3.11.22 \(\int \frac {(a+b x^4)^{3/4}}{x} \, dx\) [1022]

Optimal. Leaf size=70 \[ \frac {1}{3} \left (a+b x^4\right )^{3/4}+\frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 70, 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 = {272, 52, 65, 304, 209, 212} \begin {gather*} \frac {1}{2} a^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac {1}{3} \left (a+b x^4\right )^{3/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x,x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 272

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{3/4}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{3} \left (a+b x^4\right )^{3/4}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac {1}{3} \left (a+b x^4\right )^{3/4}+\frac {a \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{b}\\ &=\frac {1}{3} \left (a+b x^4\right )^{3/4}-\frac {1}{2} a \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=\frac {1}{3} \left (a+b x^4\right )^{3/4}+\frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 70, normalized size = 1.00 \begin {gather*} \frac {1}{3} \left (a+b x^4\right )^{3/4}+\frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x,x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x,x)

[Out]

int((b*x^4+a)^(3/4)/x,x)

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Maxima [A]
time = 0.50, size = 71, normalized size = 1.01 \begin {gather*} \frac {1}{4} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}\right )} + \frac {1}{3} \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (50) = 100\).
time = 0.39, size = 132, normalized size = 1.89 \begin {gather*} -{\left (a^{3}\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (a^{3}\right )}^{\frac {1}{4}} a^{2} - \sqrt {\sqrt {b x^{4} + a} a^{4} + \sqrt {a^{3}} a^{3}} {\left (a^{3}\right )}^{\frac {1}{4}}}{a^{3}}\right ) - \frac {1}{4} \, {\left (a^{3}\right )}^{\frac {1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} + {\left (a^{3}\right )}^{\frac {3}{4}}\right ) + \frac {1}{4} \, {\left (a^{3}\right )}^{\frac {1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} - {\left (a^{3}\right )}^{\frac {3}{4}}\right ) + \frac {1}{3} \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x,x, algorithm="fricas")

[Out]

-(a^3)^(1/4)*arctan(-((b*x^4 + a)^(1/4)*(a^3)^(1/4)*a^2 - sqrt(sqrt(b*x^4 + a)*a^4 + sqrt(a^3)*a^3)*(a^3)^(1/4
))/a^3) - 1/4*(a^3)^(1/4)*log((b*x^4 + a)^(1/4)*a^2 + (a^3)^(3/4)) + 1/4*(a^3)^(1/4)*log((b*x^4 + a)^(1/4)*a^2
- (a^3)^(3/4)) + 1/3*(b*x^4 + a)^(3/4)

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Sympy [C] Result contains complex when optimal does not.
time = 0.59, size = 44, normalized size = 0.63 \begin {gather*} - \frac {b^{\frac {3}{4}} x^{3} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x,x)

[Out]

-b**(3/4)*x**3*gamma(-3/4)*hyper((-3/4, -3/4), (1/4,), a*exp_polar(I*pi)/(b*x**4))/(4*gamma(1/4))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (50) = 100\).
time = 1.45, size = 185, normalized size = 2.64 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) - \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) + \frac {1}{3} \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.14, size = 50, normalized size = 0.71 \begin {gather*} \frac {a^{3/4}\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2}-\frac {a^{3/4}\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2}+\frac {{\left (b\,x^4+a\right )}^{3/4}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^(3/4)/x,x)

[Out]

(a^(3/4)*atan((a + b*x^4)^(1/4)/a^(1/4)))/2 - (a^(3/4)*atanh((a + b*x^4)^(1/4)/a^(1/4)))/2 + (a + b*x^4)^(3/4)
/3

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