Optimal. Leaf size=1077 \[ -\frac {\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (f x^2+d\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}} \left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right )|\frac {1}{2} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)}}+1\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {c x^2+b x+a} \sqrt {f x^2+d} \sqrt {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}} \]
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Rubi [A] time = 3.10, antiderivative size = 1077, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {993, 936, 1103} \[ -\frac {\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (f x^2+d\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}} \left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right )|\frac {1}{2} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {d b^2+\sqrt {b^2-4 a c} d b-2 a (c d-a f)}}+1\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 d c^2-2 a f c+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {c x^2+b x+a} \sqrt {f x^2+d} \sqrt {\frac {\left (4 d c^2+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (4 f a^2+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}} \]
Antiderivative was successfully verified.
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Rule 936
Rule 993
Rule 1103
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+f x^2}} \, dx &=\frac {\left (\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}\right ) \int \frac {1}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {d+f x^2}} \, dx}{\sqrt {a+b x+c x^2}}\\ &=-\frac {\left (2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {\left (4 c \left (b+\sqrt {b^2-4 a c}\right ) d+4 a \left (b+\sqrt {b^2-4 a c}\right ) f\right ) x^2}{\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f}+\frac {\left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) x^4}{\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f}}} \, dx,x,\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt {a+b x+c x^2} \sqrt {d+f x^2}}\\ &=-\frac {\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}} \left (1+\frac {\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}{\left (1+\frac {\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )|\frac {1}{2} \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (c d+a f)}{\sqrt {2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {b^2 d+b \sqrt {b^2-4 a c} d-2 a (c d-a f)}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-2 a c f+b \left (b+\sqrt {b^2-4 a c}\right ) f} \sqrt {a+b x+c x^2} \sqrt {d+f x^2} \sqrt {1-\frac {4 \left (b+\sqrt {b^2-4 a c}\right ) (c d+a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 1.64, size = 600, normalized size = 0.56 \[ -\frac {2 \sqrt {2} \left (\sqrt {f} x-i \sqrt {d}\right ) \left (\sqrt {b^2-4 a c}-b-2 c x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (\sqrt {f} x+i \sqrt {d}\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (\sqrt {f} \left (\sqrt {b^2-4 a c}+b\right )-2 i c \sqrt {d}\right )}} \sqrt {\frac {c \left (-i \sqrt {d} \left (\sqrt {b^2-4 a c}+2 c x\right )+\sqrt {f} \left (x \sqrt {b^2-4 a c}-2 a\right )+b \left (-\sqrt {f} x-i \sqrt {d}\right )\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (\sqrt {f} \left (\sqrt {b^2-4 a c}+b\right )+2 i c \sqrt {d}\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\left (\sqrt {b^2-4 a c}-b\right ) \sqrt {f}-2 i c \sqrt {d}\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 i \sqrt {d} c+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {f}\right ) \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}}\right )|\frac {c d-i \sqrt {b^2-4 a c} \sqrt {f} \sqrt {d}+a f}{c d+i \sqrt {b^2-4 a c} \sqrt {f} \sqrt {d}+a f}\right )}{\sqrt {d+f x^2} \sqrt {a+x (b+c x)} \left (\sqrt {f} \left (\sqrt {b^2-4 a c}-b\right )-2 i c \sqrt {d}\right ) \sqrt {\frac {i c \sqrt {b^2-4 a c} \left (\sqrt {d}+i \sqrt {f} x\right )}{\left (\sqrt {b^2-4 a c}-b-2 c x\right ) \left (\sqrt {f} \left (\sqrt {b^2-4 a c}+b\right )+2 i c \sqrt {d}\right )}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}}{c f x^{4} + b f x^{3} + b d x + {\left (c d + a f\right )} x^{2} + a d}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 661, normalized size = 0.61 \[ \frac {4 \left (b \,f^{2} x^{2}-4 c d f x +2 \sqrt {-d f}\, c f \,x^{2}+\sqrt {-4 a c +b^{2}}\, f^{2} x^{2}-b d f +2 \sqrt {-d f}\, b f x -2 \sqrt {-d f}\, c d -\sqrt {-4 a c +b^{2}}\, d f +2 \sqrt {-4 a c +b^{2}}\, \sqrt {-d f}\, f x \right ) \sqrt {\frac {\sqrt {-d f}\, \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right ) f}{\left (b f +2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (f x +\sqrt {-d f}\right )}}\, \sqrt {\frac {\sqrt {-d f}\, \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) f}{\left (-b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (f x +\sqrt {-d f}\right )}}\, \sqrt {-\frac {\left (b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (-f x +\sqrt {-d f}\right )}{\left (b f +2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (f x +\sqrt {-d f}\right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+d}\, \EllipticF \left (\sqrt {-\frac {\left (b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (-f x +\sqrt {-d f}\right )}{\left (b f +2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (f x +\sqrt {-d f}\right )}}, \sqrt {\frac {\left (-b f +2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (b f +2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right )}{\left (-b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \left (b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right )}}\right )}{\sqrt {\frac {\left (-f x +\sqrt {-d f}\right ) \left (f x +\sqrt {-d f}\right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{c f}}\, \sqrt {-d f}\, \left (b f -2 \sqrt {-d f}\, c +\sqrt {-4 a c +b^{2}}\, f \right ) \sqrt {c f \,x^{4}+b f \,x^{3}+a f \,x^{2}+c d \,x^{2}+b d x +a d}} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {f x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {f\,x^2+d}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {1}{\sqrt {d + f x^{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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