\(\int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx\) [366]

Optimal result

Integrand size = 19, antiderivative size = 75 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=-\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \text {arcsinh}\left (x^{5/2}\right )}{20 x^{23/2}} \]

[Out]

3/20*arcsinh(x^(5/2))*(a*x^23)^(1/2)/x^(23/2)-3/20*(a*x^23)^(1/2)*(x^5+1)^(1/2)/x^9+1/10*(a*x^23)^(1/2)*(x^5+1
)^(1/2)/x^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {15, 327, 335, 281, 221} \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {3 \sqrt {a x^{23}} \text {arcsinh}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac {3 \sqrt {x^5+1} \sqrt {a x^{23}}}{20 x^9}+\frac {\sqrt {x^5+1} \sqrt {a x^{23}}}{10 x^4} \]

[In]

Int[Sqrt[a*x^23]/Sqrt[1 + x^5],x]

[Out]

(-3*Sqrt[a*x^23]*Sqrt[1 + x^5])/(20*x^9) + (Sqrt[a*x^23]*Sqrt[1 + x^5])/(10*x^4) + (3*Sqrt[a*x^23]*ArcSinh[x^(
5/2)])/(20*x^(23/2))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a x^{23}} \int \frac {x^{23/2}}{\sqrt {1+x^5}} \, dx}{x^{23/2}} \\ & = \frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}-\frac {\left (3 \sqrt {a x^{23}}\right ) \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx}{4 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx}{8 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{4 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {\left (3 \sqrt {a x^{23}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )}{20 x^{23/2}} \\ & = -\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a x^{23}} \left (x^{5/2} \sqrt {1+x^5} \left (-3+2 x^5\right )+3 \log \left (x^{5/2}+\sqrt {1+x^5}\right )\right )}{20 x^{23/2}} \]

[In]

Integrate[Sqrt[a*x^23]/Sqrt[1 + x^5],x]

[Out]

(Sqrt[a*x^23]*(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5) + 3*Log[x^(5/2) + Sqrt[1 + x^5]]))/(20*x^(23/2))

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.64

method	result	size
meijerg	\(\frac {\sqrt {a \,x^{23}}\, \left (-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \left (-10 x^{5}+15\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{4}\right )}{5 x^{\frac {23}{2}} \sqrt {\pi }}\)	\(48\)
risch	\(\frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, \sqrt {a \,x^{23}}}{20 x^{9}}+\frac {3 \,\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{23}}\, \sqrt {a x \left (x^{5}+1\right )}}{20 \sqrt {a}\, x^{12} \sqrt {x^{5}+1}}\)	\(64\)

[In]

int((a*x^23)^(1/2)/(x^5+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(a*x^23)^(1/2)/x^(23/2)/Pi^(1/2)*(-1/20*Pi^(1/2)*x^(5/2)*(-10*x^5+15)*(x^5+1)^(1/2)+3/4*Pi^(1/2)*arcsinh(x
^(5/2)))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\left [\frac {3 \, \sqrt {a} x^{9} \log \left (-\frac {8 \, a x^{19} + 8 \, a x^{14} + a x^{9} + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a}}{x^{9}}\right ) + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{80 \, x^{9}}, -\frac {3 \, \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{19} + a x^{14}\right )}}\right ) - 2 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{40 \, x^{9}}\right ] \]

[In]

integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

[1/80*(3*sqrt(a)*x^9*log(-(8*a*x^19 + 8*a*x^14 + a*x^9 + 4*sqrt(a*x^23)*(2*x^5 + 1)*sqrt(x^5 + 1)*sqrt(a))/x^9
) + 4*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x^5 + 1))/x^9, -1/40*(3*sqrt(-a)*x^9*arctan(1/2*sqrt(a*x^23)*(2*x^5 + 1)*s
qrt(x^5 + 1)*sqrt(-a)/(a*x^19 + a*x^14)) - 2*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x^5 + 1))/x^9]

Sympy [F]

\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a x^{23}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

[In]

integrate((a*x**23)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**23)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

Maxima [F]

\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int { \frac {\sqrt {a x^{23}}}{\sqrt {x^{5} + 1}} \,d x } \]

[In]

integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^23)/sqrt(x^5 + 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (55) = 110\).

Time = 0.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a^{6} x^{5} + a^{6}} \sqrt {a x} a^{6} x^{2} {\left (\frac {2 \, x^{5}}{a^{8}} - \frac {3}{a^{8}}\right )} \mathrm {sgn}\left (x\right )}{20 \, {\left | a \right |}} - \frac {3 \, {\left (\frac {a^{\frac {5}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}} - \frac {a^{\frac {5}{2}} \log \left (a^{2} {\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}}\right )} a^{3}}{20 \, {\left | a \right |}^{4}} \]

[In]

integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

1/20*sqrt(a^6*x^5 + a^6)*sqrt(a*x)*a^6*x^2*(2*x^5/a^8 - 3/a^8)*sgn(x)/abs(a) - 3/20*(a^(5/2)*log(-sqrt(a*x)*a^
(5/2)*x^2 + sqrt(a^6*x^5 + a^6))*sgn(x)/abs(a) - a^(5/2)*log(a^2*abs(a))*sgn(x)/abs(a))*a^3/abs(a)^4

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a\,x^{23}}}{\sqrt {x^5+1}} \,d x \]

[In]

int((a*x^23)^(1/2)/(x^5 + 1)^(1/2),x)

[Out]

int((a*x^23)^(1/2)/(x^5 + 1)^(1/2), x)