Optimal. Leaf size=780 \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 5.16229, antiderivative size = 780, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1036, 1030, 208} \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{\left (a+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx &=-\frac{\int \frac{-a B e-A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (-A c e+B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (a+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx}{2 \sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}+\frac{\int \frac{-a B e-A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )+\left (-A c e+B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\left (a+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx}{2 \sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}\\ &=\frac{\left (a \left (A c e-B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2 c \left (A c e-B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )+a e x^2} \, dx,x,\frac{-a \left (A c e-B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )+c \left (a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\sqrt{d+e x+f x^2}}\right )}{\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}-\frac{\left (a \left (a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (A c e-B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2 c \left (a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) \left (A c e-B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )+a e x^2} \, dx,x,\frac{-a \left (A c e-B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )+c \left (a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x}{\sqrt{d+e x+f x^2}}\right )}{\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}\\ &=\frac{\sqrt{a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{-A c e+B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a B e+A \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{-A c e+B \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{d+e x+f x^2}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}-\frac{\sqrt{-A c e+B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right )-c \left (a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )\right ) x\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{-A c e+B \left (c d-a f-\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{a B e+A \left (c d-a f+\sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}\right )} \sqrt{d+e x+f x^2}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )}}\\ \end{align*}
Mathematica [A] time = 0.43835, size = 254, normalized size = 0.33 \[ \frac{\frac{\left (A \sqrt{c}-\sqrt{-a} B\right ) \tanh ^{-1}\left (\frac{\sqrt{c} (2 d+e x)-\sqrt{-a} (e+2 f x)}{2 \sqrt{d+x (e+f x)} \sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{-\sqrt{-a} \sqrt{c} e-a f+c d}}-\frac{\left (\sqrt{-a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{-a} (e+2 f x)+\sqrt{c} (2 d+e x)}{2 \sqrt{d+x (e+f x)} \sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{\sqrt{-a} \sqrt{c} e-a f+c d}}}{2 \sqrt{-a} \sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.355, size = 784, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\left (a + c x^{2}\right ) \sqrt{d + e x + f x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]