3.32.30 \(\int \frac {(-1+x^2) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx\)

Optimal. Leaf size=773 \[ \left (c_0 c_3 c_5{}^2-c_1 c_2 c_5{}^2\right ) \text {RootSum}\left [\text {$\#$1}^8 c_2{}^2+\text {$\#$1}^8 c_3{}^2-4 \text {$\#$1}^6 c_2{}^2 c_4-4 \text {$\#$1}^6 c_3{}^2 c_4+6 \text {$\#$1}^4 c_2{}^2 c_4{}^2+6 \text {$\#$1}^4 c_3{}^2 c_4{}^2-2 \text {$\#$1}^4 c_0 c_2 c_5{}^2-2 \text {$\#$1}^4 c_1 c_3 c_5{}^2-4 \text {$\#$1}^2 c_2{}^2 c_4{}^3-4 \text {$\#$1}^2 c_3{}^2 c_4{}^3+4 \text {$\#$1}^2 c_0 c_2 c_4 c_5{}^2+4 \text {$\#$1}^2 c_1 c_3 c_4 c_5{}^2+c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}\right )}{\text {$\#$1}^4 c_2{}^2+\text {$\#$1}^4 c_3{}^2-2 \text {$\#$1}^2 c_2{}^2 c_4-2 \text {$\#$1}^2 c_3{}^2 c_4+c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2}\& \right ]-\frac {(c_1 c_2-c_0 c_3) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{c_3 \left (\frac {c_3 (c_1 x+c_0)}{c_3 x+c_2}-c_1\right )}-\frac {(c_0 c_3-c_1 c_2) c_5 \sqrt {\sqrt {c_3} \left (\sqrt {c_1} c_5-\sqrt {c_3} c_4\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {c_1} \sqrt {c_3} c_5-c_3 c_4} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {c_3} c_4-\sqrt {c_1} c_5}\right )}{2 \sqrt {c_1} c_3{}^{3/2} \left (\sqrt {c_1} c_5-\sqrt {c_3} c_4\right )}-\frac {(c_0 c_3-c_1 c_2) c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} \tan ^{-1}\left (\frac {\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {c_3} c_4+\sqrt {c_1} c_5}\right )}{2 \sqrt {c_1} c_3{}^{3/2} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )} \]

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Rubi [F]  time = 26.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(1 + x^2),x]

[Out]

(ArcTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]
]*C[5]]]*(C[1]*C[2] - C[0]*C[3])*C[5])/(2*Sqrt[C[1]]*C[3]^(5/4)*Sqrt[Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5]]) - (Ar
cTanh[(C[3]^(1/4)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C
[5]]]*(C[1]*C[2] - C[0]*C[3])*C[5])/(2*Sqrt[C[1]]*C[3]^(5/4)*Sqrt[Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]]) + ((C[2]
+ x*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/C[3] + 8*(C[1]*C[2] - C[0]*C[3])*C[5]^2*De
fer[Subst][Defer[Int][x^4/(-(x^8*C[2]^2*(1 + C[3]^2/C[2]^2)) + 4*x^6*C[2]^2*(1 + C[3]^2/C[2]^2)*C[4] + 4*x^2*C
[2]^2*C[4]^3*(1 + (C[3]^2 - ((C[0]*C[2] + C[1]*C[3])*C[5]^2)/C[4]^2)/C[2]^2) - 6*x^4*C[2]^2*C[4]^2*(1 + (3*C[3
]^2 - ((C[0]*C[2] + C[1]*C[3])*C[5]^2)/C[4]^2)/(3*C[2]^2)) - C[2]^2*C[4]^4*(1 + (C[3]^2 + (C[5]^2*(-2*C[0]*C[2
]*C[4]^2 + C[0]^2*C[5]^2 + C[1]*(-2*C[3]*C[4]^2 + C[1]*C[5]^2)))/C[4]^4)/C[2]^2)), x], x, Sqrt[C[4] + Sqrt[(C[
0] + x*C[1])/(C[2] + x*C[3])]*C[5]]] + 8*(C[1]*C[2] - C[0]*C[3])*C[4]*C[5]^2*Defer[Subst][Defer[Int][x^2/(x^8*
C[2]^2*(1 + C[3]^2/C[2]^2) - 4*x^6*C[2]^2*(1 + C[3]^2/C[2]^2)*C[4] - 4*x^2*C[2]^2*C[4]^3*(1 + (C[3]^2 - ((C[0]
*C[2] + C[1]*C[3])*C[5]^2)/C[4]^2)/C[2]^2) + 6*x^4*C[2]^2*C[4]^2*(1 + (3*C[3]^2 - ((C[0]*C[2] + C[1]*C[3])*C[5
]^2)/C[4]^2)/(3*C[2]^2)) + C[2]^2*C[4]^4*(1 + (C[3]^2 + (C[5]^2*(-2*C[0]*C[2]*C[4]^2 + C[0]^2*C[5]^2 + C[1]*(-
2*C[3]*C[4]^2 + C[1]*C[5]^2)))/C[4]^4)/C[2]^2)), x], x, Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]
]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{1+x^2} \, dx &=(2 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x \left (-1+\frac {\left (c_0-x^2 c_2\right ){}^2}{\left (c_1-x^2 c_3\right ){}^2}\right ) \sqrt {c_4+x c_5}}{\left (c_1-x^2 c_3\right ){}^2 \left (1+\frac {\left (c_0-x^2 c_2\right ){}^2}{\left (c_1-x^2 c_3\right ){}^2}\right )} \, dx,x,\sqrt {\frac {c_0+x c_1}{c_2+x c_3}}\right )\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x^2 \left (x^2-c_4\right ) \left (-1+\frac {\left (c_0-\frac {c_2 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right )}{\left (1+\frac {\left (c_0-\frac {c_2 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right ) \left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \frac {x^2 \left (x^2-c_4\right ) \left (-1+\frac {\left (c_0-\frac {c_2 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right )}{\left (1+\frac {\left (c_0-\frac {c_2 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}{\left (c_1-\frac {c_3 \left (x^2-c_4\right ){}^2}{c_5{}^2}\right ){}^2}\right ) \left (c_1-\frac {x^4 c_3}{c_5{}^2}+\frac {2 x^2 c_3 c_4}{c_5{}^2}-\frac {c_3 c_4{}^2}{c_5{}^2}\right ){}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\frac {(4 (c_1 c_2-c_0 c_3)) \operatorname {Subst}\left (\int \left (\frac {c_5{}^4}{c_3 \left (x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2\right )}+\frac {c_5{}^4 \left (x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2\right )}{c_3 \left (x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2}+\frac {2 x^2 \left (-x^2+c_4\right ) c_5{}^4}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )}\right ) \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_5{}^2}\\ &=\left (8 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-x^2+c_4\right )}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )+\frac {\left (4 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_3}+\frac {\left (4 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^2 c_3 c_4-c_3 c_4{}^2+c_1 c_5{}^2}{\left (x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2\right ){}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_3}\\ &=\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3}+\frac {(2 (c_1 c_2-c_0 c_3) c_5) \operatorname {Subst}\left (\int \frac {1}{x^2 c_3-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{\sqrt {c_1} \sqrt {c_3}}-\frac {(2 (c_1 c_2-c_0 c_3) c_5) \operatorname {Subst}\left (\int \frac {1}{x^2 c_3-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{\sqrt {c_1} \sqrt {c_3}}+\left (8 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \left (\frac {x^4}{-x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )+4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4+4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )-6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )-c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )}+\frac {x^2 c_4}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )}\right ) \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )-\frac {(c_1 c_2-c_0 c_3) \operatorname {Subst}\left (\int \frac {6 c_1 c_3 c_5{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}{x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{2 c_1 c_3{}^2 \left (c_3 c_4{}^2-c_1 c_5{}^2\right )}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{\sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{\sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3}+\left (8 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{-x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )+4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4+4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )-6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )-c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )-\frac {\left (3 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 c_3-2 x^2 c_3 c_4+c_3 c_4{}^2-c_1 c_5{}^2} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{c_3}+\left (8 (c_1 c_2-c_0 c_3) c_4 c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{\sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{\sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3}-\frac {(3 (c_1 c_2-c_0 c_3) c_5) \operatorname {Subst}\left (\int \frac {1}{x^2 c_3-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{2 \sqrt {c_1} \sqrt {c_3}}+\frac {(3 (c_1 c_2-c_0 c_3) c_5) \operatorname {Subst}\left (\int \frac {1}{x^2 c_3-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )}{2 \sqrt {c_1} \sqrt {c_3}}+\left (8 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{-x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )+4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4+4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )-6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )-c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )+\left (8 (c_1 c_2-c_0 c_3) c_4 c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{2 \sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4-\sqrt {c_1} c_5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c_3} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}\right ) (c_1 c_2-c_0 c_3) c_5}{2 \sqrt {c_1} c_3{}^{5/4} \sqrt {\sqrt {c_3} c_4+\sqrt {c_1} c_5}}+\frac {(c_2+x c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3}+\left (8 (c_1 c_2-c_0 c_3) c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{-x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )+4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4+4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )-6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )-c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )+\left (8 (c_1 c_2-c_0 c_3) c_4 c_5{}^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{x^8 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right )-4 x^6 c_2{}^2 \left (1+\frac {c_3{}^2}{c_2{}^2}\right ) c_4-4 x^2 c_2{}^2 c_4{}^3 \left (1+\frac {c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{c_2{}^2}\right )+6 x^4 c_2{}^2 c_4{}^2 \left (1+\frac {3 c_3{}^2-\frac {(c_0 c_2+c_1 c_3) c_5{}^2}{c_4{}^2}}{3 c_2{}^2}\right )+c_2{}^2 c_4{}^4 \left (1+\frac {c_3{}^2+\frac {c_5{}^2 \left (-2 c_0 c_2 c_4{}^2+c_0{}^2 c_5{}^2+c_1 \left (-2 c_3 c_4{}^2+c_1 c_5{}^2\right )\right )}{c_4{}^4}}{c_2{}^2}\right )} \, dx,x,\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}\right )\\ \end {align*}

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Mathematica [A]  time = 4.68, size = 656, normalized size = 0.85 \begin {gather*} 4 (c_1 c_2-c_0 c_3) c_5{}^2 \left (-\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8 c_2{}^2+\text {$\#$1}^8 c_3{}^2-4 \text {$\#$1}^6 c_2{}^2 c_4-4 \text {$\#$1}^6 c_3{}^2 c_4+6 \text {$\#$1}^4 c_2{}^2 c_4{}^2+6 \text {$\#$1}^4 c_3{}^2 c_4{}^2-2 \text {$\#$1}^4 c_0 c_2 c_5{}^2-2 \text {$\#$1}^4 c_1 c_3 c_5{}^2-4 \text {$\#$1}^2 c_2{}^2 c_4{}^3-4 \text {$\#$1}^2 c_3{}^2 c_4{}^3+4 \text {$\#$1}^2 c_0 c_2 c_4 c_5{}^2+4 \text {$\#$1}^2 c_1 c_3 c_4 c_5{}^2+c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2\&,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}\right )}{\text {$\#$1}^4 c_2{}^2+\text {$\#$1}^4 c_3{}^2-2 \text {$\#$1}^2 c_2{}^2 c_4-2 \text {$\#$1}^2 c_3{}^2 c_4+c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2}\&\right ]-\frac {(c_3 x+c_2) \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{4 c_3 (c_0 c_3-c_1 c_2) c_5{}^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5}}\right )}{8 \sqrt {c_1} c_3 c_5 \sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c_3} \sqrt {c_5 \sqrt {\frac {c_1 x+c_0}{c_3 x+c_2}}+c_4}}{\sqrt {\sqrt {c_1} \sqrt {c_3} c_5-c_3 c_4}}\right )}{8 \sqrt {c_1} c_3 c_5 \sqrt {\sqrt {c_1} \sqrt {c_3} c_5-c_3 c_4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(1 + x^2),x]

[Out]

4*(C[1]*C[2] - C[0]*C[3])*C[5]^2*(ArcTan[(Sqrt[C[3]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/
Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]]/(8*Sqrt[C[1]]*C[3]*C[5]*Sqrt[-(C[3]*C[4]) - Sqrt[C[1]]*Sqrt[C
[3]]*C[5]]) - ArcTan[(Sqrt[C[3]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/Sqrt[-(C[3]*C[4]) +
Sqrt[C[1]]*Sqrt[C[3]]*C[5]]]/(8*Sqrt[C[1]]*C[3]*C[5]*Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]) - ((C[2]
+ x*C[3])*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(4*C[3]*(-(C[1]*C[2]) + C[0]*C[3])*C[5]^2)
- RootSum[C[2]^2*C[4]^4 + C[3]^2*C[4]^4 - 2*C[0]*C[2]*C[4]^2*C[5]^2 - 2*C[1]*C[3]*C[4]^2*C[5]^2 + C[0]^2*C[5]
^4 + C[1]^2*C[5]^4 - 4*C[2]^2*C[4]^3*#1^2 - 4*C[3]^2*C[4]^3*#1^2 + 4*C[0]*C[2]*C[4]*C[5]^2*#1^2 + 4*C[1]*C[3]*
C[4]*C[5]^2*#1^2 + 6*C[2]^2*C[4]^2*#1^4 + 6*C[3]^2*C[4]^2*#1^4 - 2*C[0]*C[2]*C[5]^2*#1^4 - 2*C[1]*C[3]*C[5]^2*
#1^4 - 4*C[2]^2*C[4]*#1^6 - 4*C[3]^2*C[4]*#1^6 + C[2]^2*#1^8 + C[3]^2*#1^8 & , (Log[Sqrt[C[4] + Sqrt[(C[0] + x
*C[1])/(C[2] + x*C[3])]*C[5]] - #1]*#1)/(C[2]^2*C[4]^2 + C[3]^2*C[4]^2 - C[0]*C[2]*C[5]^2 - C[1]*C[3]*C[5]^2 -
2*C[2]^2*C[4]*#1^2 - 2*C[3]^2*C[4]*#1^2 + C[2]^2*#1^4 + C[3]^2*#1^4) & ]/4)

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IntegrateAlgebraic [A]  time = 6.32, size = 773, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {-c_3 c_4+\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {c_3} c_4-\sqrt {c_1} c_5}\right ) (-c_1 c_2+c_0 c_3) c_5 \sqrt {\sqrt {c_3} \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}}{2 \sqrt {c_1} c_3{}^{3/2} \left (-\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {-c_3 c_4-\sqrt {c_1} \sqrt {c_3} c_5} \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{\sqrt {c_3} c_4+\sqrt {c_1} c_5}\right ) (-c_1 c_2+c_0 c_3) c_5 \sqrt {-\sqrt {c_3} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}}{2 \sqrt {c_1} c_3{}^{3/2} \left (\sqrt {c_3} c_4+\sqrt {c_1} c_5\right )}-\frac {(c_1 c_2-c_0 c_3) \sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}}{c_3 \left (-c_1+\frac {(c_0+x c_1) c_3}{c_2+x c_3}\right )}+\left (-c_1 c_2 c_5{}^2+c_0 c_3 c_5{}^2\right ) \text {RootSum}\left [c_2{}^2 c_4{}^4+c_3{}^2 c_4{}^4-2 c_0 c_2 c_4{}^2 c_5{}^2-2 c_1 c_3 c_4{}^2 c_5{}^2+c_0{}^2 c_5{}^4+c_1{}^2 c_5{}^4-4 c_2{}^2 c_4{}^3 \text {$\#$1}^2-4 c_3{}^2 c_4{}^3 \text {$\#$1}^2+4 c_0 c_2 c_4 c_5{}^2 \text {$\#$1}^2+4 c_1 c_3 c_4 c_5{}^2 \text {$\#$1}^2+6 c_2{}^2 c_4{}^2 \text {$\#$1}^4+6 c_3{}^2 c_4{}^2 \text {$\#$1}^4-2 c_0 c_2 c_5{}^2 \text {$\#$1}^4-2 c_1 c_3 c_5{}^2 \text {$\#$1}^4-4 c_2{}^2 c_4 \text {$\#$1}^6-4 c_3{}^2 c_4 \text {$\#$1}^6+c_2{}^2 \text {$\#$1}^8+c_3{}^2 \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c_4+\sqrt {\frac {c_0+x c_1}{c_2+x c_3}} c_5}-\text {$\#$1}\right ) \text {$\#$1}}{c_2{}^2 c_4{}^2+c_3{}^2 c_4{}^2-c_0 c_2 c_5{}^2-c_1 c_3 c_5{}^2-2 c_2{}^2 c_4 \text {$\#$1}^2-2 c_3{}^2 c_4 \text {$\#$1}^2+c_2{}^2 \text {$\#$1}^4+c_3{}^2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^2)*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(1 + x^2),x]

[Out]

-1/2*(ArcTan[(Sqrt[-(C[3]*C[4]) + Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])
]*C[5]])/(Sqrt[C[3]]*C[4] - Sqrt[C[1]]*C[5])]*(-(C[1]*C[2]) + C[0]*C[3])*C[5]*Sqrt[Sqrt[C[3]]*(-(Sqrt[C[3]]*C[
4]) + Sqrt[C[1]]*C[5])])/(Sqrt[C[1]]*C[3]^(3/2)*(-(Sqrt[C[3]]*C[4]) + Sqrt[C[1]]*C[5])) - (ArcTan[(Sqrt[-(C[3]
*C[4]) - Sqrt[C[1]]*Sqrt[C[3]]*C[5]]*Sqrt[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(Sqrt[C[3]]*C[4]
+ Sqrt[C[1]]*C[5])]*(-(C[1]*C[2]) + C[0]*C[3])*C[5]*Sqrt[-(Sqrt[C[3]]*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5]))])/
(2*Sqrt[C[1]]*C[3]^(3/2)*(Sqrt[C[3]]*C[4] + Sqrt[C[1]]*C[5])) - ((C[1]*C[2] - C[0]*C[3])*Sqrt[C[4] + Sqrt[(C[0
] + x*C[1])/(C[2] + x*C[3])]*C[5]])/(C[3]*(-C[1] + ((C[0] + x*C[1])*C[3])/(C[2] + x*C[3]))) + (-(C[1]*C[2]*C[5
]^2) + C[0]*C[3]*C[5]^2)*RootSum[C[2]^2*C[4]^4 + C[3]^2*C[4]^4 - 2*C[0]*C[2]*C[4]^2*C[5]^2 - 2*C[1]*C[3]*C[4]^
2*C[5]^2 + C[0]^2*C[5]^4 + C[1]^2*C[5]^4 - 4*C[2]^2*C[4]^3*#1^2 - 4*C[3]^2*C[4]^3*#1^2 + 4*C[0]*C[2]*C[4]*C[5]
^2*#1^2 + 4*C[1]*C[3]*C[4]*C[5]^2*#1^2 + 6*C[2]^2*C[4]^2*#1^4 + 6*C[3]^2*C[4]^2*#1^4 - 2*C[0]*C[2]*C[5]^2*#1^4
- 2*C[1]*C[3]*C[5]^2*#1^4 - 4*C[2]^2*C[4]*#1^6 - 4*C[3]^2*C[4]*#1^6 + C[2]^2*#1^8 + C[3]^2*#1^8 & , (Log[Sqrt
[C[4] + Sqrt[(C[0] + x*C[1])/(C[2] + x*C[3])]*C[5]] - #1]*#1)/(C[2]^2*C[4]^2 + C[3]^2*C[4]^2 - C[0]*C[2]*C[5]^
2 - C[1]*C[3]*C[5]^2 - 2*C[2]^2*C[4]*#1^2 - 2*C[3]^2*C[4]*#1^2 + C[2]^2*#1^4 + C[3]^2*#1^4) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right ) \sqrt {\textit {\_C4} +\sqrt {\frac {\textit {\_C1} x +\textit {\_C0}}{\textit {\_C3} x +\textit {\_C2}}}\, \textit {\_C5}}}{x^{2}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)

[Out]

int((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} - 1\right )} \sqrt {\_{C_{5}} \sqrt {\frac {\_{C_{1}} x + \_{C_{0}}}{\_{C_{3}} x + \_{C_{2}}}} + \_{C_{4}}}}{x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))^(1/2)*_C5)^(1/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)*sqrt(_C5*sqrt((_C1*x + _C0)/(_C3*x + _C2)) + _C4)/(x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {_{\mathrm {C4}}+_{\mathrm {C5}}\,\sqrt {\frac {_{\mathrm {C0}}+_{\mathrm {C1}}\,x}{_{\mathrm {C2}}+_{\mathrm {C3}}\,x}}}}{x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 1)*(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2))/(x^2 + 1),x)

[Out]

int(((x^2 - 1)*(_C4 + _C5*((_C0 + _C1*x)/(_C2 + _C3*x))^(1/2))^(1/2))/(x^2 + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)*(_C4+((_C1*x+_C0)/(_C3*x+_C2))**(1/2)*_C5)**(1/2)/(x**2+1),x)

[Out]

Timed out

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