Optimal. Leaf size=284 \[ \frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.63039, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{e+f x^2} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{d} \sqrt{c+d x^2} (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-3 a c f+a d e+2 b c e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 90.132, size = 257, normalized size = 0.9 \[ - \frac{x \sqrt{e + f x^{2}} \left (a d - b c\right )}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (c f - d e\right )} + \frac{\sqrt{e} \sqrt{f} \sqrt{c + d x^{2}} \left (3 a c f - a d e - 2 b c e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 c^{2} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (c f - d e\right )^{2}} + \frac{\sqrt{e + f x^{2}} \left (- 4 a c d f + 2 a d^{2} e + b c^{2} f + b c d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 c^{\frac{3}{2}} \sqrt{d} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (c f - d e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 2.13819, size = 302, normalized size = 1.06 \[ \frac{x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (a d \left (-5 c^2 f+c d \left (3 e-4 f x^2\right )+2 d^2 e x^2\right )+b c \left (2 c^2 f+c d f x^2+d^2 e x^2\right )\right )+i \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-3 a c f+2 a d e+b c e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (c f+d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 c^2 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (d e-c f)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/((c + d*x^2)^(5/2)*Sqrt[e + f*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.049, size = 1352, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{2} + a}{{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)/((d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)),x, algorithm="giac")
[Out]