Optimal. Leaf size=797 \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (a B \left (-f b^2+2 c^2 d+2 a c f\right )+A \left (b^3 f-b c (c d+3 a f)\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]
[Out]
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Rubi [A] time = 4.31787, antiderivative size = 796, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (B \sqrt{d}-A \sqrt{f}\right ) \tanh ^{-1}\left (\frac{-2 \sqrt{f} a+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x+b \sqrt{d}}{2 \sqrt{-\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (-\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}+\frac{\left (\sqrt{f} A+B \sqrt{d}\right ) \tanh ^{-1}\left (\frac{2 \sqrt{f} a+\left (\sqrt{f} b+2 c \sqrt{d}\right ) x+b \sqrt{d}}{2 \sqrt{\sqrt{d} \sqrt{f} b+c d+a f} \sqrt{c x^2+b x+a}}\right ) f^{3/2}}{2 \sqrt{d} \left (\sqrt{d} \sqrt{f} b+c d+a f\right )^{5/2}}-\frac{2 \left (3 B d f^2 b^6-A f^2 (7 c d+6 a f) b^5-B f \left (7 c^2 d^2+14 a c f d-3 a^2 f^2\right ) b^4+A c f \left (15 c^2 d^2+46 a c f d+43 a^2 f^2\right ) b^3+2 B c \left (2 c^3 d^3+5 a c^2 f d^2+4 a^2 c f^2 d-11 a^3 f^3\right ) b^2-4 A c^2 \left (2 c^3 d^3+9 a c^2 f d^2+24 a^2 c f^2 d+17 a^3 f^3\right ) b+24 a^2 B c^2 f (c d+a f)^2+c \left (3 B d f^2 b^5-2 A f^2 (4 c d+3 a f) b^4-B f \left (17 c^2 d^2+10 a c f d-3 a^2 f^2\right ) b^3+2 A c f \left (15 c^2 d^2+22 a c f d+19 a^2 f^2\right ) b^2+4 B c \left (2 c^3 d^3+11 a c^2 f d^2+4 a^2 c f^2 d-5 a^3 f^3\right ) b-8 A c^2 (c d+a f)^2 (2 c d+5 a f)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c^2 d^2+2 a c f d-f \left (b^2 d-a^2 f\right )\right )^2 \sqrt{c x^2+b x+a}}-\frac{2 \left (A f b^3-A c (c d+3 a f) b+a B \left (-f b^2+2 c^2 d+2 a c f\right )+c \left (A f b^2+B (c d-a f) b-2 A c (c d+a f)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \left (c x^2+b x+a\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^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+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)
[Out]
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Mathematica [B] time = 7.37118, size = 1847, normalized size = 2.32 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)),x]
[Out]
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Maple [B] time = 0.04, size = 6422, normalized size = 8.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(c*x^2+b*x+a)^(5/2)/(-f*x^2+d),x)
[Out]
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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(-(B*x + A)/((c*x^2 + b*x + a)^(5/2)*(f*x^2 - d)),x, algorithm="maxima")
[Out]
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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(-(B*x + A)/((c*x^2 + b*x + a)^(5/2)*(f*x^2 - d)),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+A)/(c*x**2+b*x+a)**(5/2)/(-f*x**2+d),x)
[Out]
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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(-(B*x + A)/((c*x^2 + b*x + a)^(5/2)*(f*x^2 - d)),x, algorithm="giac")
[Out]