3.11.2 \(\int x^6 \sqrt [4]{a+b x^4} \, dx\) [1002]

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

[Out]

1/32*a*x^3*(b*x^4+a)^(1/4)/b+1/8*x^7*(b*x^4+a)^(1/4)+3/64*a^2*arctan(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(7/4)-3/64*a
^2*arctanh(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(7/4)

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Rubi [A]
time = 0.03, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 338, 304, 209, 212} \begin {gather*} \frac {3 a^2 \text {ArcTan}\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} \end {gather*}

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 209

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

Rule 212

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

Rule 285

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 304

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 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 338

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]

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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.27, size = 87, normalized size = 0.84 \begin {gather*} \frac {2 b^{3/4} x^3 \sqrt [4]{a+b x^4} \left (a+4 b x^4\right )+3 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-3 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x^4)^(1/4),x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{6} \left (b \,x^{4}+a \right )^{\frac {1}{4}}\, dx\]

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 [A]
time = 0.50, size = 152, normalized size = 1.48 \begin {gather*} \frac {\frac {3 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b}{x} + \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{x^{5}}}{32 \, {\left (b^{3} - \frac {2 \, {\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2} b}{x^{8}}\right )}} - \frac {3 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{128 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(3*(b*x^4 + a)^(1/4)*a^2*b/x + (b*x^4 + a)^(5/4)*a^2/x^5)/(b^3 - 2*(b*x^4 + a)*b^2/x^4 + (b*x^4 + a)^2*b/
x^8) - 3/128*(2*a^2*arctan((b*x^4 + a)^(1/4)/(b^(1/4)*x))/b^(3/4) - a^2*log(-(b^(1/4) - (b*x^4 + a)^(1/4)/x)/(
b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(3/4))/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (79) = 158\).
time = 0.38, size = 220, normalized size = 2.14 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 1.08, size = 39, normalized size = 0.38 \begin {gather*} \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 {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^4 + a)^(1/4)*x^6, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,{\left (b\,x^4+a\right )}^{1/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x^4)^(1/4),x)

[Out]

int(x^6*(a + b*x^4)^(1/4), x)

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