Optimal. Leaf size=369 \[ \frac{f x \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f))}{3 c d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (3 a d f-4 b c f+3 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b c-3 a d)}{3 c d^2}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
[Out]
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Rubi [A] time = 1.12367, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{f x \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f))}{3 c d^3 \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (b c (7 d e-8 c f)-3 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d^3 \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{e^{3/2} \sqrt{c+d x^2} (3 a d f-4 b c f+3 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c d^2 \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \sqrt{c+d x^2} \sqrt{e+f x^2} (4 b c-3 a d)}{3 c d^2}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 114.324, size = 345, normalized size = 0.93 \[ \frac{\sqrt{c} \sqrt{e + f x^{2}} \left (3 a d f - 4 b c f + 3 b d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{3 d^{\frac{5}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \left (e + f x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} - \frac{f x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (3 a d - 4 b c\right )}{3 c d^{2}} + \frac{\sqrt{e} \sqrt{f} \sqrt{c + d x^{2}} \left (- 6 a c d f + 3 a d^{2} e + 8 b c^{2} f - 7 b c d e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{3 c d^{3} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} - \frac{f x \sqrt{c + d x^{2}} \left (- 6 a c d f + 3 a d^{2} e + 8 b c^{2} f - 7 b c d e\right )}{3 c d^{3} \sqrt{e + f x^{2}}} \]
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)**(3/2),x)
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Mathematica [C] time = 1.19675, size = 248, normalized size = 0.67 \[ \frac{\sqrt{\frac{d}{c}} \left (x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (3 a d (d e-c f)+b c \left (4 c f-3 d e+d f x^2\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (4 b c-3 a d) (c f-d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (3 a d (d e-2 c f)+b c (8 c f-7 d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{3 d^3 \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.038, size = 671, normalized size = 1.8 \[ -{\frac{1}{3\,{d}^{2} \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) c}\sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c} \left ( -{x}^{5}bcd{f}^{2}\sqrt{-{\frac{d}{c}}}+3\,{x}^{3}acd{f}^{2}\sqrt{-{\frac{d}{c}}}-3\,{x}^{3}a{d}^{2}ef\sqrt{-{\frac{d}{c}}}-4\,{x}^{3}b{c}^{2}{f}^{2}\sqrt{-{\frac{d}{c}}}+2\,{x}^{3}bcdef\sqrt{-{\frac{d}{c}}}+3\,{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) acdef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-3\,{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-4\,{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) b{c}^{2}ef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+4\,{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcd{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-6\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) acdef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+3\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+8\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) b{c}^{2}ef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-7\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcd{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+3\,xacdef\sqrt{-{\frac{d}{c}}}-3\,xa{d}^{2}{e}^{2}\sqrt{-{\frac{d}{c}}}-4\,xb{c}^{2}ef\sqrt{-{\frac{d}{c}}}+3\,xbcd{e}^{2}\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]
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)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\right )} \sqrt{f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]
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)^(3/2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac{3}{2}}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
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)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2),x, algorithm="giac")
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