Optimal. Leaf size=400 \[ \frac{e^{3/2} \sqrt{c+d x^2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]
[Out]
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Rubi [A] time = 1.21625, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 d^3 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{e^{3/2} \sqrt{c+d x^2} (2 b c (3 d e-2 c f)-5 a d (3 d e-c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac{b x \sqrt{c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]
[Out]
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Rubi in Sympy [A] time = 114.798, size = 389, normalized size = 0.97 \[ \frac{b x \sqrt{c + d x^{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}{5 d} + \frac{\sqrt{c} \sqrt{e + f x^{2}} \left (- 5 a c d f + 15 a d^{2} e + 4 b c^{2} f - 6 b c d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{15 d^{\frac{5}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (5 a d f - 4 b c f + 3 b d e\right )}{15 d^{2}} - \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (- 10 a c d f^{2} + 20 a d^{2} e f + 8 b c^{2} f^{2} - 13 b c d e f + 3 b d^{2} e^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{15 d^{3} \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} + \frac{x \sqrt{c + d x^{2}} \left (- 10 a c d f^{2} + 20 a d^{2} e f + 8 b c^{2} f^{2} - 13 b c d e f + 3 b d^{2} e^{2}\right )}{15 d^{3} \sqrt{e + f x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(1/2),x)
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Mathematica [C] time = 1.51848, size = 275, normalized size = 0.69 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \left (-\sqrt{\frac{d}{c}}\right ) \left (c+d x^2\right ) \left (e+f x^2\right ) \left (-5 a d f+4 b c f-3 b d \left (2 e+f x^2\right )\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (-5 a d f+4 b c f-3 b d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{15 c^2 f \left (\frac{d}{c}\right )^{5/2} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/Sqrt[c + d*x^2],x]
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Maple [B] time = 0.027, size = 870, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\right )} \sqrt{f x^{2} + e}}{\sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac{3}{2}}}{\sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/sqrt(d*x^2 + c),x, algorithm="giac")
[Out]