3.984 \(\int \sqrt{x} (A+B x) \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} x^{7/2} (A c+b B)+\frac{2}{9} B c x^{9/2} \]

[Out]

(2*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(5/2))/5 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B
*c*x^(9/2))/9

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Rubi [A]  time = 0.063651, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{2}{5} x^{5/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{7} x^{7/2} (A c+b B)+\frac{2}{9} B c x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2),x]

[Out]

(2*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(5/2))/5 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B
*c*x^(9/2))/9

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Rubi in Sympy [A]  time = 8.39736, size = 60, normalized size = 1.09 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{9}{2}}}{9} + x^{\frac{7}{2}} \left (\frac{2 A c}{7} + \frac{2 B b}{7}\right ) + x^{\frac{5}{2}} \left (\frac{2 A b}{5} + \frac{2 B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)*x**(1/2),x)

[Out]

2*A*a*x**(3/2)/3 + 2*B*c*x**(9/2)/9 + x**(7/2)*(2*A*c/7 + 2*B*b/7) + x**(5/2)*(2
*A*b/5 + 2*B*a/5)

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Mathematica [A]  time = 0.0341012, size = 47, normalized size = 0.85 \[ \frac{2}{315} x^{3/2} (21 a (5 A+3 B x)+x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2),x]

[Out]

(2*x^(3/2)*(21*a*(5*A + 3*B*x) + x*(9*A*(7*b + 5*c*x) + 5*B*x*(9*b + 7*c*x))))/3
15

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Maple [A]  time = 0.006, size = 42, normalized size = 0.8 \[{\frac{70\,Bc{x}^{3}+90\,Ac{x}^{2}+90\,Bb{x}^{2}+126\,Abx+126\,aBx+210\,aA}{315}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)*x^(1/2),x)

[Out]

2/315*x^(3/2)*(35*B*c*x^3+45*A*c*x^2+45*B*b*x^2+63*A*b*x+63*B*a*x+105*A*a)

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Maxima [A]  time = 0.711556, size = 53, normalized size = 0.96 \[ \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b + A c\right )} x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)*sqrt(x),x, algorithm="maxima")

[Out]

2/9*B*c*x^(9/2) + 2/7*(B*b + A*c)*x^(7/2) + 2/3*A*a*x^(3/2) + 2/5*(B*a + A*b)*x^
(5/2)

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Fricas [A]  time = 0.278685, size = 57, normalized size = 1.04 \[ \frac{2}{315} \,{\left (35 \, B c x^{4} + 45 \,{\left (B b + A c\right )} x^{3} + 105 \, A a x + 63 \,{\left (B a + A b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)*sqrt(x),x, algorithm="fricas")

[Out]

2/315*(35*B*c*x^4 + 45*(B*b + A*c)*x^3 + 105*A*a*x + 63*(B*a + A*b)*x^2)*sqrt(x)

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Sympy [A]  time = 2.9127, size = 53, normalized size = 0.96 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{9}{2}}}{9} + \frac{2 x^{\frac{7}{2}} \left (A c + B b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (A b + B a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)*x**(1/2),x)

[Out]

2*A*a*x**(3/2)/3 + 2*B*c*x**(9/2)/9 + 2*x**(7/2)*(A*c + B*b)/7 + 2*x**(5/2)*(A*b
+ B*a)/5

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GIAC/XCAS [A]  time = 0.267833, size = 58, normalized size = 1.05 \[ \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)*sqrt(x),x, algorithm="giac")

[Out]

2/9*B*c*x^(9/2) + 2/7*B*b*x^(7/2) + 2/7*A*c*x^(7/2) + 2/5*B*a*x^(5/2) + 2/5*A*b*
x^(5/2) + 2/3*A*a*x^(3/2)