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

Optimal. Leaf size=55 \[ \frac {2}{9} x^{9/2} (a B+A b)+\frac {2}{7} a A x^{7/2}+\frac {2}{11} x^{11/2} (A c+b B)+\frac {2}{13} B c x^{13/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{9} x^{9/2} (a B+A b)+\frac {2}{7} a A x^{7/2}+\frac {2}{11} x^{11/2} (A c+b B)+\frac {2}{13} B c x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 48, normalized size = 0.87 \begin {gather*} \frac {2 x^{7/2} (143 a (9 A+7 B x)+7 x (13 A (11 b+9 c x)+9 B x (13 b+11 c x)))}{9009} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(143*a*(9*A + 7*B*x) + 7*x*(13*A*(11*b + 9*c*x) + 9*B*x*(13*b + 11*c*x))))/9009

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 59, normalized size = 1.07 \begin {gather*} \frac {2 \left (1287 a A x^{7/2}+1001 a B x^{9/2}+1001 A b x^{9/2}+819 A c x^{11/2}+819 b B x^{11/2}+693 B c x^{13/2}\right )}{9009} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1287*a*A*x^(7/2) + 1001*A*b*x^(9/2) + 1001*a*B*x^(9/2) + 819*b*B*x^(11/2) + 819*A*c*x^(11/2) + 693*B*c*x^(
13/2)))/9009

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 44, normalized size = 0.80 \begin {gather*} \frac {2}{9009} \, {\left (693 \, B c x^{6} + 819 \, {\left (B b + A c\right )} x^{5} + 1287 \, A a x^{3} + 1001 \, {\left (B a + A b\right )} x^{4}\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+a),x, algorithm="fricas")

[Out]

2/9009*(693*B*c*x^6 + 819*(B*b + A*c)*x^5 + 1287*A*a*x^3 + 1001*(B*a + A*b)*x^4)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 43, normalized size = 0.78 \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} \, B a x^{\frac {9}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} + \frac {2}{7} \, A a x^{\frac {7}{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+a),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*B*a*x^(9/2) + 2/9*A*b*x^(9/2) + 2/7*A*a*x^(7/2
)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 42, normalized size = 0.76 \begin {gather*} \frac {2 \left (693 B c \,x^{3}+819 A c \,x^{2}+819 B b \,x^{2}+1001 A b x +1001 B a x +1287 A a \right ) x^{\frac {7}{2}}}{9009} \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+a),x)

[Out]

2/9009*x^(7/2)*(693*B*c*x^3+819*A*c*x^2+819*B*b*x^2+1001*A*b*x+1001*B*a*x+1287*A*a)

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 39, normalized size = 0.71 \begin {gather*} \frac {2}{13} \, B c x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b + A c\right )} x^{\frac {11}{2}} + \frac {2}{7} \, A a x^{\frac {7}{2}} + \frac {2}{9} \, {\left (B a + A b\right )} 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+a),x, algorithm="maxima")

[Out]

2/13*B*c*x^(13/2) + 2/11*(B*b + A*c)*x^(11/2) + 2/7*A*a*x^(7/2) + 2/9*(B*a + A*b)*x^(9/2)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 41, normalized size = 0.75 \begin {gather*} x^{9/2}\,\left (\frac {2\,A\,b}{9}+\frac {2\,B\,a}{9}\right )+x^{11/2}\,\left (\frac {2\,A\,c}{11}+\frac {2\,B\,b}{11}\right )+\frac {2\,A\,a\,x^{7/2}}{7}+\frac {2\,B\,c\,x^{13/2}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(9/2)*((2*A*b)/9 + (2*B*a)/9) + x^(11/2)*((2*A*c)/11 + (2*B*b)/11) + (2*A*a*x^(7/2))/7 + (2*B*c*x^(13/2))/13

________________________________________________________________________________________

sympy [A]  time = 3.97, size = 70, normalized size = 1.27 \begin {gather*} \frac {2 A a x^{\frac {7}{2}}}{7} + \frac {2 A b x^{\frac {9}{2}}}{9} + \frac {2 A c x^{\frac {11}{2}}}{11} + \frac {2 B a x^{\frac {9}{2}}}{9} + \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+a),x)

[Out]

2*A*a*x**(7/2)/7 + 2*A*b*x**(9/2)/9 + 2*A*c*x**(11/2)/11 + 2*B*a*x**(9/2)/9 + 2*B*b*x**(11/2)/11 + 2*B*c*x**(1
3/2)/13

________________________________________________________________________________________