Optimal. Leaf size=76 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x} \sqrt {a k^2+a+b}}{\sqrt {a} (x-1) \left (k^2 x-1\right )}\right )}{\sqrt {a} \sqrt {a k^2+a+b}} \]
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Rubi [C] time = 3.22, antiderivative size = 299, normalized size of antiderivative = 3.93, number of steps used = 16, number of rules used = 9, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {6718, 6688, 6728, 714, 115, 934, 12, 168, 537} \begin {gather*} \frac {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b-\sqrt {b^2-4 a^2 k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b+\sqrt {b^2-4 a^2 k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {x-x^2} \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 115
Rule 168
Rule 537
Rule 714
Rule 934
Rule 6688
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \int \frac {-1+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (a+b x+a k^2 x^2\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (a+b x+a k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {-1+k^2 x^2}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a k^2 x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {1}{a \sqrt {1-k^2 x} \sqrt {x-x^2}}-\frac {2 a+b x}{a \sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a k^2 x^2\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \sqrt {x-x^2}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {2 a+b x}{\sqrt {1-k^2 x} \sqrt {x-x^2} \left (a+b x+a k^2 x^2\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {b-\sqrt {b^2-4 a^2 k^2}}{\sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}}+\frac {b+\sqrt {b^2-4 a^2 k^2}}{\sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}}\right ) \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right ) \sqrt {x-x^2}} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\sqrt {2} \left (b-\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\sqrt {2} \left (b+\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2} \sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (\left (b-\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (b-\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}-\frac {\left (\left (b+\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {2-2 x} \sqrt {-x} \sqrt {1-k^2 x} \left (b+\sqrt {b^2-4 a^2 k^2}+2 a k^2 x\right )} \, dx}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (b-\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (b-\sqrt {b^2-4 a^2 k^2}-2 a k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {\left (2 \left (b+\sqrt {b^2-4 a^2 k^2}\right ) \sqrt {2-2 x} \sqrt {1-x} \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+2 x^2} \sqrt {1+k^2 x^2} \left (b+\sqrt {b^2-4 a^2 k^2}-2 a k^2 x^2\right )} \, dx,x,\sqrt {-x}\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ &=\frac {2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{a \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b-\sqrt {b^2-4 a^2 k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}+\frac {2 (1-x) \sqrt {-x} \sqrt {x} \sqrt {1-k^2 x} \Pi \left (-\frac {2 a}{b+\sqrt {b^2-4 a^2 k^2}};\sin ^{-1}\left (\sqrt {-k^2} \sqrt {-x}\right )|\frac {1}{k^2}\right )}{a \sqrt {-k^2} \sqrt {(1-x) x \left (1-k^2 x\right )} \sqrt {x-x^2}}\\ \end {align*}
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Mathematica [C] time = 6.04, size = 345, normalized size = 4.54 \begin {gather*} \frac {i \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \left (2 a \left (k^2-1\right ) \sqrt {b^2-4 a^2 k^2} F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (b \sqrt {b^2-4 a^2 k^2}+2 a \left (\sqrt {b^2-4 a^2 k^2}-2 a k^2\right )+b^2\right ) \Pi \left (\frac {2 \left (a k^2+a+b\right )}{2 a k^2+b-\sqrt {b^2-4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (b \sqrt {b^2-4 a^2 k^2}+2 a \left (\sqrt {b^2-4 a^2 k^2}+2 a k^2\right )-b^2\right ) \Pi \left (\frac {2 \left (a k^2+a+b\right )}{2 a k^2+b+\sqrt {b^2-4 a^2 k^2}};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{a \sqrt {(x-1) x \left (k^2 x-1\right )} \left (a k^2+a+b\right ) \sqrt {b^2-4 a^2 k^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 76, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x} \sqrt {a k^2+a+b}}{\sqrt {a} (x-1) \left (k^2 x-1\right )}\right )}{\sqrt {a} \sqrt {a k^2+a+b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 413, normalized size = 5.43 \begin {gather*} \left [-\frac {\sqrt {-a^{2} k^{2} - a^{2} - a b} \log \left (\frac {a^{2} k^{4} x^{4} - 2 \, {\left (4 \, a^{2} k^{4} + {\left (4 \, a^{2} + 3 \, a b\right )} k^{2}\right )} x^{3} + {\left (8 \, a^{2} k^{4} + 2 \, {\left (9 \, a^{2} + 4 \, a b\right )} k^{2} + 8 \, a^{2} + 8 \, a b + b^{2}\right )} x^{2} - 4 \, {\left (a k^{2} x^{2} - {\left (2 \, a k^{2} + 2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {-a^{2} k^{2} - a^{2} - a b} + a^{2} - 2 \, {\left (4 \, a^{2} k^{2} + 4 \, a^{2} + 3 \, a b\right )} x}{a^{2} k^{4} x^{4} + 2 \, a b k^{2} x^{3} + 2 \, a b x + {\left (2 \, a^{2} k^{2} + b^{2}\right )} x^{2} + a^{2}}\right )}{2 \, {\left (a^{2} k^{2} + a^{2} + a b\right )}}, \frac {\arctan \left (\frac {{\left (a k^{2} x^{2} - {\left (2 \, a k^{2} + 2 \, a + b\right )} x + a\right )} \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} \sqrt {a^{2} k^{2} + a^{2} + a b}}{2 \, {\left ({\left (a^{2} k^{4} + {\left (a^{2} + a b\right )} k^{2}\right )} x^{3} - {\left (a^{2} k^{4} + {\left (2 \, a^{2} + a b\right )} k^{2} + a^{2} + a b\right )} x^{2} + {\left (a^{2} k^{2} + a^{2} + a b\right )} x\right )}}\right )}{\sqrt {a^{2} k^{2} + a^{2} + a b}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{2} - 1}{{\left (a k^{2} x^{2} + b x + a\right )} \sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 1178, normalized size = 15.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.76, size = 90, normalized size = 1.18 \begin {gather*} \frac {\ln \left (\frac {a-2\,a\,x-b\,x-2\,a\,k^2\,x+a\,k^2\,x^2+\sqrt {a\,\left (a\,k^2+a+b\right )}\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}\,2{}\mathrm {i}}{a\,k^2\,x^2+b\,x+a}\right )\,1{}\mathrm {i}}{\sqrt {a^2\,k^2+a^2+b\,a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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