3.1309 \(\int \frac{x^{13/2}}{\sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=57 \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*b^(3/2))

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Rubi [A]  time = 0.0358856, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {321, 329, 275, 217, 206} \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(13/2)/Sqrt[a + b*x^5],x]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*b^(3/2))

Rule 321

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 329

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]

Rule 275

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 217

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

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0215827, size = 57, normalized size = 1. \[ \frac{x^{5/2} \sqrt{a+b x^5}}{5 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/Sqrt[a + b*x^5],x]

[Out]

(x^(5/2)*Sqrt[a + b*x^5])/(5*b) - (a*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*b^(3/2))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{{x}^{{\frac{13}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.01799, size = 344, normalized size = 6.04 \begin{align*} \left [\frac{4 \, \sqrt{b x^{5} + a} b x^{\frac{5}{2}} + a \sqrt{b} \log \left (-8 \, b^{2} x^{10} - 8 \, a b x^{5} + 4 \,{\left (2 \, b x^{7} + a x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{b} \sqrt{x} - a^{2}\right )}{20 \, b^{2}}, \frac{2 \, \sqrt{b x^{5} + a} b x^{\frac{5}{2}} + a \sqrt{-b} \arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right )}{10 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/20*(4*sqrt(b*x^5 + a)*b*x^(5/2) + a*sqrt(b)*log(-8*b^2*x^10 - 8*a*b*x^5 + 4*(2*b*x^7 + a*x^2)*sqrt(b*x^5 +
a)*sqrt(b)*sqrt(x) - a^2))/b^2, 1/10*(2*sqrt(b*x^5 + a)*b*x^(5/2) + a*sqrt(-b)*arctan(2*sqrt(b*x^5 + a)*sqrt(-
b)*x^(5/2)/(2*b*x^5 + a)))/b^2]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.25429, size = 59, normalized size = 1.04 \begin{align*} \frac{\sqrt{b x^{5} + a} x^{\frac{5}{2}}}{5 \, b} + \frac{a \log \left ({\left | -\sqrt{b} x^{\frac{5}{2}} + \sqrt{b x^{5} + a} \right |}\right )}{5 \, b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(b*x^5 + a)*x^(5/2)/b + 1/5*a*log(abs(-sqrt(b)*x^(5/2) + sqrt(b*x^5 + a)))/b^(3/2)