Optimal. Leaf size=1077 \[ \text{result too large to display} \]
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Rubi [A] time = 3.09899, antiderivative size = 1077, normalized size of antiderivative = 1., 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 993
Rule 936
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*}
Mathematica [C] time = 1.52761, size = 600, normalized size = 0.56 \[ -\frac{2 \sqrt{2} \left (\sqrt{f} x-i \sqrt{d}\right ) \left (\sqrt{b^2-4 a c}-b-2 c x\right ) \sqrt{-\frac{c \sqrt{b^2-4 a c} \left (\sqrt{f} x+i \sqrt{d}\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (\sqrt{f} \left (\sqrt{b^2-4 a c}+b\right )-2 i c \sqrt{d}\right )}} \sqrt{\frac{c \left (-i \sqrt{d} \left (\sqrt{b^2-4 a c}+2 c x\right )+\sqrt{f} \left (x \sqrt{b^2-4 a c}-2 a\right )+b \left (-\sqrt{f} x-i \sqrt{d}\right )\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (\sqrt{f} \left (\sqrt{b^2-4 a c}+b\right )+2 i c \sqrt{d}\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\sqrt{f} \left (\sqrt{b^2-4 a c}-b\right )-2 i c \sqrt{d}\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (\sqrt{f} \left (\sqrt{b^2-4 a c}+b\right )+2 i c \sqrt{d}\right )}}\right ),\frac{-i \sqrt{d} \sqrt{f} \sqrt{b^2-4 a c}+a f+c d}{i \sqrt{d} \sqrt{f} \sqrt{b^2-4 a c}+a f+c d}\right )}{\sqrt{d+f x^2} \sqrt{a+x (b+c x)} \left (\sqrt{f} \left (\sqrt{b^2-4 a c}-b\right )-2 i c \sqrt{d}\right ) \sqrt{\frac{i c \sqrt{b^2-4 a c} \left (\sqrt{d}+i \sqrt{f} x\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (\sqrt{f} \left (\sqrt{b^2-4 a c}+b\right )+2 i c \sqrt{d}\right )}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.618, size = 661, normalized size = 0.6 \begin{align*} 4\,{\frac{ \left ( b{f}^{2}{x}^{2}+2\,{x}^{2}cf\sqrt{-df}+\sqrt{-4\,ac+{b}^{2}}{f}^{2}{x}^{2}+2\,xbf\sqrt{-df}-4\,cxfd+2\,xf\sqrt{-4\,ac+{b}^{2}}\sqrt{-df}-bdf-2\,cd\sqrt{-df}-\sqrt{-4\,ac+{b}^{2}}df \right ) \sqrt{c{x}^{2}+bx+a}\sqrt{f{x}^{2}+d}}{\sqrt{-df} \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c+bf \right ) \sqrt{cf{x}^{4}+bf{x}^{3}+af{x}^{2}+cd{x}^{2}+bdx+ad}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c+bf \right ) \left ( -fx+\sqrt{-df} \right ) }{ \left ( f\sqrt{-4\,ac+{b}^{2}}+2\,\sqrt{-df}c+bf \right ) \left ( fx+\sqrt{-df} \right ) }}},\sqrt{{\frac{ \left ( f\sqrt{-4\,ac+{b}^{2}}+2\,\sqrt{-df}c-bf \right ) \left ( f\sqrt{-4\,ac+{b}^{2}}+2\,\sqrt{-df}c+bf \right ) }{ \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c-bf \right ) \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c+bf \right ) }}} \right ) \sqrt{{\frac{\sqrt{-df} \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) f}{ \left ( f\sqrt{-4\,ac+{b}^{2}}+2\,\sqrt{-df}c+bf \right ) \left ( fx+\sqrt{-df} \right ) }}}\sqrt{{\frac{\sqrt{-df} \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) f}{ \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c-bf \right ) \left ( fx+\sqrt{-df} \right ) }}}\sqrt{-{\frac{ \left ( f\sqrt{-4\,ac+{b}^{2}}-2\,\sqrt{-df}c+bf \right ) \left ( -fx+\sqrt{-df} \right ) }{ \left ( f\sqrt{-4\,ac+{b}^{2}}+2\,\sqrt{-df}c+bf \right ) \left ( fx+\sqrt{-df} \right ) }}}{\frac{1}{\sqrt{{\frac{ \left ( -fx+\sqrt{-df} \right ) \left ( fx+\sqrt{-df} \right ) \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{cf}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{f x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d + f x^{2}} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{f x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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