3.8.76 \(\int x^4 \sqrt {a+c x^4} \, dx\) [776]

Optimal. Leaf size=127 \[ \frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4}-\frac {a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 c^{5/4} \sqrt {a+c x^4}} \]

[Out]

2/21*a*x*(c*x^4+a)^(1/2)/c+1/7*x^5*(c*x^4+a)^(1/2)-1/21*a^(7/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos
(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((
c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(5/4)/(c*x^4+a)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {285, 327, 226} \begin {gather*} -\frac {a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 c^{5/4} \sqrt {a+c x^4}}+\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[a + c*x^4],x]

[Out]

(2*a*x*Sqrt[a + c*x^4])/(21*c) + (x^5*Sqrt[a + c*x^4])/7 - (a^(7/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(
Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(21*c^(5/4)*Sqrt[a + c*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 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]

Rubi steps

\begin {align*} \int x^4 \sqrt {a+c x^4} \, dx &=\frac {1}{7} x^5 \sqrt {a+c x^4}+\frac {1}{7} (2 a) \int \frac {x^4}{\sqrt {a+c x^4}} \, dx\\ &=\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4}-\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{21 c}\\ &=\frac {2 a x \sqrt {a+c x^4}}{21 c}+\frac {1}{7} x^5 \sqrt {a+c x^4}-\frac {a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{21 c^{5/4} \sqrt {a+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.50, size = 62, normalized size = 0.49 \begin {gather*} \frac {x \sqrt {a+c x^4} \left (a+c x^4-\frac {a \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )}{\sqrt {1+\frac {c x^4}{a}}}\right )}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[a + c*x^4],x]

[Out]

(x*Sqrt[a + c*x^4]*(a + c*x^4 - (a*Hypergeometric2F1[-1/2, 1/4, 5/4, -((c*x^4)/a)])/Sqrt[1 + (c*x^4)/a]))/(7*c
)

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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 108, normalized size = 0.85

method	result	size
risch	\(\frac {x \left (3 x^{4} c +2 a \right ) \sqrt {x^{4} c +a}}{21 c}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(103\)
default	\(\frac {x^{5} \sqrt {x^{4} c +a}}{7}+\frac {2 a x \sqrt {x^{4} c +a}}{21 c}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(108\)
elliptic	\(\frac {x^{5} \sqrt {x^{4} c +a}}{7}+\frac {2 a x \sqrt {x^{4} c +a}}{21 c}-\frac {2 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/7*x^5*(c*x^4+a)^(1/2)+2/21*a*x*(c*x^4+a)^(1/2)/c-2/21*a^2/c/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x
^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^4 + a)*x^4, x)

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Fricas [A]
time = 0.07, size = 57, normalized size = 0.45 \begin {gather*} -\frac {2 \, a \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (3 \, c x^{5} + 2 \, a x\right )} \sqrt {c x^{4} + a}}{21 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(c*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

-1/21*(2*a*sqrt(c)*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) - (3*c*x^5 + 2*a*x)*sqrt(c*x^4 + a))/c

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Sympy [C] Result contains complex when optimal does not.
time = 0.41, size = 39, normalized size = 0.31 \begin {gather*} \frac {\sqrt {a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(c*x**4+a)**(1/2),x)

[Out]

sqrt(a)*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4))

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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^4*(c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + a)*x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,\sqrt {c\,x^4+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + c*x^4)^(1/2),x)

[Out]

int(x^4*(a + c*x^4)^(1/2), x)

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