac {2 \, {\left (x^{4} + 5\right )}}{x^{4}} - \frac {{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac {75}{16} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {75}{16} \, \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(x*(3*x^2+2)*(x^4+5)^(3/2),x, algorithm="maxima")

[Out]

3/10*(x^4 + 5)^(5/2) + 25/8*(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) + 75/16*log(sqrt(x^4 + 5)/x^2 + 1) - 75/16*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]
time = 0.38, size = 48, normalized size = 0.80 \begin {gather*} \frac {1}{40} \, {\left (12 \, x^{8} + 10 \, x^{6} + 120 \, x^{4} + 125 \, x^{2} + 300\right )} \sqrt {x^{4} + 5} - \frac {75}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(12*x^8 + 10*x^6 + 120*x^4 + 125*x^2 + 300)*sqrt(x^4 + 5) - 75/8*log(-x^2 + sqrt(x^4 + 5))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (54) = 108\).
time = 4.25, size = 109, normalized size = 1.82 \begin {gather*} \frac {x^{10}}{4 \sqrt {x^{4} + 5}} + \frac {3 x^{8} \sqrt {x^{4} + 5}}{10} + \frac {35 x^{6}}{8 \sqrt {x^{4} + 5}} + \frac {x^{4} \sqrt {x^{4} + 5}}{2} + \frac {125 x^{2}}{8 \sqrt {x^{4} + 5}} + \frac {5 \left (x^{4} + 5\right )^{\frac {3}{2}}}{2} - 5 \sqrt {x^{4} + 5} + \frac {75 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**10/(4*sqrt(x**4 + 5)) + 3*x**8*sqrt(x**4 + 5)/10 + 35*x**6/(8*sqrt(x**4 + 5)) + x**4*sqrt(x**4 + 5)/2 + 125
*x**2/(8*sqrt(x**4 + 5)) + 5*(x**4 + 5)**(3/2)/2 - 5*sqrt(x**4 + 5) + 75*asinh(sqrt(5)*x**2/5)/8

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Giac [A]
time = 4.85, size = 57, normalized size = 0.95 \begin {gather*} \frac {1}{8} \, {\left (2 \, x^{4} + 5\right )} \sqrt {x^{4} + 5} x^{2} + \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} + \frac {5}{2} \, \sqrt {x^{4} + 5} x^{2} - \frac {75}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*x^4 + 5)*sqrt(x^4 + 5)*x^2 + 3/10*(x^4 + 5)^(5/2) + 5/2*sqrt(x^4 + 5)*x^2 - 75/8*log(-x^2 + sqrt(x^4 +
5))

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Mupad [B]
time = 0.18, size = 42, normalized size = 0.70 \begin {gather*} \frac {75\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{8}+\sqrt {x^4+5}\,\left (\frac {3\,x^8}{10}+\frac {x^6}{4}+3\,x^4+\frac {25\,x^2}{8}+\frac {15}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^4 + 5)^(3/2)*(3*x^2 + 2),x)

[Out]

(75*asinh((5^(1/2)*x^2)/5))/8 + (x^4 + 5)^(1/2)*((25*x^2)/8 + 3*x^4 + x^6/4 + (3*x^8)/10 + 15/2)

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