Optimal. Leaf size=191 \[ \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b^2 c}{a^2 d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+d x^2} \sqrt {a^2 f+b^2 e}}{\sqrt {e+f x^2} \sqrt {a^2 d+b^2 c}}\right )}{\sqrt {a^2 d+b^2 c} \sqrt {a^2 f+b^2 e}} \]
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Rubi [A] time = 0.51, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2113, 538, 537, 571, 93, 208} \[ \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \Pi \left (-\frac {b^2 c}{a^2 d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+d x^2} \sqrt {a^2 f+b^2 e}}{\sqrt {e+f x^2} \sqrt {a^2 d+b^2 c}}\right )}{\sqrt {a^2 d+b^2 c} \sqrt {a^2 f+b^2 e}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rule 537
Rule 538
Rule 571
Rule 2113
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx &=a \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\\ &=-\left (\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{\left (a^2-b^2 x\right ) \sqrt {c+d x} \sqrt {e+f x}} \, dx,x,x^2\right )\right )+\frac {\left (a \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {e+f x^2}} \, dx}{\sqrt {c+d x^2}}\\ &=-\left (b \operatorname {Subst}\left (\int \frac {1}{b^2 c+a^2 d-\left (b^2 e+a^2 f\right ) x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {e+f x^2}}\right )\right )+\frac {\left (a \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{\sqrt {c+d x^2} \sqrt {e+f x^2}}\\ &=-\frac {b \tanh ^{-1}\left (\frac {\sqrt {b^2 e+a^2 f} \sqrt {c+d x^2}}{\sqrt {b^2 c+a^2 d} \sqrt {e+f x^2}}\right )}{\sqrt {b^2 c+a^2 d} \sqrt {b^2 e+a^2 f}}+\frac {\sqrt {-c} \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b^2 c}{a^2 d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {-c}}\right )|\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}\\ \end {align*}
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Mathematica [C] time = 1.62, size = 772, normalized size = 4.04 \[ \frac {2 \sqrt {d} \left (\sqrt {c}+i \sqrt {d} x\right ) \left (\sqrt {e}+i \sqrt {f} x\right ) \sqrt {\frac {\left (\sqrt {d} x+i \sqrt {c}\right ) \left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right )}{\left (\sqrt {d} x-i \sqrt {c}\right ) \left (\sqrt {c} \sqrt {f}+\sqrt {d} \sqrt {e}\right )}} \sqrt {\frac {\sqrt {c} \sqrt {d} \left (\sqrt {f} x+i \sqrt {e}\right )}{\left (\sqrt {d} x-i \sqrt {c}\right ) \left (\sqrt {c} \sqrt {f}-\sqrt {d} \sqrt {e}\right )}} \left (\left (b \sqrt {c}+i a \sqrt {d}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (\sqrt {d} x+i \sqrt {c}\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (\sqrt {d} x-i \sqrt {c}\right )}}\right )|\frac {\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )^2}{\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right )^2}\right )-2 b \sqrt {c} \Pi \left (\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) \left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )}{\left (i \sqrt {d} a+b \sqrt {c}\right ) \left (\sqrt {c} \sqrt {f}-\sqrt {d} \sqrt {e}\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (\sqrt {d} x+i \sqrt {c}\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (\sqrt {d} x-i \sqrt {c}\right )}}\right )|\frac {\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )^2}{\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right )^2}\right )\right )}{\sqrt {c+d x^2} \sqrt {e+f x^2} \left (b \sqrt {c}-i a \sqrt {d}\right ) \left (b \sqrt {c}+i a \sqrt {d}\right ) \left (\sqrt {c} \sqrt {f}-\sqrt {d} \sqrt {e}\right ) \sqrt {\frac {\sqrt {c} \sqrt {d} \left (\sqrt {e}+i \sqrt {f} x\right )}{\left (\sqrt {c}+i \sqrt {d} x\right ) \left (\sqrt {c} \sqrt {f}+\sqrt {d} \sqrt {e}\right )}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 353, normalized size = 1.85 \[ \frac {\left (-\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, a \arctanh \left (\frac {2 a^{2} d f \,x^{2}+b^{2} c f \,x^{2}+b^{2} d e \,x^{2}+a^{2} c f +a^{2} d e +2 b^{2} c e}{2 \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, b^{2}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}\, b \EllipticPi \left (\sqrt {-\frac {d}{c}}\, x , -\frac {b^{2} c}{a^{2} d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )\right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{2 \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} c e}{b^{4}}}\, \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e} {\left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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