Optimal. Leaf size=167 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (A b-3 a B)}{8 b^4}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (2 A b-3 a B)}{7 b^4}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 b^4}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]
[Out]
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Rubi [A] time = 0.353063, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (A b-3 a B)}{8 b^4}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (2 A b-3 a B)}{7 b^4}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 b^4}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4} \]
Antiderivative was successfully verified.
[In] Int[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 22.8026, size = 163, normalized size = 0.98 \[ \frac{B x^{3} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 b} + \frac{a^{2} \left (2 a + 2 b x\right ) \left (3 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{144 b^{4}} - \frac{a \left (3 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{84 b^{4}} + \frac{x^{2} \left (2 a + 2 b x\right ) \left (3 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0794585, size = 125, normalized size = 0.75 \[ \frac{x^3 \sqrt{(a+b x)^2} \left (42 a^5 (4 A+3 B x)+126 a^4 b x (5 A+4 B x)+168 a^3 b^2 x^2 (6 A+5 B x)+120 a^2 b^3 x^3 (7 A+6 B x)+45 a b^4 x^4 (8 A+7 B x)+7 b^5 x^5 (9 A+8 B x)\right )}{504 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.011, size = 140, normalized size = 0.8 \[{\frac{{x}^{3} \left ( 56\,B{b}^{5}{x}^{6}+63\,{x}^{5}A{b}^{5}+315\,{x}^{5}Ba{b}^{4}+360\,{x}^{4}Aa{b}^{4}+720\,{x}^{4}B{a}^{2}{b}^{3}+840\,{x}^{3}A{a}^{2}{b}^{3}+840\,{x}^{3}B{a}^{3}{b}^{2}+1008\,A{a}^{3}{b}^{2}{x}^{2}+504\,B{a}^{4}b{x}^{2}+630\,xA{a}^{4}b+126\,xB{a}^{5}+168\,A{a}^{5} \right ) }{504\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(B*x+A)*(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^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.301682, size = 159, normalized size = 0.95 \[ \frac{1}{9} \, B b^{5} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{1}{8} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac{5}{7} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} +{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282169, size = 297, normalized size = 1.78 \[ \frac{1}{9} \, B b^{5} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{8} \, B a b^{4} x^{8}{\rm sign}\left (b x + a\right ) + \frac{1}{8} \, A b^{5} x^{8}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, B a^{2} b^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{7} \, A a b^{4} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, B a^{3} b^{2} x^{6}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, A a^{2} b^{3} x^{6}{\rm sign}\left (b x + a\right ) + B a^{4} b x^{5}{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} b^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, B a^{5} x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, A a^{4} b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, A a^{5} x^{3}{\rm sign}\left (b x + a\right ) - \frac{{\left (B a^{9} - 3 \, A a^{8} b\right )}{\rm sign}\left (b x + a\right )}{504 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^2,x, algorithm="giac")
[Out]