3.1002 \(\int x^6 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=103 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b} \]

[Out]

(a*x^3*(a + b*x^4)^(1/4))/(32*b) + (x^7*(a + b*x^4)^(1/4))/8 + (3*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(
64*b^(7/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(7/4))

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Rubi [A]  time = 0.0382135, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 321, 331, 298, 203, 206} \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x^4)^(1/4),x]

[Out]

(a*x^3*(a + b*x^4)^(1/4))/(32*b) + (x^7*(a + b*x^4)^(1/4))/8 + (3*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(
64*b^(7/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(7/4))

Rule 279

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

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 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 x^6 \sqrt [4]{a+b x^4} \, dx &=\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{1}{8} a \int \frac{x^6}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}-\frac{\left (3 a^2\right ) \int \frac{x^2}{\left (a+b x^4\right )^{3/4}} \, dx}{32 b}\\ &=\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{32 b}\\ &=\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/2}}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/2}}\\ &=\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}\\ \end{align*}

Mathematica [C]  time = 0.0470745, size = 64, normalized size = 0.62 \[ \frac{x^3 \sqrt [4]{a+b x^4} \left (-\frac{a \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}}+a+b x^4\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^4)^(1/4),x]

[Out]

(x^3*(a + b*x^4)^(1/4)*(a + b*x^4 - (a*Hypergeometric2F1[-1/4, 3/4, 7/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^(1/4))
)/(8*b)

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{x}^{6}\sqrt [4]{b{x}^{4}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x^4+a)^(1/4),x)

[Out]

int(x^6*(b*x^4+a)^(1/4),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^6*(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6137, size = 502, normalized size = 4.87 \begin{align*} \frac{12 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} \left (\frac{a^{8}}{b^{7}}\right )^{\frac{3}{4}} b^{5} - \left (\frac{a^{8}}{b^{7}}\right )^{\frac{3}{4}} b^{5} x \sqrt{\frac{\sqrt{\frac{a^{8}}{b^{7}}} b^{4} x^{2} + \sqrt{b x^{4} + a} a^{4}}{x^{2}}}}{a^{8} x}\right ) - 3 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\frac{3 \,{\left (\left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 3 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (-\frac{3 \,{\left (\left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{7} + a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

1/128*(12*(a^8/b^7)^(1/4)*b*arctan(-((b*x^4 + a)^(1/4)*a^2*(a^8/b^7)^(3/4)*b^5 - (a^8/b^7)^(3/4)*b^5*x*sqrt((s
qrt(a^8/b^7)*b^4*x^2 + sqrt(b*x^4 + a)*a^4)/x^2))/(a^8*x)) - 3*(a^8/b^7)^(1/4)*b*log(3*((a^8/b^7)^(1/4)*b^2*x
+ (b*x^4 + a)^(1/4)*a^2)/x) + 3*(a^8/b^7)^(1/4)*b*log(-3*((a^8/b^7)^(1/4)*b^2*x - (b*x^4 + a)^(1/4)*a^2)/x) +
4*(4*b*x^7 + a*x^3)*(b*x^4 + a)^(1/4))/b

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Sympy [C]  time = 2.27357, size = 39, normalized size = 0.38 \begin{align*} \frac{\sqrt [4]{a} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**7*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(11/4))

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Giac [B]  time = 1.17719, size = 348, normalized size = 3.38 \begin{align*} \frac{1}{256} \,{\left (\frac{8 \, x^{8}{\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )}}{x} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x}\right )}}{a^{2} b} - \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \log \left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}} + \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \log \left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}}\right )} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

1/256*(8*x^8*((b*x^4 + a)^(1/4)*(b + a/x^4)/x + 3*(b*x^4 + a)^(1/4)*b/x)/(a^2*b) - 6*sqrt(2)*(-b)^(1/4)*arctan
(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b^2 - 6*sqrt(2)*(-b)^(1/4)*arctan(-1/2*s
qrt(2)*(sqrt(2)*(-b)^(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b^2 - 3*sqrt(2)*(-b)^(1/4)*log(sqrt(-b) + sqrt
(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b^2 + 3*sqrt(2)*(-b)^(1/4)*log(sqrt(-b) - sqrt(2)*(b
*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b^2)*a^2