Optimal. Leaf size=250 \[ -\frac{(b d-a e)^3 (3 a B e-8 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{5/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (3 a B e-8 A b e+5 b B d)}{64 b^2 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (3 a B e-8 A b e+5 b B d)}{96 b^2 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (3 a B e-8 A b e+5 b B d)}{24 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e} \]
[Out]
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Rubi [A] time = 0.500813, antiderivative size = 250, 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 \[ -\frac{(b d-a e)^3 (3 a B e-8 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{64 b^{5/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2 (3 a B e-8 A b e+5 b B d)}{64 b^2 e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e) (3 a B e-8 A b e+5 b B d)}{96 b^2 e^2}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (3 a B e-8 A b e+5 b B d)}{24 b^2 e}+\frac{B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(3/2)*(A + B*x)*Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 42.0466, size = 241, normalized size = 0.96 \[ \frac{B \left (a + b x\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}}}{4 b e} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (8 A b e - 3 B a e - 5 B b d\right )}{24 b e^{2}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (8 A b e - 3 B a e - 5 B b d\right )}{32 b e^{3}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )^{2} \left (8 A b e - 3 B a e - 5 B b d\right )}{64 b^{2} e^{3}} - \frac{\left (a e - b d\right )^{3} \left (8 A b e - 3 B a e - 5 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{64 b^{\frac{5}{2}} e^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.355982, size = 249, normalized size = 1. \[ \frac{2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x} \left (-9 a^3 B e^3+3 a^2 b e^2 (8 A e+3 B d+2 B e x)+a b^2 e \left (16 A e (4 d+7 e x)+B \left (-31 d^2+20 d e x+72 e^2 x^2\right )\right )+b^3 \left (8 A e \left (-3 d^2+2 d e x+8 e^2 x^2\right )+B \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )\right )-3 (b d-a e)^3 (3 a B e-8 A b e+5 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{384 b^{5/2} e^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(3/2)*(A + B*x)*Sqrt[d + e*x],x]
[Out]
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Maple [B] time = 0.021, size = 1150, normalized size = 4.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(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)*(b*x + a)^(3/2)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271801, 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*x + a)^(3/2)*sqrt(e*x + 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)**(3/2)*(B*x+A)*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.30049, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)*sqrt(e*x + d),x, algorithm="giac")
[Out]