Optimal. Leaf size=484 \[ -\frac{\left (c e \left (e-\sqrt{e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (-a^2 f^3+2 a c d f^2+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\left (c e \left (\sqrt{e^2-4 d f}+e\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )-2 f \left (-a^2 f^3+2 a c d f^2+c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{2 f^3}-\frac{c \sqrt{a+c x^2} (2 e-f x)}{2 f^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 8.82711, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (-2 a^2 f^4-c e \left (e-\sqrt{e^2-4 d f}\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\left (-2 a^2 f^4-c e \left (\sqrt{e^2-4 d f}+e\right ) \left (2 a f^2+c \left (e^2-2 d f\right )\right )+4 a c d f^3+2 c^2 d f \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} f^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-d f\right )\right )}{2 f^3}-\frac{c \sqrt{a+c x^2} (2 e-f x)}{2 f^2} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^(3/2)/(d + e*x + f*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 2.71635, size = 932, normalized size = 1.93 \[ \frac{c f \sqrt{c x^2+a} (f x-2 e)-\frac{\sqrt{2} \left (-2 a^2 f^4+2 a c \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) f^2+c^2 \left (-e^4+\sqrt{e^2-4 d f} e^3+4 d f e^2-2 d f \sqrt{e^2-4 d f} e-2 d^2 f^2\right )\right ) \log \left (-e-2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}-\frac{\sqrt{2} \left (2 a^2 f^4+2 a c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^2+c^2 \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right )\right ) \log \left (e+2 f x+\sqrt{e^2-4 d f}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}+\sqrt{c} \left (3 a f^2+2 c \left (e^2-d f\right )\right ) \log \left (c x+\sqrt{c} \sqrt{c x^2+a}\right )+\frac{\sqrt{2} \left (-2 a^2 f^4+2 a c \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right ) f^2+c^2 \left (-e^4+\sqrt{e^2-4 d f} e^3+4 d f e^2-2 d f \sqrt{e^2-4 d f} e-2 d^2 f^2\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f+c \left (e^2-\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )}}+\frac{\sqrt{2} \left (2 a^2 f^4+2 a c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right ) f^2+c^2 \left (e^4+\sqrt{e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt{e^2-4 d f} e+2 d^2 f^2\right )\right ) \log \left (2 a \sqrt{e^2-4 d f} f-c \left (e^2+\sqrt{e^2-4 d f} e-4 d f\right ) x+\sqrt{2} \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )}}}{2 f^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^(3/2)/(d + e*x + f*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 8954, normalized size = 18.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(3/2)/(f*x^2+e*x+d),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)/(f*x^2 + e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)/(f*x^2 + e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{d + e x + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)/(f*x^2 + e*x + d),x, algorithm="giac")
[Out]