Optimal. Leaf size=359 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e^3 (a+b x) (-7 a B e-A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac{5 e^2 \sqrt{d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e-A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.783216, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^{7/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e^3 (a+b x) (-7 a B e-A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac{5 e^2 \sqrt{d+e x} (-7 a B e-A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{5 e (d+e x)^{3/2} (-7 a B e-A b e+8 b B d)}{96 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 1.02379, size = 235, normalized size = 0.65 \[ \frac{(a+b x) \left (\frac{5 e^3 (7 a B e+A b e-8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} \left (3 e^2 (a+b x)^3 (-93 a B e+5 A b e+88 b B d)+8 (a+b x) (b d-a e)^2 (-25 a B e+17 A b e+8 b B d)+2 e (a+b x)^2 (b d-a e) (-163 a B e+59 A b e+104 b B d)+48 (A b-a B) (b d-a e)^3\right )}{3 b^4 (a+b x)^4 (b d-a e)}\right )}{64 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.033, size = 1273, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),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)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.296219, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),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+A)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.358208, size = 879, normalized size = 2.45 \[ -\frac{5 \,{\left (8 \, B b d e^{3} - 7 \, B a e^{4} - A b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{64 \,{\left (b^{5} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{264 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{4} d e^{3} - 584 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} d^{2} e^{3} + 440 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e^{3} - 120 \, \sqrt{x e + d} B b^{4} d^{4} e^{3} - 279 \,{\left (x e + d\right )}^{\frac{7}{2}} B a b^{3} e^{4} + 15 \,{\left (x e + d\right )}^{\frac{7}{2}} A b^{4} e^{4} + 1095 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{3} d e^{4} + 73 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{4} d e^{4} - 1265 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{4} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{4} + 465 \, \sqrt{x e + d} B a b^{3} d^{3} e^{4} + 15 \, \sqrt{x e + d} A b^{4} d^{3} e^{4} - 511 \,{\left (x e + d\right )}^{\frac{5}{2}} B a^{2} b^{2} e^{5} - 73 \,{\left (x e + d\right )}^{\frac{5}{2}} A a b^{3} e^{5} + 1210 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{5} + 110 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{5} - 675 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{5} - 45 \, \sqrt{x e + d} A a b^{3} d^{2} e^{5} - 385 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{6} - 55 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{6} + 435 \, \sqrt{x e + d} B a^{3} b d e^{6} + 45 \, \sqrt{x e + d} A a^{2} b^{2} d e^{6} - 105 \, \sqrt{x e + d} B a^{4} e^{7} - 15 \, \sqrt{x e + d} A a^{3} b e^{7}}{192 \,{\left (b^{5} d{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a b^{4} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]