3.983 \(\int x^{3/2} (A+B x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{11} B c x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.0245567, 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, Rules used = {765} \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{11} B c x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x^{3/2} (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{5/2}+(b B+A c) x^{7/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{5} a A x^{5/2}+\frac{2}{7} (A b+a B) x^{7/2}+\frac{2}{9} (b B+A c) x^{9/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0473992, size = 48, normalized size = 0.87 \[ \frac{2 x^{5/2} (99 a (7 A+5 B x)+5 x (11 A (9 b+7 c x)+7 B x (11 b+9 c x)))}{3465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(99*a*(7*A + 5*B*x) + 5*x*(11*A*(9*b + 7*c*x) + 7*B*x*(11*b + 9*c*x))))/3465

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Maple [A]  time = 0.004, size = 42, normalized size = 0.8 \begin{align*}{\frac{630\,Bc{x}^{3}+770\,Ac{x}^{2}+770\,Bb{x}^{2}+990\,Abx+990\,aBx+1386\,aA}{3465}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(B*x+A)*(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.08479, size = 53, normalized size = 0.96 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b + A c\right )} x^{\frac{9}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} + \frac{2}{7} \,{\left (B a + A b\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.28996, size = 119, normalized size = 2.16 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c x^{5} + 385 \,{\left (B b + A c\right )} x^{4} + 693 \, A a x^{2} + 495 \,{\left (B a + A b\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.45297, size = 70, normalized size = 1.27 \begin{align*} \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{7}{2}}}{7} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x+a),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*a*x**(7/2)/7 + 2*B*b*x**(9/2)/9 + 2*B*c*x**(11/2)
/11

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Giac [A]  time = 1.19284, size = 58, normalized size = 1.05 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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