3.4.4 \(\int x^2 (3+4 x^4)^{5/4} \, dx\) [304]

Optimal. Leaf size=93 \[ \frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}} \]

[Out]

15/32*x^3*(4*x^4+3)^(1/4)+1/8*x^3*(4*x^4+3)^(5/4)-45/256*arctan(x*2^(1/2)/(4*x^4+3)^(1/4))*2^(1/2)+45/256*arct
anh(x*2^(1/2)/(4*x^4+3)^(1/4))*2^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 338, 304, 209, 212} \begin {gather*} -\frac {45 \text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt {2}}+\frac {1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac {15}{32} \sqrt [4]{4 x^4+3} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(3 + 4*x^4)^(5/4),x]

[Out]

(15*x^3*(3 + 4*x^4)^(1/4))/32 + (x^3*(3 + 4*x^4)^(5/4))/8 - (45*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(128*Sq
rt[2]) + (45*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(128*Sqrt[2])

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 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^2 \left (3+4 x^4\right )^{5/4} \, dx &=\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {15}{8} \int x^2 \sqrt [4]{3+4 x^4} \, dx\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{32} \int \frac {x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{32} \text {Subst}\left (\int \frac {x^2}{1-4 x^4} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac {45}{128} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )-\frac {45}{128} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac {15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^3 \left (3+4 x^4\right )^{5/4}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 82, normalized size = 0.88 \begin {gather*} \frac {1}{32} x^3 \sqrt [4]{3+4 x^4} \left (27+16 x^4\right )-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(3 + 4*x^4)^(5/4),x]

[Out]

(x^3*(3 + 4*x^4)^(1/4)*(27 + 16*x^4))/32 - (45*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(128*Sqrt[2]) + (45*ArcT
anh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(128*Sqrt[2])

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 2.11, size = 19, normalized size = 0.20

method	result	size
meijerg	\(3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [-\frac {5}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )\)	\(19\)
risch	\(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {5 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{32}\)	\(42\)
trager	\(\frac {x^{3} \left (16 x^{4}+27\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{32}+\frac {45 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{512}+\frac {45 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}+2\right ) \left (4 x^{4}+3\right )^{\frac {3}{4}} x -4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}} x^{3}-4 \sqrt {4 x^{4}+3}\, x^{2}+8 x^{4}+3\right )}{512}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(4*x^4+3)^(5/4),x,method=_RETURNVERBOSE)

[Out]

3^(1/4)*x^3*hypergeom([-5/4,3/4],[7/4],-4/3*x^4)

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Maxima [A]
time = 1.36, size = 130, normalized size = 1.40 \begin {gather*} \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) + \frac {9 \, {\left (\frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} - \frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{32 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(4*x^4+3)^(5/4),x, algorithm="maxima")

[Out]

45/256*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) - 45/512*sqrt(2)*log(-(sqrt(2) - (4*x^4 + 3)^(1/4)/x)/(
sqrt(2) + (4*x^4 + 3)^(1/4)/x)) + 9/32*(20*(4*x^4 + 3)^(1/4)/x - 9*(4*x^4 + 3)^(5/4)/x^5)/(8*(4*x^4 + 3)/x^4 -
(4*x^4 + 3)^2/x^8 - 16)

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Fricas [A]
time = 1.55, size = 105, normalized size = 1.13 \begin {gather*} \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {45}{512} \, \sqrt {2} \log \left (8 \, x^{4} + 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} + 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{32} \, {\left (16 \, x^{7} + 27 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(4*x^4+3)^(5/4),x, algorithm="fricas")

[Out]

45/256*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) + 45/512*sqrt(2)*log(8*x^4 + 4*sqrt(2)*(4*x^4 + 3)^(1/4
)*x^3 + 4*sqrt(4*x^4 + 3)*x^2 + 2*sqrt(2)*(4*x^4 + 3)^(3/4)*x + 3) + 1/32*(16*x^7 + 27*x^3)*(4*x^4 + 3)^(1/4)

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Sympy [C] Result contains complex when optimal does not.
time = 0.95, size = 41, normalized size = 0.44 \begin {gather*} \frac {3 \cdot \sqrt [4]{3} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(4*x**4+3)**(5/4),x)

[Out]

3*3**(1/4)*x**3*gamma(3/4)*hyper((-5/4, 3/4), (7/4,), 4*x**4*exp_polar(I*pi)/3)/(4*gamma(7/4))

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Giac [A]
time = 0.74, size = 110, normalized size = 1.18 \begin {gather*} \frac {1}{32} \, x^{8} {\left (\frac {9 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} - \frac {20 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} + \frac {45}{256} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {45}{512} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(4*x^4+3)^(5/4),x, algorithm="giac")

[Out]

1/32*x^8*(9*(4*x^4 + 3)^(1/4)*(3/x^4 + 4)/x - 20*(4*x^4 + 3)^(1/4)/x) + 45/256*sqrt(2)*arctan(1/2*sqrt(2)*(4*x
^4 + 3)^(1/4)/x) - 45/512*sqrt(2)*log(-(sqrt(2) - (4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (4\,x^4+3\right )}^{5/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(4*x^4 + 3)^(5/4),x)

[Out]

int(x^2*(4*x^4 + 3)^(5/4), x)

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