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

Optimal. Leaf size=85 \[ \frac {2}{11} A b^3 x^{11/2}+\frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{17} c^2 x^{17/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{19} B c^3 x^{19/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} \frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{11} A b^3 x^{11/2}+\frac {2}{17} c^2 x^{17/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{19} B c^3 x^{19/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(11/2))/11 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (2*b*c*(b*B + A*c)*x^(15/2))/5 + (2*c^2*(3*b*B + A
*c)*x^(17/2))/17 + (2*B*c^3*x^(19/2))/19

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 (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{9/2}+b^2 (b B+3 A c) x^{11/2}+3 b c (b B+A c) x^{13/2}+c^2 (3 b B+A c) x^{15/2}+B c^3 x^{17/2}\right ) \, dx\\ &=\frac {2}{11} A b^3 x^{11/2}+\frac {2}{13} b^2 (b B+3 A c) x^{13/2}+\frac {2}{5} b c (b B+A c) x^{15/2}+\frac {2}{17} c^2 (3 b B+A c) x^{17/2}+\frac {2}{19} B c^3 x^{19/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{11/2} (b+c x)^4-\frac {x^{11/2} \left (1105 b^3+2805 b^2 c x+2431 b c^2 x^2+715 c^3 x^3\right ) (11 b B-19 A c)}{12155}\right )}{19 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(11/2)*(b + c*x)^4 - ((11*b*B - 19*A*c)*x^(11/2)*(1105*b^3 + 2805*b^2*c*x + 2431*b*c^2*x^2 + 715*c^3*x
^3))/12155))/(19*c)

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IntegrateAlgebraic [A]  time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (20995 A b^3 x^{11/2}+53295 A b^2 c x^{13/2}+46189 A b c^2 x^{15/2}+13585 A c^3 x^{17/2}+17765 b^3 B x^{13/2}+46189 b^2 B c x^{15/2}+40755 b B c^2 x^{17/2}+12155 B c^3 x^{19/2}\right )}{230945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(20995*A*b^3*x^(11/2) + 17765*b^3*B*x^(13/2) + 53295*A*b^2*c*x^(13/2) + 46189*b^2*B*c*x^(15/2) + 46189*A*b*
c^2*x^(15/2) + 40755*b*B*c^2*x^(17/2) + 13585*A*c^3*x^(17/2) + 12155*B*c^3*x^(19/2)))/230945

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fricas [A]  time = 0.41, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{230945} \, {\left (12155 \, B c^{3} x^{9} + 20995 \, A b^{3} x^{5} + 13585 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + 46189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + 17765 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\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+b*x)^3,x, algorithm="fricas")

[Out]

2/230945*(12155*B*c^3*x^9 + 20995*A*b^3*x^5 + 13585*(3*B*b*c^2 + A*c^3)*x^8 + 46189*(B*b^2*c + A*b*c^2)*x^7 +
17765*(B*b^3 + 3*A*b^2*c)*x^6)*sqrt(x)

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giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b c^{2} x^{\frac {17}{2}} + \frac {2}{17} \, A c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, A b c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A b^{2} c x^{\frac {13}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*c^3*x^(19/2) + 6/17*B*b*c^2*x^(17/2) + 2/17*A*c^3*x^(17/2) + 2/5*B*b^2*c*x^(15/2) + 2/5*A*b*c^2*x^(15/2
) + 2/13*B*b^3*x^(13/2) + 6/13*A*b^2*c*x^(13/2) + 2/11*A*b^3*x^(11/2)

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maple [A]  time = 0.06, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (12155 B \,c^{3} x^{4}+13585 A \,c^{3} x^{3}+40755 B b \,c^{2} x^{3}+46189 A b \,c^{2} x^{2}+46189 B \,b^{2} c \,x^{2}+53295 A \,b^{2} c x +17765 B \,b^{3} x +20995 A \,b^{3}\right ) x^{\frac {11}{2}}}{230945} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/230945*x^(11/2)*(12155*B*c^3*x^4+13585*A*c^3*x^3+40755*B*b*c^2*x^3+46189*A*b*c^2*x^2+46189*B*b^2*c*x^2+53295
*A*b^2*c*x+17765*B*b^3*x+20995*A*b^3)

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maxima [A]  time = 0.57, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {2}{17} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {17}{2}} + \frac {2}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{13} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {13}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*B*c^3*x^(19/2) + 2/11*A*b^3*x^(11/2) + 2/17*(3*B*b*c^2 + A*c^3)*x^(17/2) + 2/5*(B*b^2*c + A*b*c^2)*x^(15/
2) + 2/13*(B*b^3 + 3*A*b^2*c)*x^(13/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,b^3}{13}+\frac {6\,A\,c\,b^2}{13}\right )+x^{17/2}\,\left (\frac {2\,A\,c^3}{17}+\frac {6\,B\,b\,c^2}{17}\right )+\frac {2\,A\,b^3\,x^{11/2}}{11}+\frac {2\,B\,c^3\,x^{19/2}}{19}+\frac {2\,b\,c\,x^{15/2}\,\left (A\,c+B\,b\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(13/2)*((2*B*b^3)/13 + (6*A*b^2*c)/13) + x^(17/2)*((2*A*c^3)/17 + (6*B*b*c^2)/17) + (2*A*b^3*x^(11/2))/11 +
(2*B*c^3*x^(19/2))/19 + (2*b*c*x^(15/2)*(A*c + B*b))/5

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sympy [A]  time = 8.27, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {6 A b^{2} c x^{\frac {13}{2}}}{13} + \frac {2 A b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {17}{2}}}{17} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B c^{3} x^{\frac {19}{2}}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**3*x**(11/2)/11 + 6*A*b**2*c*x**(13/2)/13 + 2*A*b*c**2*x**(15/2)/5 + 2*A*c**3*x**(17/2)/17 + 2*B*b**3*x*
*(13/2)/13 + 2*B*b**2*c*x**(15/2)/5 + 6*B*b*c**2*x**(17/2)/17 + 2*B*c**3*x**(19/2)/19

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