3.32.19 \(\int \frac {x^2 (x^2 c_3-c_4) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{(-x+x^2 c_3+c_4) (x+x^2 c_3+c_4) (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\)

Optimal. Leaf size=685 \[ \frac {1}{4} (c_0-c_1) \text {RootSum}\left [\text {$\#$1}^8+\text {$\#$1}^8 c_1{}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^4 c_0 c_1+1+c_0{}^2\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}\right )}{\text {$\#$1}^4+\text {$\#$1}^4 c_1{}^2-1-c_0 c_1}\& \right ]+\frac {\sqrt [4]{1+c_0} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{1+c_0}}{\sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\sqrt [4]{-1-c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\sqrt {2} \sqrt [4]{1+c_0}}}{\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}\right )}{2 \sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\sqrt [4]{-1+c_0} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{-1+c_0}}{\sqrt {2} \sqrt [4]{1-c_1}}-\frac {\sqrt [4]{1-c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\sqrt {2} \sqrt [4]{-1+c_0}}}{\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}\right )}{2 \sqrt {2} \sqrt [4]{1-c_1}}-\frac {\sqrt [4]{1+c_0} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1+c_0} \sqrt [4]{-1-c_1} \sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\sqrt {-1-c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}+\sqrt {1+c_0}}\right )}{2 \sqrt {2} \sqrt [4]{-1-c_1}}+\frac {\sqrt [4]{-1+c_0} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+c_0} \sqrt [4]{1-c_1} \sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\sqrt {1-c_1} \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}+\sqrt {-1+c_0}}\right )}{2 \sqrt {2} \sqrt [4]{1-c_1}} \]

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Rubi [F]  time = 30.48, 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 {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx &=\int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^4 c_3{}^2+c_4{}^2+x^2 (1+2 c_3 c_4)\right )} \, dx\\ &=\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{x c_0+x^2 c_3+c_4}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \sqrt [4]{x c_1+x^2 c_3+c_4} \left (x^4 c_3{}^2+c_4{}^2+x^2 (1+2 c_3 c_4)\right )} \, dx}{\sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \left (\frac {(-1+2 x c_3) \sqrt [4]{x c_0+x^2 c_3+c_4}}{4 \left (-x+x^2 c_3+c_4\right ) \sqrt [4]{x c_1+x^2 c_3+c_4}}+\frac {(-1-2 x c_3) \sqrt [4]{x c_0+x^2 c_3+c_4}}{4 \left (x+x^2 c_3+c_4\right ) \sqrt [4]{x c_1+x^2 c_3+c_4}}+\frac {\left (-x^2 c_3+c_4\right ) \sqrt [4]{x c_0+x^2 c_3+c_4}}{2 \sqrt [4]{x c_1+x^2 c_3+c_4} \left (x^4 c_3{}^2+c_4{}^2+x^2 (1+2 c_3 c_4)\right )}\right ) \, dx}{\sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {(-1+2 x c_3) \sqrt [4]{x c_0+x^2 c_3+c_4}}{\left (-x+x^2 c_3+c_4\right ) \sqrt [4]{x c_1+x^2 c_3+c_4}} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {(-1-2 x c_3) \sqrt [4]{x c_0+x^2 c_3+c_4}}{\left (x+x^2 c_3+c_4\right ) \sqrt [4]{x c_1+x^2 c_3+c_4}} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\left (-x^2 c_3+c_4\right ) \sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (x^4 c_3{}^2+c_4{}^2+x^2 (1+2 c_3 c_4)\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \left (\frac {2 c_3 \sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}+\frac {2 c_3 \sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \left (-\frac {2 c_3 \sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}-\frac {2 c_3 \sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \left (\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \left (-c_3-c_3 \sqrt {1+4 c_3 c_4}\right )}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x^2 c_3{}^2+2 c_3 c_4+\sqrt {1+4 c_3 c_4}\right )}+\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \left (-c_3+c_3 \sqrt {1+4 c_3 c_4}\right )}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x^2 c_3{}^2+2 c_3 c_4-\sqrt {1+4 c_3 c_4}\right )}\right ) \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1-\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x^2 c_3{}^2+2 c_3 c_4-\sqrt {1+4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x^2 c_3{}^2+2 c_3 c_4+\sqrt {1+4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1-\sqrt {1+4 c_3 c_4}\right )\right ) \int \left (\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}}{2 \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 c_3 c_4-\sqrt {1+4 c_3 c_4}\right ) \left (-\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}\right )}+\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}}{2 \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 c_3 c_4-\sqrt {1+4 c_3 c_4}\right ) \left (\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}\right )}\right ) \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+\sqrt {1+4 c_3 c_4}\right )\right ) \int \left (\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}}{2 \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 c_3 c_4+\sqrt {1+4 c_3 c_4}\right ) \left (-\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}\right )}+\frac {\sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}}{2 \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 c_3 c_4+\sqrt {1+4 c_3 c_4}\right ) \left (\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}\right )}\right ) \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}\\ &=\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}-\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4}\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt [4]{x c_0+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}\right )} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1+\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}\right )} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4-\sqrt {1+4 c_3 c_4}}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1-\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (-\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}\right )} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}}+\frac {\left (c_3 \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt [4]{x c_1+x^2 c_3+c_4} \left (1-\sqrt {1+4 c_3 c_4}\right )\right ) \int \frac {\sqrt [4]{x c_0+x^2 c_3+c_4}}{\sqrt [4]{x c_1+x^2 c_3+c_4} \left (\sqrt {2} x c_3+\sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}\right )} \, dx}{4 \sqrt [4]{x c_0+x^2 c_3+c_4} \sqrt {-1-2 c_3 c_4+\sqrt {1+4 c_3 c_4}}}\\ \end {align*}

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 9.52, size = 685, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1+c_0} \sqrt [4]{-1-c_1} \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {1+c_0}-\sqrt {-1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{1+c_0}}{2 \sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{1+c_0}}{\sqrt {2} \sqrt [4]{-1-c_1}}+\frac {\sqrt [4]{-1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {2} \sqrt [4]{1+c_0}}}{\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{1+c_0}}{2 \sqrt {2} \sqrt [4]{-1-c_1}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+c_0} \sqrt [4]{1-c_1} \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {-1+c_0}-\sqrt {1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{-1+c_0}}{2 \sqrt {2} \sqrt [4]{1-c_1}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{-1+c_0}}{\sqrt {2} \sqrt [4]{1-c_1}}+\frac {\sqrt [4]{1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {2} \sqrt [4]{-1+c_0}}}{\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{-1+c_0}}{2 \sqrt {2} \sqrt [4]{1-c_1}}+\frac {1}{4} (c_0-c_1) \text {RootSum}\left [1+c_0{}^2-2 \text {$\#$1}^4-2 c_0 c_1 \text {$\#$1}^4+\text {$\#$1}^8+c_1{}^2 \text {$\#$1}^8\&,\frac {\log \left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}}{-1-c_0 c_1+\text {$\#$1}^4+c_1{}^2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(ArcTan[(Sqrt[2]*(1 + C[0])^(1/4)*(-1 - C[1])^(1/4)*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]
))^(1/4))/(Sqrt[1 + C[0]] - Sqrt[-1 - C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])]*(1 +
C[0])^(1/4))/(Sqrt[2]*(-1 - C[1])^(1/4)) - (ArcTanh[((1 + C[0])^(1/4)/(Sqrt[2]*(-1 - C[1])^(1/4)) + ((-1 - C[
1])^(1/4)*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(Sqrt[2]*(1 + C[0])^(1/4)))/((x*C[0] +
x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4)]*(1 + C[0])^(1/4))/(2*Sqrt[2]*(-1 - C[1])^(1/4)) + (ArcTan[
(Sqrt[2]*(-1 + C[0])^(1/4)*(1 - C[1])^(1/4)*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4))/(Sq
rt[-1 + C[0]] - Sqrt[1 - C[1]]*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])]*(-1 + C[0])^(1/4)
)/(2*Sqrt[2]*(1 - C[1])^(1/4)) + (ArcTanh[((-1 + C[0])^(1/4)/(Sqrt[2]*(1 - C[1])^(1/4)) + ((1 - C[1])^(1/4)*Sq
rt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])])/(Sqrt[2]*(-1 + C[0])^(1/4)))/((x*C[0] + x^2*C[3] +
C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4)]*(-1 + C[0])^(1/4))/(2*Sqrt[2]*(1 - C[1])^(1/4)) + ((C[0] - C[1])*Root
Sum[1 + C[0]^2 - 2*#1^4 - 2*C[0]*C[1]*#1^4 + #1^8 + C[1]^2*#1^8 & , (Log[((x*C[0] + x^2*C[3] + C[4])/(x*C[1] +
x^2*C[3] + C[4]))^(1/4) - #1]*#1)/(-1 - C[0]*C[1] + #1^4 + C[1]^2*#1^4) & ])/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*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_
C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_
C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, algorithm="giac")

[Out]

integrate((_C3*x^2 - _C4)*x^2*((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))^(1/4)/((_C3^2*x^4 + 2*_C3*_C4*
x^2 + _C4^2 + x^2)*(_C3*x^2 + _C4 + x)*(_C3*x^2 + _C4 - x)), x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (\textit {\_C3} \,x^{2}-\textit {\_C4} \right ) \left (\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}\right )^{\frac {1}{4}}}{\left (\textit {\_C3} \,x^{2}+\textit {\_C4} -x \right ) \left (\textit {\_C3} \,x^{2}+\textit {\_C4} +x \right ) \left (\textit {\_C3}^{2} x^{4}+2 \textit {\_C3} \textit {\_C4} \,x^{2}+\textit {\_C4}^{2}+x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x
^4+2*_C3*_C4*x^2+_C4^2+x^2),x)

[Out]

int(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x
^4+2*_C3*_C4*x^2+_C4^2+x^2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_
C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, algorithm="maxima")

[Out]

integrate((_C3*x^2 - _C4)*x^2*((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))^(1/4)/((_C3^2*x^4 + 2*_C3*_C4*
x^2 + _C4^2 + x^2)*(_C3*x^2 + _C4 + x)*(_C3*x^2 + _C4 - x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^2\,\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )\,{\left (\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}\right )}^{1/4}}{\left (_{\mathrm {C3}}\,x^2-x+_{\mathrm {C4}}\right )\,\left (_{\mathrm {C3}}\,x^2+x+_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2+x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(_C4 - _C3*x^2)*((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/4))/((_C4 - x + _C3*x^2)*(_C4 +
x + _C3*x^2)*(_C4^2 + x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)),x)

[Out]

int(-(x^2*(_C4 - _C3*x^2)*((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/4))/((_C4 - x + _C3*x^2)*(_C4 +
x + _C3*x^2)*(_C4^2 + x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)), 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*(_C3*x**2-_C4)*((_C3*x**2+_C0*x+_C4)/(_C3*x**2+_C1*x+_C4))**(1/4)/(_C3*x**2+_C4-x)/(_C3*x**2+_C
4+x)/(_C3**2*x**4+2*_C3*_C4*x**2+_C4**2+x**2),x)

[Out]

Timed out

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