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

Optimal. Leaf size=45 \[ \frac{2}{3} a A x^{3/2}+\frac{2}{5} a B x^{5/2}+\frac{2}{7} A c x^{7/2}+\frac{2}{9} B c x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.0386267, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2}{3} a A x^{3/2}+\frac{2}{5} a B x^{5/2}+\frac{2}{7} A c x^{7/2}+\frac{2}{9} B c x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.02973, size = 46, normalized size = 1.02 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{7}{2}}}{7} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0191145, size = 35, normalized size = 0.78 \[ \frac{2}{315} x^{3/2} \left (21 a (5 A+3 B x)+5 c x^2 (9 A+7 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(21*a*(5*A + 3*B*x) + 5*c*x^2*(9*A + 7*B*x)))/315

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Maple [A]  time = 0.005, size = 30, normalized size = 0.7 \[{\frac{70\,Bc{x}^{3}+90\,Ac{x}^{2}+126\,aBx+210\,aA}{315}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.684977, size = 39, normalized size = 0.87 \[ \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276821, size = 43, normalized size = 0.96 \[ \frac{2}{315} \,{\left (35 \, B c x^{4} + 45 \, A c x^{3} + 63 \, B a x^{2} + 105 \, A a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.52079, size = 46, normalized size = 1.02 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{7}{2}}}{7} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268115, size = 39, normalized size = 0.87 \[ \frac{2}{9} \, B c x^{\frac{9}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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