3.2.41 \(\int x^{5/2} (A+B x) (b x+c x^2) \, dx\)

Optimal. Leaf size=39 \[ \frac {2}{11} x^{11/2} (A c+b B)+\frac {2}{9} A b x^{9/2}+\frac {2}{13} B c x^{13/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {2}{11} x^{11/2} (A c+b B)+\frac {2}{9} A b x^{9/2}+\frac {2}{13} B c x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

(2*A*b*x^(9/2))/9 + (2*(b*B + A*c)*x^(11/2))/11 + (2*B*c*x^(13/2))/13

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^{5/2} (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{7/2}+(b B+A c) x^{9/2}+B c x^{11/2}\right ) \, dx\\ &=\frac {2}{9} A b x^{9/2}+\frac {2}{11} (b B+A c) x^{11/2}+\frac {2}{13} B c x^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.85 \begin {gather*} \frac {2 x^{9/2} (13 A (11 b+9 c x)+9 B x (13 b+11 c x))}{1287} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

(2*x^(9/2)*(13*A*(11*b + 9*c*x) + 9*B*x*(13*b + 11*c*x)))/1287

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 1.05 \begin {gather*} \frac {2 \left (143 A b x^{9/2}+117 A c x^{11/2}+117 b B x^{11/2}+99 B c x^{13/2}\right )}{1287} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)*(A + B*x)*(b*x + c*x^2),x]

[Out]

(2*(143*A*b*x^(9/2) + 117*b*B*x^(11/2) + 117*A*c*x^(11/2) + 99*B*c*x^(13/2)))/1287

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fricas [A]  time = 0.40, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{1287} \, {\left (99 \, B c x^{6} + 143 \, A b x^{4} + 117 \, {\left (B b + A c\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

2/1287*(99*B*c*x^6 + 143*A*b*x^4 + 117*(B*b + A*c)*x^5)*sqrt(x)

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giac [A]  time = 0.19, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{13} \, B c x^{\frac {13}{2}} + \frac {2}{11} \, B b x^{\frac {11}{2}} + \frac {2}{11} \, A c x^{\frac {11}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/13*B*c*x^(13/2) + 2/11*B*b*x^(11/2) + 2/11*A*c*x^(11/2) + 2/9*A*b*x^(9/2)

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maple [A]  time = 0.05, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (99 B c \,x^{2}+117 A c x +117 B b x +143 A b \right ) x^{\frac {9}{2}}}{1287} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(c*x^2+b*x),x)

[Out]

2/1287*x^(9/2)*(99*B*c*x^2+117*A*c*x+117*B*b*x+143*A*b)

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maxima [A]  time = 0.56, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, B c x^{\frac {13}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} + \frac {2}{11} \, {\left (B b + A c\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

2/13*B*c*x^(13/2) + 2/9*A*b*x^(9/2) + 2/11*(B*b + A*c)*x^(11/2)

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mupad [B]  time = 0.04, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{9/2}\,\left (143\,A\,b+117\,A\,c\,x+117\,B\,b\,x+99\,B\,c\,x^2\right )}{1287} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x + c*x^2)*(A + B*x),x)

[Out]

(2*x^(9/2)*(143*A*b + 117*A*c*x + 117*B*b*x + 99*B*c*x^2))/1287

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sympy [A]  time = 3.86, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {9}{2}}}{9} + \frac {2 A c x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {11}{2}}}{11} + \frac {2 B c x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x),x)

[Out]

2*A*b*x**(9/2)/9 + 2*A*c*x**(11/2)/11 + 2*B*b*x**(11/2)/11 + 2*B*c*x**(13/2)/13

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