3.386 \(\int x^{3/2} (A+B x) \left (a+c x^2\right ) \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]

[Out]

(2*a*A*x^(5/2))/5 + (2*a*B*x^(7/2))/7 + (2*A*c*x^(9/2))/9 + (2*B*c*x^(11/2))/11

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Rubi [A]  time = 0.0413456, 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 \[ \frac{2}{5} a A x^{5/2}+\frac{2}{7} a B x^{7/2}+\frac{2}{9} A c x^{9/2}+\frac{2}{11} B c x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*A*x^(5/2))/5 + (2*a*B*x^(7/2))/7 + (2*A*c*x^(9/2))/9 + (2*B*c*x^(11/2))/11

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Rubi in Sympy [A]  time = 5.01982, size = 46, normalized size = 1.02 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)*(c*x**2+a),x)

[Out]

2*A*a*x**(5/2)/5 + 2*A*c*x**(9/2)/9 + 2*B*a*x**(7/2)/7 + 2*B*c*x**(11/2)/11

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Mathematica [A]  time = 0.0273934, size = 37, normalized size = 0.82 \[ \frac{2}{35} a x^{5/2} (7 A+5 B x)+\frac{2}{99} c x^{9/2} (11 A+9 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*a*x^(5/2)*(7*A + 5*B*x))/35 + (2*c*x^(9/2)*(11*A + 9*B*x))/99

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Maple [A]  time = 0.005, size = 30, normalized size = 0.7 \[{\frac{630\,Bc{x}^{3}+770\,Ac{x}^{2}+990\,aBx+1386\,aA}{3465}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)*(c*x^2+a),x)

[Out]

2/3465*x^(5/2)*(315*B*c*x^3+385*A*c*x^2+495*B*a*x+693*A*a)

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Maxima [A]  time = 0.68001, size = 39, normalized size = 0.87 \[ \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*x^(3/2),x, algorithm="maxima")

[Out]

2/11*B*c*x^(11/2) + 2/9*A*c*x^(9/2) + 2/7*B*a*x^(7/2) + 2/5*A*a*x^(5/2)

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Fricas [A]  time = 0.275363, size = 46, normalized size = 1.02 \[ \frac{2}{3465} \,{\left (315 \, B c x^{5} + 385 \, A c x^{4} + 495 \, B a x^{3} + 693 \, A a x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*x^(3/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c*x^5 + 385*A*c*x^4 + 495*B*a*x^3 + 693*A*a*x^2)*sqrt(x)

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Sympy [A]  time = 3.4709, size = 46, normalized size = 1.02 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)*(c*x**2+a),x)

[Out]

2*A*a*x**(5/2)/5 + 2*A*c*x**(9/2)/9 + 2*B*a*x**(7/2)/7 + 2*B*c*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.266713, size = 39, normalized size = 0.87 \[ \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*x^(3/2),x, algorithm="giac")

[Out]

2/11*B*c*x^(11/2) + 2/9*A*c*x^(9/2) + 2/7*B*a*x^(7/2) + 2/5*A*a*x^(5/2)