Optimal. Leaf size=1432 \[ \text{result too large to display} \]
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Rubi [A] time = 6.21881, antiderivative size = 1432, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {992, 935, 1103} \[ -\frac{\sqrt [4]{d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \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+e x+d\right )}{\left (4 f a^2-2 \left (b+\sqrt{b^2-4 a c}\right ) e a+\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{f b^2-c e b+2 c^2 d-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\sqrt{d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt{b^2-4 a c}\right )}+1\right ) \sqrt{\frac{\frac{\left (4 d c^2-2 \left (b+\sqrt{b^2-4 a c}\right ) e c+\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^2-2 \left (b+\sqrt{b^2-4 a c}\right ) e a+\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{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}+1}{\left (\frac{\sqrt{f b^2-c e b+2 c^2 d-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\sqrt{d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt{b^2-4 a c}\right )}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \sqrt{2 a+\left (b+\sqrt{b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \sqrt{b+2 c x+\sqrt{b^2-4 a c}}}\right )|\frac{1}{4} \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt{d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \sqrt{2 d c^2-\left (b e+\sqrt{b^2-4 a c} e+2 a f\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) f}}+2\right )\right )}{\left (4 a c-\left (b+\sqrt{b^2-4 a c}\right )^2\right ) \sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \sqrt{c x^2+b x+a} \sqrt{f x^2+e x+d} \sqrt{\frac{\left (4 d c^2-2 \left (b+\sqrt{b^2-4 a c}\right ) e c+\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^2-2 \left (b+\sqrt{b^2-4 a c}\right ) e a+\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{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt{b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}+1}} \]
Antiderivative was successfully verified.
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Rule 992
Rule 935
Rule 1103
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x+c x^2} \sqrt{d+e x+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+e x+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+e x+f x^2\right )}{\left (\left (b+\sqrt{b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+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 c e-\left (b+\sqrt{b^2-4 a c}\right )^2 e+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-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+4 a^2 f}+\frac{\left (4 c^2 d-2 c \left (b+\sqrt{b^2-4 a c}\right ) e+\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-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+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+e x+f x^2}}\\ &=-\frac{\sqrt [4]{b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \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+e x+f x^2\right )}{\left (\left (b+\sqrt{b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+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-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\sqrt{b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}\right ) \sqrt{\frac{1-\frac{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt{b^2-4 a c}+2 c x\right )}+\frac{\left (4 c^2 d-2 c \left (b+\sqrt{b^2-4 a c}\right ) e+\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-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+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-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\sqrt{b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \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-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \sqrt{2 a+\left (b+\sqrt{b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \sqrt{b+\sqrt{b^2-4 a c}+2 c x}}\right )|\frac{1}{4} \left (2+\frac{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt{b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )} \sqrt{2 c^2 d+b \left (b+\sqrt{b^2-4 a c}\right ) f-c \left (b e+\sqrt{b^2-4 a c} e+2 a f\right )}}\right )\right )}{\left (4 a c-\left (b+\sqrt{b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt{b^2-4 a c} (c e-b f)} \sqrt{a+b x+c x^2} \sqrt{d+e x+f x^2} \sqrt{1-\frac{\left (b+\sqrt{b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt{b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt{b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt{b^2-4 a c}+2 c x\right )}+\frac{\left (4 c^2 d-2 c \left (b+\sqrt{b^2-4 a c}\right ) e+\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-2 a \left (b+\sqrt{b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt{b^2-4 a c}+2 c x\right )^2}}}\\ \end{align*}
Mathematica [A] time = 2.487, size = 670, normalized size = 0.47 \[ -\frac{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (-\sqrt{e^2-4 d f}+e+2 f x\right ) \sqrt{-\frac{c \sqrt{b^2-4 a c} \left (\sqrt{e^2-4 d f}+e+2 f x\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt{b^2-4 a c}+b\right )-c \left (\sqrt{e^2-4 d f}+e\right )\right )}} \sqrt{-\frac{c \left (\sqrt{b^2-4 a c} \sqrt{e^2-4 d f}-e \left (\sqrt{b^2-4 a c}+2 c x\right )-2 f x \sqrt{b^2-4 a c}+4 a f+b \left (\sqrt{e^2-4 d f}-e+2 f x\right )+2 c x \sqrt{e^2-4 d f}\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt{b^2-4 a c}+b\right )+c \left (\sqrt{e^2-4 d f}-e\right )\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (f \left (\sqrt{b^2-4 a c}-b\right )+c \left (e-\sqrt{e^2-4 d f}\right )\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt{b^2-4 a c}+b\right )+c \left (\sqrt{e^2-4 d f}-e\right )\right )}}\right ),\frac{-\sqrt{b^2-4 a c} \sqrt{e^2-4 d f}+2 a f-b e+2 c d}{\sqrt{b^2-4 a c} \sqrt{e^2-4 d f}+2 a f-b e+2 c d}\right )}{\sqrt{a+x (b+c x)} \sqrt{d+x (e+f x)} \left (f \left (\sqrt{b^2-4 a c}-b\right )+c \left (e-\sqrt{e^2-4 d f}\right )\right ) \sqrt{\frac{c \sqrt{b^2-4 a c} \left (\sqrt{e^2-4 d f}-e-2 f x\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (f \left (\sqrt{b^2-4 a c}+b\right )+c \left (\sqrt{e^2-4 d f}-e\right )\right )}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.836, size = 929, normalized size = 0.7 \begin{align*} \text{result too large to display} \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} + e x + 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} + e x + d}}{c f x^{4} +{\left (c e + b f\right )} x^{3} +{\left (c d + b e + a f\right )} x^{2} + a d +{\left (b d + a e\right )} x}, 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{a + b x + c x^{2}} \sqrt{d + e x + f 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} + e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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