3.31.87 \(\int \frac {(-q+p x^4) \sqrt {q+p x^4}}{x^2 (a q+b x^2+a p x^4)} \, dx\) [3087]

3.31.87.1 Optimal result

Integrand size = 41, antiderivative size = 524 \[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\frac {\sqrt {q+p x^4}}{a x}+\frac {\left (\sqrt {2} b \sqrt {b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}-a \sqrt {p} \sqrt {q}}-i \sqrt {2} \sqrt {b} \sqrt {b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}-a \sqrt {p} \sqrt {q}} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}-a \sqrt {p} \sqrt {q}} x}{\sqrt {a} \left (\sqrt {q}+\sqrt {p} x^2+\sqrt {q+p x^4}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}}-\frac {i \left (-i \sqrt {2} b \sqrt {-b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}+a \sqrt {p} \sqrt {q}}+\sqrt {2} \sqrt {b} \sqrt {-b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}+a \sqrt {p} \sqrt {q}} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b+i \sqrt {b} \sqrt {-b+2 a \sqrt {p} \sqrt {q}}+a \sqrt {p} \sqrt {q}} x}{\sqrt {a} \left (\sqrt {q}+\sqrt {p} x^2+\sqrt {q+p x^4}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}} \]

output

(p*x^4+q)^(1/2)/a/x+1/2*(2^(1/2)*b*(b+I*b^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^( 
1/2)-a*p^(1/2)*q^(1/2))^(1/2)-I*2^(1/2)*b^(1/2)*(b+I*b^(1/2)*(-b+2*a*p^(1/ 
2)*q^(1/2))^(1/2)-a*p^(1/2)*q^(1/2))^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^(1/2)) 
*arctan(2^(1/2)*(b+I*b^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^(1/2)-a*p^(1/2)*q^(1 
/2))^(1/2)*x/a^(1/2)/(q^(1/2)+p^(1/2)*x^2+(p*x^4+q)^(1/2)))/a^(5/2)/p^(1/2 
)/q^(1/2)-1/2*I*(-I*2^(1/2)*b*(-b+I*b^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^(1/2) 
+a*p^(1/2)*q^(1/2))^(1/2)+2^(1/2)*b^(1/2)*(-b+I*b^(1/2)*(-b+2*a*p^(1/2)*q^ 
(1/2))^(1/2)+a*p^(1/2)*q^(1/2))^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^(1/2))*arct 
anh(2^(1/2)*(-b+I*b^(1/2)*(-b+2*a*p^(1/2)*q^(1/2))^(1/2)+a*p^(1/2)*q^(1/2) 
)^(1/2)*x/a^(1/2)/(q^(1/2)+p^(1/2)*x^2+(p*x^4+q)^(1/2)))/a^(5/2)/p^(1/2)/q 
^(1/2)
 

3.31.87.2 Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.10 \[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\frac {\sqrt {q+p x^4}}{a x}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {q+p x^4}}\right )}{a^{3/2}} \]

input

Integrate[((-q + p*x^4)*Sqrt[q + p*x^4])/(x^2*(a*q + b*x^2 + a*p*x^4)),x]
 

output

Sqrt[q + p*x^4]/(a*x) + (Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^ 
4])])/a^(3/2)
 

3.31.87.3 Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.48 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {7279, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (p x^4-q\right ) \sqrt {p x^4+q}}{x^2 \left (a p x^4+a q+b x^2\right )} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {\sqrt {p x^4+q} \left (2 a p x^2+b\right )}{a \left (a p x^4+a q+b x^2\right )}-\frac {\sqrt {p x^4+q}}{a x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {p x^4+q}}\right )}{a^{3/2}}+\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \left (b-\sqrt {b^2-4 a^2 p q}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right ),\frac {1}{2}\right )}{a \sqrt {p x^4+q} \left (-\sqrt {b^2-4 a^2 p q}-2 a \sqrt {p} \sqrt {q}+b\right )}+\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \left (\sqrt {b^2-4 a^2 p q}+b\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right ),\frac {1}{2}\right )}{a \sqrt {p x^4+q} \left (\sqrt {b^2-4 a^2 p q}-2 a \sqrt {p} \sqrt {q}+b\right )}-\frac {b \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \left (-\sqrt {b^2-4 a^2 p q}+2 a \sqrt {p} \sqrt {q}+b\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right ),2 \arctan \left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right ),\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q} \left (-\sqrt {b^2-4 a^2 p q}-2 a \sqrt {p} \sqrt {q}+b\right )}-\frac {b \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \left (\sqrt {b^2-4 a^2 p q}+2 a \sqrt {p} \sqrt {q}+b\right ) \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right ),2 \arctan \left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right ),\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q} \left (\sqrt {b^2-4 a^2 p q}-2 a \sqrt {p} \sqrt {q}+b\right )}-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right ),\frac {1}{2}\right )}{a \sqrt {p x^4+q}}+\frac {\sqrt {p x^4+q}}{a x}\)

input

Int[((-q + p*x^4)*Sqrt[q + p*x^4])/(x^2*(a*q + b*x^2 + a*p*x^4)),x]
 

output

Sqrt[q + p*x^4]/(a*x) + (Sqrt[b]*ArcTan[(Sqrt[b]*x)/(Sqrt[a]*Sqrt[q + p*x^ 
4])])/a^(3/2) - (p^(1/4)*q^(1/4)*(Sqrt[q] + Sqrt[p]*x^2)*Sqrt[(q + p*x^4)/ 
(Sqrt[q] + Sqrt[p]*x^2)^2]*EllipticF[2*ArcTan[(p^(1/4)*x)/q^(1/4)], 1/2])/ 
(a*Sqrt[q + p*x^4]) + (p^(1/4)*q^(1/4)*(b - Sqrt[b^2 - 4*a^2*p*q])*(Sqrt[q 
] + Sqrt[p]*x^2)*Sqrt[(q + p*x^4)/(Sqrt[q] + Sqrt[p]*x^2)^2]*EllipticF[2*A 
rcTan[(p^(1/4)*x)/q^(1/4)], 1/2])/(a*(b - 2*a*Sqrt[p]*Sqrt[q] - Sqrt[b^2 - 
 4*a^2*p*q])*Sqrt[q + p*x^4]) + (p^(1/4)*q^(1/4)*(b + Sqrt[b^2 - 4*a^2*p*q 
])*(Sqrt[q] + Sqrt[p]*x^2)*Sqrt[(q + p*x^4)/(Sqrt[q] + Sqrt[p]*x^2)^2]*Ell 
ipticF[2*ArcTan[(p^(1/4)*x)/q^(1/4)], 1/2])/(a*(b - 2*a*Sqrt[p]*Sqrt[q] + 
Sqrt[b^2 - 4*a^2*p*q])*Sqrt[q + p*x^4]) - (b*(b + 2*a*Sqrt[p]*Sqrt[q] - Sq 
rt[b^2 - 4*a^2*p*q])*(Sqrt[q] + Sqrt[p]*x^2)*Sqrt[(q + p*x^4)/(Sqrt[q] + S 
qrt[p]*x^2)^2]*EllipticPi[(2 - b/(a*Sqrt[p]*Sqrt[q]))/4, 2*ArcTan[(p^(1/4) 
*x)/q^(1/4)], 1/2])/(4*a^2*p^(1/4)*q^(1/4)*(b - 2*a*Sqrt[p]*Sqrt[q] - Sqrt 
[b^2 - 4*a^2*p*q])*Sqrt[q + p*x^4]) - (b*(b + 2*a*Sqrt[p]*Sqrt[q] + Sqrt[b 
^2 - 4*a^2*p*q])*(Sqrt[q] + Sqrt[p]*x^2)*Sqrt[(q + p*x^4)/(Sqrt[q] + Sqrt[ 
p]*x^2)^2]*EllipticPi[(2 - b/(a*Sqrt[p]*Sqrt[q]))/4, 2*ArcTan[(p^(1/4)*x)/ 
q^(1/4)], 1/2])/(4*a^2*p^(1/4)*q^(1/4)*(b - 2*a*Sqrt[p]*Sqrt[q] + Sqrt[b^2 
 - 4*a^2*p*q])*Sqrt[q + p*x^4])
 

3.31.87.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

3.31.87.4 Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.09

method	result	size
risch	\(\frac {\sqrt {p \,x^{4}+q}}{a x}-\frac {b \arctan \left (\frac {a \sqrt {p \,x^{4}+q}}{x \sqrt {a b}}\right )}{a \sqrt {a b}}\)	\(49\)
default	\(\frac {-b \arctan \left (\frac {a \sqrt {p \,x^{4}+q}}{x \sqrt {a b}}\right ) x +\sqrt {p \,x^{4}+q}\, \sqrt {a b}}{a x \sqrt {a b}}\)	\(53\)
pseudoelliptic	\(\frac {-b \arctan \left (\frac {a \sqrt {p \,x^{4}+q}}{x \sqrt {a b}}\right ) x +\sqrt {p \,x^{4}+q}\, \sqrt {a b}}{a x \sqrt {a b}}\)	\(53\)
elliptic	\(\frac {\left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{a x}-\frac {b \sqrt {2}\, \arctan \left (\frac {a \sqrt {p \,x^{4}+q}}{x \sqrt {a b}}\right )}{a \sqrt {a b}}\right ) \sqrt {2}}{2}\)	\(60\)

input

int((p*x^4-q)*(p*x^4+q)^(1/2)/x^2/(a*p*x^4+b*x^2+a*q),x,method=_RETURNVERB 
OSE)
 

output

(p*x^4+q)^(1/2)/a/x-b/a/(a*b)^(1/2)*arctan(a*(p*x^4+q)^(1/2)/x/(a*b)^(1/2) 
)
 

3.31.87.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\text {Timed out} \]

input

integrate((p*x^4-q)*(p*x^4+q)^(1/2)/x^2/(a*p*x^4+b*x^2+a*q),x, algorithm=" 
fricas")
 

output

Timed out
 

3.31.87.6 Sympy [F]

\[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\int \frac {\left (p x^{4} - q\right ) \sqrt {p x^{4} + q}}{x^{2} \left (a p x^{4} + a q + b x^{2}\right )}\, dx \]

input

integrate((p*x**4-q)*(p*x**4+q)**(1/2)/x**2/(a*p*x**4+b*x**2+a*q),x)
 

output

Integral((p*x**4 - q)*sqrt(p*x**4 + q)/(x**2*(a*p*x**4 + a*q + b*x**2)), x 
)
 

3.31.87.7 Maxima [F]

\[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\int { \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )}}{{\left (a p x^{4} + b x^{2} + a q\right )} x^{2}} \,d x } \]

input

integrate((p*x^4-q)*(p*x^4+q)^(1/2)/x^2/(a*p*x^4+b*x^2+a*q),x, algorithm=" 
maxima")
 

output

integrate(sqrt(p*x^4 + q)*(p*x^4 - q)/((a*p*x^4 + b*x^2 + a*q)*x^2), x)
 

3.31.87.8 Giac [F]

\[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\int { \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )}}{{\left (a p x^{4} + b x^{2} + a q\right )} x^{2}} \,d x } \]

input

integrate((p*x^4-q)*(p*x^4+q)^(1/2)/x^2/(a*p*x^4+b*x^2+a*q),x, algorithm=" 
giac")
 

output

integrate(sqrt(p*x^4 + q)*(p*x^4 - q)/((a*p*x^4 + b*x^2 + a*q)*x^2), x)
 

3.31.87.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx=\int -\frac {\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{x^2\,\left (a\,p\,x^4+b\,x^2+a\,q\right )} \,d x \]

input

int(-((q + p*x^4)^(1/2)*(q - p*x^4))/(x^2*(a*q + b*x^2 + a*p*x^4)),x)
 

output

int(-((q + p*x^4)^(1/2)*(q - p*x^4))/(x^2*(a*q + b*x^2 + a*p*x^4)), x)