Optimal. Leaf size=246 \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]
[Out]
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Rubi [A] time = 0.522734, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{9/2} e^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (7 a B e-8 A b e+b B d)}{64 b^4 e}-\frac{5 \sqrt{a+b x} (d+e x)^{3/2} (b d-a e) (7 a B e-8 A b e+b B d)}{96 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{5/2} (7 a B e-8 A b e+b B d)}{24 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{7/2}}{4 b e} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(5/2))/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 40.617, size = 238, normalized size = 0.97 \[ \frac{B \sqrt{a + b x} \left (d + e x\right )^{\frac{7}{2}}}{4 b e} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (8 A b e - 7 B a e - B b d\right )}{24 b^{2} e} - \frac{5 \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (8 A b e - 7 B a e - B b d\right )}{96 b^{3} e} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - 7 B a e - B b d\right )}{64 b^{4} e} - \frac{5 \left (a e - b d\right )^{3} \left (8 A b e - 7 B a e - B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{64 b^{\frac{9}{2}} e^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.411643, size = 244, normalized size = 0.99 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^3 B e^3+5 a^2 b e^2 (24 A e+53 B d+14 B e x)-a b^2 e \left (80 A e (4 d+e x)+B \left (191 d^2+172 d e x+56 e^2 x^2\right )\right )+b^3 \left (8 A e \left (33 d^2+26 d e x+8 e^2 x^2\right )+B \left (15 d^3+118 d^2 e x+136 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^4 e}-\frac{5 (b d-a e)^3 (7 a B e-8 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{128 b^{9/2} e^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(5/2))/Sqrt[a + b*x],x]
[Out]
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Maple [B] time = 0.036, size = 968, normalized size = 3.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(5/2)/(b*x+a)^(1/2),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)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.627192, 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)/sqrt(b*x + a),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*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.368254, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(b*x + a),x, algorithm="giac")
[Out]