Optimal. Leaf size=501 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right )}{15 e f^3 \sqrt{e+f x^2}}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}+\frac{d x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]
[Out]
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Rubi [A] time = 1.69633, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{e} \sqrt{c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 \sqrt{e} f^{7/2} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{x \sqrt{c+d x^2} \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right )}{15 e f^3 \sqrt{e+f x^2}}-\frac{d x \sqrt{c+d x^2} \sqrt{e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{15 e f^3}+\frac{d x \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (6 b e-5 a f)}{5 e f^2}-\frac{x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]
[Out]
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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((b*x**2+a)*(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)
[Out]
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Mathematica [C] time = 2.05784, size = 369, normalized size = 0.74 \[ \frac{-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \left (5 a d f (9 c f-8 d e)+b \left (15 c^2 f^2-64 c d e f+48 d^2 e^2\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )-5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+f x \sqrt{\frac{d}{c}} \left (c+d x^2\right ) \left (5 a f \left (3 c^2 f^2-6 c d e f+d^2 e \left (4 e+f x^2\right )\right )+b e \left (-15 c^2 f^2+c d f \left (41 e+11 f x^2\right )-3 d^2 \left (8 e^2+2 e f x^2-f^2 x^4\right )\right )\right )}{15 e f^4 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)*(c + d*x^2)^(5/2))/(e + f*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.069, size = 1169, normalized size = 2.3 \[ \text{result too large to display} \]
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)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b d^{2} x^{6} +{\left (2 \, b c d + a d^{2}\right )} x^{4} + a c^{2} +{\left (b c^{2} + 2 \, a c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}, x\right ) \]
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)^(3/2),x, algorithm="fricas")
[Out]
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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)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]
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)^(3/2),x, algorithm="giac")
[Out]