3.1189 \(\int x^6 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=263 \[ \frac{3 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a-b x^4}-\frac{a x^3 \sqrt [4]{a-b x^4}}{32 b} \]

[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[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x
^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/
4)) + (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(7
/4)) - (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(
7/4))

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Rubi [A]  time = 0.157805, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {279, 321, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a-b x^4}-\frac{a x^3 \sqrt [4]{a-b x^4}}{32 b} \]

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[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x
^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/
4)) + (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(7
/4)) - (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*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 297

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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-\sqrt{b} x^2}{1+b x^4} \, 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+\sqrt{b} x^2}{1+b x^4} \, 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{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^2}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^2}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}+2 x}{-\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}-2 x}{-\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}\\ &=-\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 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}\\ &=-\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 (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{7/4}}\\ \end{align*}

Mathematica [C]  time = 0.0513212, size = 66, normalized size = 0.25 \[ \frac{x^3 \sqrt [4]{a-b x^4} \left (\frac{a \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{\sqrt [4]{1-\frac{b x^4}{a}}}-a+b x^4\right )}{8 b} \]

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 = 0.019, 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 [A]  time = 1.86189, size = 521, normalized size = 1.98 \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*sq
rt((sqrt(-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 = 2.3454, size = 41, normalized size = 0.16 \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^{2 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(2*I*pi)/a)/(4*gamma(11/4))

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Giac [A]  time = 1.21963, size = 312, normalized size = 1.19 \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} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{7}{4}}} - \frac{6 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{7}{4}}} - \frac{3 \, \sqrt{2} \log \left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{7}{4}}} + \frac{3 \, \sqrt{2} \log \left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{7}{4}}}\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)*arctan(1/2*sqrt
(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(7/4) - 6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/
4) - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(7/4) - 3*sqrt(2)*log(sqrt(b) + sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x +
sqrt(-b*x^4 + a)/x^2)/b^(7/4) + 3*sqrt(2)*log(sqrt(b) - sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 +
a)/x^2)/b^(7/4))*a^2