\(\int \frac {(a+c x^4)^{3/2}}{x^2} \, dx\) [801]

Optimal result

Integrand size = 15, antiderivative size = 251 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx=\frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{x}-\frac {12 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {6 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 \sqrt {a+c x^4}} \]

[Out]

-(c*x^4+a)^(3/2)/x+6/5*c*x^3*(c*x^4+a)^(1/2)+12/5*a*x*c^(1/2)*(c*x^4+a)^(1/2)/(a^(1/2)+x^2*c^(1/2))-12/5*a^(5/
4)*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(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*x^4+a)^(1/
2)+6/5*a^(5/4)*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(s
in(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*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 251, 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 = {283, 285, 311, 226, 1210} \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx=\frac {6 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {12 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )} \]

[In]

Int[(a + c*x^4)^(3/2)/x^2,x]

[Out]

(6*c*x^3*Sqrt[a + c*x^4])/5 + (12*a*Sqrt[c]*x*Sqrt[a + c*x^4])/(5*(Sqrt[a] + Sqrt[c]*x^2)) - (a + c*x^4)^(3/2)
/x - (12*a^(5/4)*c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTa
n[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*Sqrt[a + c*x^4]) + (6*a^(5/4)*c^(1/4)*(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])/(5*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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 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 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 {\left (a+c x^4\right )^{3/2}}{x}+(6 c) \int x^2 \sqrt {a+c x^4} \, dx \\ & = \frac {6}{5} c x^3 \sqrt {a+c x^4}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {1}{5} (12 a c) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx \\ & = \frac {6}{5} c x^3 \sqrt {a+c x^4}-\frac {\left (a+c x^4\right )^{3/2}}{x}+\frac {1}{5} \left (12 a^{3/2} \sqrt {c}\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx-\frac {1}{5} \left (12 a^{3/2} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx \\ & = \frac {6}{5} c x^3 \sqrt {a+c x^4}+\frac {12 a \sqrt {c} x \sqrt {a+c x^4}}{5 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a+c x^4\right )^{3/2}}{x}-\frac {12 a^{5/4} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+c x^4}}+\frac {6 a^{5/4} \sqrt [4]{c} \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 )}{5 \sqrt {a+c x^4}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 9.68 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.20 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx=-\frac {a \sqrt {a+c x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},\frac {3}{4},-\frac {c x^4}{a}\right )}{x \sqrt {1+\frac {c x^4}{a}}} \]

[In]

Integrate[(a + c*x^4)^(3/2)/x^2,x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.49

method	result	size
risch	\(-\frac {\sqrt {x^{4} c +a}\, \left (-x^{4} c +5 a \right )}{5 x}+\frac {12 i a^{\frac {3}{2}} \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(122\)
default	\(-\frac {a \sqrt {x^{4} c +a}}{x}+\frac {c \,x^{3} \sqrt {x^{4} c +a}}{5}+\frac {12 i a^{\frac {3}{2}} \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(128\)
elliptic	\(-\frac {a \sqrt {x^{4} c +a}}{x}+\frac {c \,x^{3} \sqrt {x^{4} c +a}}{5}+\frac {12 i a^{\frac {3}{2}} \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\)	\(128\)

[In]

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

[Out]

-1/5*(c*x^4+a)^(1/2)*(-c*x^4+5*a)/x+12/5*I*a^(3/2)*c^(1/2)/(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)-EllipticE(x*(
I/a^(1/2)*c^(1/2))^(1/2),I))

Fricas [F]

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

[In]

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

[Out]

integral((c*x^4 + a)^(3/2)/x^2, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.16 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx=\frac {a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]

[In]

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

[Out]

a**(3/2)*gamma(-1/4)*hyper((-3/2, -1/4), (3/4,), c*x**4*exp_polar(I*pi)/a)/(4*x*gamma(3/4))

Maxima [F]

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

[In]

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

[Out]

integrate((c*x^4 + a)^(3/2)/x^2, x)

Giac [F]

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

[In]

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

[Out]

integrate((c*x^4 + a)^(3/2)/x^2, x)

Mupad [B] (verification not implemented)

Time = 5.97 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.16 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^2} \, dx=\frac {{\left (c\,x^4+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {5}{4};\ -\frac {1}{4};\ -\frac {a}{c\,x^4}\right )}{5\,x\,{\left (\frac {a}{c\,x^4}+1\right )}^{3/2}} \]

[In]

int((a + c*x^4)^(3/2)/x^2,x)

[Out]

((a + c*x^4)^(3/2)*hypergeom([-3/2, -5/4], -1/4, -a/(c*x^4)))/(5*x*(a/(c*x^4) + 1)^(3/2))