Integrand size = 47, antiderivative size = 125 \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=-\frac {4 \arctan \left (\frac {\sqrt {1+k+k^2} x}{1-\sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}-\frac {2 \arctan \left (\frac {(-1+k) x}{1+2 \sqrt {k} x+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{3 (-1+k)} \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.55 (sec) , antiderivative size = 709, normalized size of antiderivative = 5.67, number of steps used = 28, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.277, Rules used = {1976, 6857, 1117, 1738, 1224, 1712, 209, 1261, 738, 210, 1230, 1720, 212} \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=-\frac {\arctan \left (\frac {(1-k) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {2 \arctan \left (\frac {\sqrt {k^2+k+1} x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {k^2+k+1}}+\frac {\arctan \left (\frac {(1-k) \left (k x^2+1\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 (1-k)}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}+\frac {\left (k x^2+1\right ) \sqrt {\frac {k^2 x^4-\left (k^2+1\right ) x^2+1}{\left (k x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(k+1)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}-\frac {(-1)^{2/3} \sqrt {2} \text {arctanh}\left (\frac {-\left (\sqrt [3]{-1} \left (k^2+1\right )+2 k\right ) k x^2+k^2+2 \sqrt [3]{-1} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1+i \sqrt {3}\right ) \left (k^2+k+1\right )}}+\frac {\sqrt [3]{-1} \sqrt {2} \text {arctanh}\left (\frac {-\left (2 k-(-1)^{2/3} \left (k^2+1\right )\right ) k x^2+k^2-2 (-1)^{2/3} k+1}{\sqrt {2} \sqrt {k} \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{3 \sqrt {\left (1-i \sqrt {3}\right ) \left (k^2+k+1\right )}} \]
[In]
[Out]
Rule 209
Rule 210
Rule 212
Rule 738
Rule 1117
Rule 1224
Rule 1230
Rule 1261
Rule 1712
Rule 1720
Rule 1738
Rule 1976
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+k^{3/2} x^3}{\left (1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {2}{\left (1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+k^{3/2} x^3\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx\right )+\int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-2 \int \left (-\frac {1}{3 \left (-1-\sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {1}{3 \left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {1}{3 \left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}+\frac {2}{3} \int \frac {1}{\left (-1-\sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {2}{3} \int \frac {1}{\left (-1+\sqrt [3]{-1} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {2}{3} \int \frac {1}{\left (-1-(-1)^{2/3} \sqrt {k} x\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {2}{3} \int \frac {1}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2}{3} \int \frac {1}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {1}{3} \left (2 \sqrt {k}\right ) \int \frac {x}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \left (2 \sqrt [3]{-1} \sqrt {k}\right ) \int \frac {x}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {1}{3} \left (2 (-1)^{2/3} \sqrt {k}\right ) \int \frac {x}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \int \frac {1+k x^2}{\left (1-k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {2 \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{3 \left (1-\sqrt [3]{-1}\right )}-\frac {1}{3} \left (2 \left (1-\sqrt [3]{-1}\right )\right ) \int \frac {1}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx-\frac {1}{3} \left (2 \left (1+(-1)^{2/3}\right )\right ) \int \frac {1+k x^2}{\left (1-(-1)^{2/3} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\frac {1}{3} \sqrt {k} \text {Subst}\left (\int \frac {1}{(1-k x) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (\sqrt [3]{-1} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{\left (1-(-1)^{2/3} k x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left ((-1)^{2/3} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{-1} k x\right ) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right )+\frac {\left (2 \sqrt [3]{-1} k \left (k+\sqrt [3]{-1} k\right )\right ) \int \frac {1+k x^2}{\left (1+\sqrt [3]{-1} k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{3 \left (k^2-(-1)^{2/3} k^2\right )} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {1+k+k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\left (-1+2 k-k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )-\frac {1}{3} \left (2 \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{8 k^2+4 k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1-2 k+k^2+(1-k)^2 k x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )+\frac {1}{3} \left (2 \sqrt [3]{-1} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{4 k^2-4 \sqrt [3]{-1} k^2+4 (-1)^{2/3} k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1-2 (-1)^{2/3} k+k^2-\left (2 k^2+(-1)^{2/3} k \left (-1-k^2\right )\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )-\frac {1}{3} \left (2 (-1)^{2/3} \sqrt {k}\right ) \text {Subst}\left (\int \frac {1}{4 k^2+4 (-1)^{2/3} k^2-4 \sqrt [3]{-1} k \left (-1-k^2\right )-x^2} \, dx,x,\frac {1+2 \sqrt [3]{-1} k+k^2-\left (2 k^2-\sqrt [3]{-1} k \left (-1-k^2\right )\right ) x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1-k) x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 (1-k)}-\frac {2 \arctan \left (\frac {\sqrt {1+k+k^2} x}{\sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {1+k+k^2}}+\frac {\arctan \left (\frac {(1-k) \left (1+k x^2\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 (1-k)}-\frac {(-1)^{2/3} \sqrt {2} \text {arctanh}\left (\frac {1+2 \sqrt [3]{-1} k+k^2-k \left (2 k+\sqrt [3]{-1} \left (1+k^2\right )\right ) x^2}{\sqrt {2} \sqrt {k} \sqrt {\left (1+i \sqrt {3}\right ) \left (1+k+k^2\right )} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {\left (1+i \sqrt {3}\right ) \left (1+k+k^2\right )}}+\frac {\sqrt [3]{-1} \sqrt {2} \text {arctanh}\left (\frac {1-2 (-1)^{2/3} k+k^2-k \left (2 k-(-1)^{2/3} \left (1+k^2\right )\right ) x^2}{\sqrt {2} \sqrt {k} \sqrt {\left (1-i \sqrt {3}\right ) \left (1+k+k^2\right )} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{3 \sqrt {\left (1-i \sqrt {3}\right ) \left (1+k+k^2\right )}}+\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \left (1-\sqrt [3]{-1}\right ) \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\left (1-\sqrt [3]{-1}\right ) \left (1+k x^2\right ) \sqrt {\frac {1-\left (1+k^2\right ) x^2+k^2 x^4}{\left (1+k x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {k} x\right ),\frac {(1+k)^2}{4 k}\right )}{3 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 16.25 (sec) , antiderivative size = 558, normalized size of antiderivative = 4.46 \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=\frac {(-1-i) \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \left (-\frac {(1-i) \arctan \left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )}{-1+k}+\frac {\left (-1-i \sqrt {3}+k-i \sqrt {3} k\right ) \arctan \left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2+2 i \sqrt {3}}-4 i k+\left (-i+\sqrt {3}\right ) k^2}}+\frac {\left (1-i \sqrt {3}+\left (-1-i \sqrt {3}\right ) k\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {1+k+k^2} \sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2} \sqrt {-1+x^2}}\right )}{\sqrt {1+k+k^2} \sqrt {-\sqrt {2-2 i \sqrt {3}}+4 i k+\left (i+\sqrt {3}\right ) k^2}}\right )+3 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (k,\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (-\sqrt [3]{-1} k,\arcsin (x),k^2\right )-2 \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (\frac {1}{2} i \left (i+\sqrt {3}\right ) k,\arcsin (x),k^2\right )}{3 \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
[In]
[Out]
Time = 1.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.74
method | result | size |
pseudoelliptic | \(-\frac {2 \sqrt {-\left (-1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-k^{2}-k -1}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+k^{\frac {3}{2}} x^{2}+\sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-\sqrt {k}\, x +k \,x^{2}}\right )+\sqrt {-k^{2}-k -1}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )+2 \ln \left (2\right ) \left (\sqrt {-\left (-1+k \right )^{2}}+\frac {\sqrt {-k^{2}-k -1}}{2}\right )}{3 \sqrt {-k^{2}-k -1}\, \sqrt {-\left (-1+k \right )^{2}}}\) | \(217\) |
elliptic | \(\text {Expression too large to display}\) | \(1050\) |
default | \(\text {Expression too large to display}\) | \(1400\) |
[In]
[Out]
none
Time = 0.54 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.70 \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=-\frac {2 \, \sqrt {k^{2} + k + 1} {\left (k - 1\right )} \arctan \left (\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} \sqrt {k^{2} + k + 1} {\left ({\left (k^{2} + 2 \, k + 1\right )} x + {\left (k x^{2} + 1\right )} \sqrt {k}\right )}}{k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} + 4 \, k^{2} + 4 \, k + 1\right )} x^{2} + k}\right ) - {\left (k^{2} + k + 1\right )} \arctan \left (-\frac {\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1} {\left ({\left (k^{3} + k^{2} - k - 1\right )} x - 2 \, {\left ({\left (k^{2} - k\right )} x^{2} + k - 1\right )} \sqrt {k}\right )}}{4 \, k^{3} x^{4} - {\left (k^{4} + 4 \, k^{3} - 2 \, k^{2} + 4 \, k + 1\right )} x^{2} + 4 \, k}\right )}{3 \, {\left (k^{3} - 1\right )}} \]
[In]
[Out]
\[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=\int \frac {\left (\sqrt {k} x - 1\right ) \left (\sqrt {k} x + k x^{2} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (\sqrt {k} x + 1\right ) \left (- \sqrt {k} x + k x^{2} + 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=\int { \frac {k^{\frac {3}{2}} x^{3} - 1}{{\left (k^{\frac {3}{2}} x^{3} + 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {-1+k^{3/2} x^3}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1+k^{3/2} x^3\right )} \, dx=\int \frac {k^{3/2}\,x^3-1}{\left (k^{3/2}\,x^3+1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \]
[In]
[Out]