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

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} \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} \end {gather*}

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 37, normalized size = 0.82 \begin {gather*} \frac {2}{35} a x^{5/2} (7 A+5 B x)+\frac {2}{99} c x^{9/2} (11 A+9 B x) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \frac {2 \left (693 a A x^{5/2}+495 a B x^{7/2}+385 A c x^{9/2}+315 B c x^{11/2}\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(693*a*A*x^(5/2) + 495*a*B*x^(7/2) + 385*A*c*x^(9/2) + 315*B*c*x^(11/2)))/3465

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fricas [A]  time = 0.39, size = 34, normalized size = 0.76 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(c*x^2+a),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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giac [A]  time = 0.15, size = 29, normalized size = 0.64 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.06, size = 30, normalized size = 0.67 \begin {gather*} \frac {2 \left (315 B c \,x^{3}+385 A c \,x^{2}+495 B a x +693 a A \right ) x^{\frac {5}{2}}}{3465} \end {gather*}

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 = 0.49, size = 29, normalized size = 0.64 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(c*x^2+a),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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mupad [B]  time = 0.04, size = 29, normalized size = 0.64 \begin {gather*} \frac {2\,A\,a\,x^{5/2}}{5}+\frac {2\,B\,a\,x^{7/2}}{7}+\frac {2\,A\,c\,x^{9/2}}{9}+\frac {2\,B\,c\,x^{11/2}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + c*x^2)*(A + B*x),x)

[Out]

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

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sympy [A]  time = 1.66, size = 46, normalized size = 1.02 \begin {gather*} \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} \end {gather*}

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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