\(\int \frac {x^{10}}{(a+b x^4)^{3/2}} \, dx\) [868]

Optimal result

Integrand size = 15, antiderivative size = 258 \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}-\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}} \]

[Out]

-1/2*x^7/b/(b*x^4+a)^(1/2)+7/10*x^3*(b*x^4+a)^(1/2)/b^2-21/10*a*x*(b*x^4+a)^(1/2)/b^(5/2)/(a^(1/2)+x^2*b^(1/2)
)+21/10*a^(5/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*ar
ctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/b^(11/4)
/(b*x^4+a)^(1/2)-21/20*a^(5/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Ell
ipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)
^(1/2)/b^(11/4)/(b*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {294, 327, 311, 226, 1210} \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {x^7}{2 b \sqrt {a+b x^4}} \]

[In]

Int[x^10/(a + b*x^4)^(3/2),x]

[Out]

-1/2*x^7/(b*Sqrt[a + b*x^4]) + (7*x^3*Sqrt[a + b*x^4])/(10*b^2) - (21*a*x*Sqrt[a + b*x^4])/(10*b^(5/2)*(Sqrt[a
] + Sqrt[b]*x^2)) + (21*a^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[
2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(10*b^(11/4)*Sqrt[a + b*x^4]) - (21*a^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[
(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(20*b^(11/4)*Sqrt[a + b*
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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

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

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 1210

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 \int \frac {x^6}{\sqrt {a+b x^4}} \, dx}{2 b} \\ & = -\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {(21 a) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{10 b^2} \\ & = -\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {\left (21 a^{3/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{10 b^{5/2}}+\frac {\left (21 a^{3/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{10 b^{5/2}} \\ & = -\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}-\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.26 \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x^3 \left (-7 a+b x^4+7 a \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{5 b^2 \sqrt {a+b x^4}} \]

[In]

Integrate[x^10/(a + b*x^4)^(3/2),x]

[Out]

(x^3*(-7*a + b*x^4 + 7*a*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^4)/a)]))/(5*b^2*Sqrt[a +
b*x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.58 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {a \,x^{3}}{2 b^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{5 b^{2}}-\frac {21 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 b^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(137\)
elliptic	\(\frac {a \,x^{3}}{2 b^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{5 b^{2}}-\frac {21 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{10 b^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\)	\(137\)
risch	\(\frac {x^{3} \sqrt {b \,x^{4}+a}}{5 b^{2}}-\frac {a \left (8 b \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+3 a \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{5 b^{2}}\)	\(268\)

[In]

int(x^10/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2/b^2*a*x^3/((x^4+a/b)*b)^(1/2)+1/5*x^3*(b*x^4+a)^(1/2)/b^2-21/10*I*a^(3/2)/b^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2
)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*(EllipticF(x*(I/a^(1/2)*b^(1
/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.51 \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=-\frac {21 \, {\left (a b x^{5} + a^{2} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 21 \, {\left (a b x^{5} + a^{2} x\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (2 \, b^{2} x^{8} - 14 \, a b x^{4} - 21 \, a^{2}\right )} \sqrt {b x^{4} + a}}{10 \, {\left (b^{4} x^{5} + a b^{3} x\right )}} \]

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

-1/10*(21*(a*b*x^5 + a^2*x)*sqrt(b)*(-a/b)^(3/4)*elliptic_e(arcsin((-a/b)^(1/4)/x), -1) - 21*(a*b*x^5 + a^2*x)
*sqrt(b)*(-a/b)^(3/4)*elliptic_f(arcsin((-a/b)^(1/4)/x), -1) - (2*b^2*x^8 - 14*a*b*x^4 - 21*a^2)*sqrt(b*x^4 +
a))/(b^4*x^5 + a*b^3*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.14 \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=\frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \]

[In]

integrate(x**10/(b*x**4+a)**(3/2),x)

[Out]

x**11*gamma(11/4)*hyper((3/2, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(15/4))

Maxima [F]

\[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^10/(b*x^4 + a)^(3/2), x)

Giac [F]

\[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=\int { \frac {x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^10/(b*x^4 + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx=\int \frac {x^{10}}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \]

[In]

int(x^10/(a + b*x^4)^(3/2),x)

[Out]

int(x^10/(a + b*x^4)^(3/2), x)