Optimal. Leaf size=280 \[ -\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{a B e-5 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{3 e (a+b x) (a B e-5 A b e+4 b B d)}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e (a+b x) (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
[Out]
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Rubi [A] time = 0.632146, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{a B e-5 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{3 e (a+b x) (a B e-5 A b e+4 b B d)}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e (a+b x) (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 1.15753, size = 174, normalized size = 0.62 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (\frac{2 (a B-A b) (b d-a e)}{(a+b x)^2}+\frac{-3 a B e+7 A b e-4 b B d}{a+b x}+\frac{8 e (A e-B d)}{d+e x}\right )}{(b d-a e)^3}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{7/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.037, size = 681, normalized size = 2.4 \[ -{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{3}} \left ( 15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}{e}^{2}-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}de+30\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}xa{b}^{2}{e}^{2}-6\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}x{a}^{2}b{e}^{2}-24\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}xa{b}^{2}de+15\,A\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}{e}^{2}+15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{2}b{e}^{2}-3\,B\sqrt{b \left ( ae-bd \right ) }{x}^{2}ab{e}^{2}-12\,B\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{2}de-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{3}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \sqrt{ex+d}{a}^{2}bde+25\,A\sqrt{b \left ( ae-bd \right ) }xab{e}^{2}+5\,A\sqrt{b \left ( ae-bd \right ) }x{b}^{2}de-5\,B\sqrt{b \left ( ae-bd \right ) }x{a}^{2}{e}^{2}-21\,B\sqrt{b \left ( ae-bd \right ) }xabde-4\,B\sqrt{b \left ( ae-bd \right ) }x{b}^{2}{d}^{2}+8\,A\sqrt{b \left ( ae-bd \right ) }{a}^{2}{e}^{2}+9\,A\sqrt{b \left ( ae-bd \right ) }abde-2\,A\sqrt{b \left ( ae-bd \right ) }{b}^{2}{d}^{2}-13\,B\sqrt{b \left ( ae-bd \right ) }{a}^{2}de-2\,B\sqrt{b \left ( ae-bd \right ) }ab{d}^{2} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}{\frac{1}{\sqrt{ex+d}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.31, 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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(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+A)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.325318, size = 848, normalized size = 3.03 \[ \frac{3 \,{\left (4 \, B b d e^{2} + B a e^{3} - 5 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{x e + d}} + \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{2} - 4 \, \sqrt{x e + d} B b^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{3} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{3} - \sqrt{x e + d} B a b d e^{3} + 9 \, \sqrt{x e + d} A b^{2} d e^{3} + 5 \, \sqrt{x e + d} B a^{2} e^{4} - 9 \, \sqrt{x e + d} A a b e^{4}}{4 \,{\left (b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 3 \, a^{2} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - a^{3} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]