3.10.6 \(\int x^{7/2} (A+B x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac {2}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} x^{13/2} (A c+b B)+\frac {2}{15} B c x^{15/2} \]

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Rubi [A]  time = 0.03, 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}{11} x^{11/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{13} x^{13/2} (A c+b B)+\frac {2}{15} B c x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*(a + b*x + c*x^2),x]

[Out]

(2*a*A*x^(9/2))/9 + (2*(A*b + a*B)*x^(11/2))/11 + (2*(b*B + A*c)*x^(13/2))/13 + (2*B*c*x^(15/2))/15

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^{7/2} (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A x^{7/2}+(A b+a B) x^{9/2}+(b B+A c) x^{11/2}+B c x^{13/2}\right ) \, dx\\ &=\frac {2}{9} a A x^{9/2}+\frac {2}{11} (A b+a B) x^{11/2}+\frac {2}{13} (b B+A c) x^{13/2}+\frac {2}{15} B c x^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 47, normalized size = 0.85 \begin {gather*} \frac {2 x^{9/2} \left (65 a (11 A+9 B x)+45 A x (13 b+11 c x)+33 B x^2 (15 b+13 c x)\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x)*(a + b*x + c*x^2),x]

[Out]

(2*x^(9/2)*(65*a*(11*A + 9*B*x) + 45*A*x*(13*b + 11*c*x) + 33*B*x^2*(15*b + 13*c*x)))/6435

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IntegrateAlgebraic [A]  time = 0.03, size = 59, normalized size = 1.07 \begin {gather*} \frac {2 \left (715 a A x^{9/2}+585 a B x^{11/2}+585 A b x^{11/2}+495 A c x^{13/2}+495 b B x^{13/2}+429 B c x^{15/2}\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(A + B*x)*(a + b*x + c*x^2),x]

[Out]

(2*(715*a*A*x^(9/2) + 585*A*b*x^(11/2) + 585*a*B*x^(11/2) + 495*b*B*x^(13/2) + 495*A*c*x^(13/2) + 429*B*c*x^(1
5/2)))/6435

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fricas [A]  time = 0.42, size = 44, normalized size = 0.80 \begin {gather*} \frac {2}{6435} \, {\left (429 \, B c x^{7} + 495 \, {\left (B b + A c\right )} x^{6} + 715 \, A a x^{4} + 585 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

2/6435*(429*B*c*x^7 + 495*(B*b + A*c)*x^6 + 715*A*a*x^4 + 585*(B*a + A*b)*x^5)*sqrt(x)

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giac [A]  time = 0.15, size = 43, normalized size = 0.78 \begin {gather*} \frac {2}{15} \, B c x^{\frac {15}{2}} + \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{13} \, A c x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{11} \, A b x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2/15*B*c*x^(15/2) + 2/13*B*b*x^(13/2) + 2/13*A*c*x^(13/2) + 2/11*B*a*x^(11/2) + 2/11*A*b*x^(11/2) + 2/9*A*a*x^
(9/2)

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maple [A]  time = 0.05, size = 42, normalized size = 0.76 \begin {gather*} \frac {2 \left (429 B c \,x^{3}+495 A c \,x^{2}+495 B b \,x^{2}+585 A b x +585 B a x +715 A a \right ) x^{\frac {9}{2}}}{6435} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*(c*x^2+b*x+a),x)

[Out]

2/6435*x^(9/2)*(429*B*c*x^3+495*A*c*x^2+495*B*b*x^2+585*A*b*x+585*B*a*x+715*A*a)

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maxima [A]  time = 0.60, size = 39, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, B c x^{\frac {15}{2}} + \frac {2}{13} \, {\left (B b + A c\right )} x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} + \frac {2}{11} \, {\left (B a + A b\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

2/15*B*c*x^(15/2) + 2/13*(B*b + A*c)*x^(13/2) + 2/9*A*a*x^(9/2) + 2/11*(B*a + A*b)*x^(11/2)

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mupad [B]  time = 1.26, size = 41, normalized size = 0.75 \begin {gather*} x^{11/2}\,\left (\frac {2\,A\,b}{11}+\frac {2\,B\,a}{11}\right )+x^{13/2}\,\left (\frac {2\,A\,c}{13}+\frac {2\,B\,b}{13}\right )+\frac {2\,A\,a\,x^{9/2}}{9}+\frac {2\,B\,c\,x^{15/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(A + B*x)*(a + b*x + c*x^2),x)

[Out]

x^(11/2)*((2*A*b)/11 + (2*B*a)/11) + x^(13/2)*((2*A*c)/13 + (2*B*b)/13) + (2*A*a*x^(9/2))/9 + (2*B*c*x^(15/2))
/15

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sympy [A]  time = 8.08, size = 70, normalized size = 1.27 \begin {gather*} \frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {11}{2}}}{11} + \frac {2 A c x^{\frac {13}{2}}}{13} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B b x^{\frac {13}{2}}}{13} + \frac {2 B c x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(9/2)/9 + 2*A*b*x**(11/2)/11 + 2*A*c*x**(13/2)/13 + 2*B*a*x**(11/2)/11 + 2*B*b*x**(13/2)/13 + 2*B*c*x
**(15/2)/15

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