3.4.67 \(\int \frac {\sqrt {a x^{13}}}{\sqrt {1+x^5}} \, dx\) [367]

Optimal. Leaf size=50 \[ \frac {\sqrt {a x^{13}} \sqrt {1+x^5}}{5 x^4}-\frac {\sqrt {a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]

[Out]

-1/5*arcsinh(x^(5/2))*(a*x^13)^(1/2)/x^(13/2)+1/5*(a*x^13)^(1/2)*(x^5+1)^(1/2)/x^4

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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {15, 327, 335, 281, 221} \begin {gather*} \frac {\sqrt {x^5+1} \sqrt {a x^{13}}}{5 x^4}-\frac {\sqrt {a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^13]/Sqrt[1 + x^5],x]

[Out]

(Sqrt[a*x^13]*Sqrt[1 + x^5])/(5*x^4) - (Sqrt[a*x^13]*ArcSinh[x^(5/2)])/(5*x^(13/2))

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

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Mathematica [A]
time = 0.15, size = 52, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a x^{13}} \left (x^{5/2} \sqrt {1+x^5}-\tanh ^{-1}\left (\frac {x^{5/2}}{\sqrt {1+x^5}}\right )\right )}{5 x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^13]/Sqrt[1 + x^5],x]

[Out]

(Sqrt[a*x^13]*(x^(5/2)*Sqrt[1 + x^5] - ArcTanh[x^(5/2)/Sqrt[1 + x^5]]))/(5*x^(13/2))

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Maple [A]
time = 0.22, size = 40, normalized size = 0.80

method	result	size
meijerg	\(\frac {\sqrt {a \,x^{13}}\, \left (\sqrt {\pi }\, x^{\frac {5}{2}} \sqrt {x^{5}+1}-\sqrt {\pi }\, \arcsinh \left (x^{\frac {5}{2}}\right )\right )}{5 x^{\frac {13}{2}} \sqrt {\pi }}\)	\(40\)
risch	\(\frac {\sqrt {a \,x^{13}}\, \sqrt {x^{5}+1}}{5 x^{4}}-\frac {\arcsinh \left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{13}}\, \sqrt {a x \left (x^{5}+1\right )}}{5 \sqrt {a}\, x^{7} \sqrt {x^{5}+1}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^13)^(1/2)/(x^5+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(a*x^13)^(1/2)/x^(13/2)/Pi^(1/2)*(Pi^(1/2)*x^(5/2)*(x^5+1)^(1/2)-Pi^(1/2)*arcsinh(x^(5/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^13)^(1/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^13)/sqrt(x^5 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (36) = 72\).
time = 0.43, size = 153, normalized size = 3.06 \begin {gather*} \left [\frac {\sqrt {a} x^{4} \log \left (-\frac {8 \, a x^{14} + 8 \, a x^{9} + a x^{4} - 4 \, \sqrt {a x^{13}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a}}{x^{4}}\right ) + 4 \, \sqrt {a x^{13}} \sqrt {x^{5} + 1}}{20 \, x^{4}}, \frac {\sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {a x^{13}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{14} + a x^{9}\right )}}\right ) + 2 \, \sqrt {a x^{13}} \sqrt {x^{5} + 1}}{10 \, x^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^13)^(1/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

[1/20*(sqrt(a)*x^4*log(-(8*a*x^14 + 8*a*x^9 + a*x^4 - 4*sqrt(a*x^13)*(2*x^5 + 1)*sqrt(x^5 + 1)*sqrt(a))/x^4) +
4*sqrt(a*x^13)*sqrt(x^5 + 1))/x^4, 1/10*(sqrt(-a)*x^4*arctan(1/2*sqrt(a*x^13)*(2*x^5 + 1)*sqrt(x^5 + 1)*sqrt(
-a)/(a*x^14 + a*x^9)) + 2*sqrt(a*x^13)*sqrt(x^5 + 1))/x^4]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{13}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**13)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**13)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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Giac [A]
time = 4.53, size = 68, normalized size = 1.36 \begin {gather*} \frac {a^{\frac {11}{2}} \log \left (-\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}}\right )}{5 \, {\left | a \right |}^{5}} + \frac {\sqrt {a^{6} x^{5} + a^{6}} \sqrt {a x} x^{2}}{5 \, a^{2} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^13)^(1/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

1/5*a^(11/2)*log(-sqrt(a*x)*a^(5/2)*x^2 + sqrt(a^6*x^5 + a^6))/abs(a)^5 + 1/5*sqrt(a^6*x^5 + a^6)*sqrt(a*x)*x^
2/(a^2*abs(a))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a\,x^{13}}}{\sqrt {x^5+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^13)^(1/2)/(x^5 + 1)^(1/2),x)

[Out]

int((a*x^13)^(1/2)/(x^5 + 1)^(1/2), x)

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