Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
[Out]
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Rubi [A] time = 0.0641828, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.93751, size = 66, normalized size = 0.96 \[ \frac{B \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0543402, size = 83, normalized size = 1.2 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^3 (2 A+B x)+10 a^2 b x (3 A+2 B x)+5 a b^2 x^2 (4 A+3 B x)+b^3 x^3 (5 A+4 B x)\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 90, normalized size = 1.3 \[{\frac{x \left ( 4\,B{b}^{3}{x}^{4}+5\,A{b}^{3}{x}^{3}+15\,{x}^{3}a{b}^{2}B+20\,Aa{b}^{2}{x}^{2}+20\,B{a}^{2}b{x}^{2}+30\,xA{a}^{2}b+10\,{a}^{3}Bx+20\,A{a}^{3} \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(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^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303939, size = 93, normalized size = 1.35 \[ \frac{1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} +{\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
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),x)
[Out]
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GIAC/XCAS [A] time = 0.273779, size = 194, normalized size = 2.81 \[ \frac{1}{5} \, B b^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, B a b^{2} x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A b^{3} x^{4}{\rm sign}\left (b x + a\right ) + B a^{2} b x^{3}{\rm sign}\left (b x + a\right ) + A a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, A a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + A a^{3} x{\rm sign}\left (b x + a\right ) - \frac{{\left (B a^{5} - 5 \, A a^{4} b\right )}{\rm sign}\left (b x + a\right )}{20 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A),x, algorithm="giac")
[Out]