3.2.59 \(\int x^{5/2} (A+B x) (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{13} A b^3 x^{13/2}+\frac {2}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{21} B c^3 x^{21/2} \]

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Rubi [A]  time = 0.05, 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}{15} b^2 x^{15/2} (3 A c+b B)+\frac {2}{13} A b^3 x^{13/2}+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{21} B c^3 x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(15/2))/15 + (6*b*c*(b*B + A*c)*x^(17/2))/17 + (2*c^2*(3*b*B +
A*c)*x^(19/2))/19 + (2*B*c^3*x^(21/2))/21

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 )^3 \, dx &=\int \left (A b^3 x^{11/2}+b^2 (b B+3 A c) x^{13/2}+3 b c (b B+A c) x^{15/2}+c^2 (3 b B+A c) x^{17/2}+B c^3 x^{19/2}\right ) \, dx\\ &=\frac {2}{13} A b^3 x^{13/2}+\frac {2}{15} b^2 (b B+3 A c) x^{15/2}+\frac {6}{17} b c (b B+A c) x^{17/2}+\frac {2}{19} c^2 (3 b B+A c) x^{19/2}+\frac {2}{21} B c^3 x^{21/2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{13/2} (b+c x)^4-\frac {x^{13/2} \left (1615 b^3+4199 b^2 c x+3705 b c^2 x^2+1105 c^3 x^3\right ) (13 b B-21 A c)}{20995}\right )}{21 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*(B*x^(13/2)*(b + c*x)^4 - ((13*b*B - 21*A*c)*x^(13/2)*(1615*b^3 + 4199*b^2*c*x + 3705*b*c^2*x^2 + 1105*c^3*
x^3))/20995))/(21*c)

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IntegrateAlgebraic [A]  time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (33915 A b^3 x^{13/2}+88179 A b^2 c x^{15/2}+77805 A b c^2 x^{17/2}+23205 A c^3 x^{19/2}+29393 b^3 B x^{15/2}+77805 b^2 B c x^{17/2}+69615 b B c^2 x^{19/2}+20995 B c^3 x^{21/2}\right )}{440895} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*(33915*A*b^3*x^(13/2) + 29393*b^3*B*x^(15/2) + 88179*A*b^2*c*x^(15/2) + 77805*b^2*B*c*x^(17/2) + 77805*A*b*
c^2*x^(17/2) + 69615*b*B*c^2*x^(19/2) + 23205*A*c^3*x^(19/2) + 20995*B*c^3*x^(21/2)))/440895

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fricas [A]  time = 0.41, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{440895} \, {\left (20995 \, B c^{3} x^{10} + 33915 \, A b^{3} x^{6} + 23205 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 77805 \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + 29393 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7}\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)^3,x, algorithm="fricas")

[Out]

2/440895*(20995*B*c^3*x^10 + 33915*A*b^3*x^6 + 23205*(3*B*b*c^2 + A*c^3)*x^9 + 77805*(B*b^2*c + A*b*c^2)*x^8 +
29393*(B*b^3 + 3*A*b^2*c)*x^7)*sqrt(x)

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giac [A]  time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b c^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, A b c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{5} \, A b^{2} c x^{\frac {15}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{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)^3,x, algorithm="giac")

[Out]

2/21*B*c^3*x^(21/2) + 6/19*B*b*c^2*x^(19/2) + 2/19*A*c^3*x^(19/2) + 6/17*B*b^2*c*x^(17/2) + 6/17*A*b*c^2*x^(17
/2) + 2/15*B*b^3*x^(15/2) + 2/5*A*b^2*c*x^(15/2) + 2/13*A*b^3*x^(13/2)

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maple [A]  time = 0.05, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (20995 B \,c^{3} x^{4}+23205 A \,c^{3} x^{3}+69615 B b \,c^{2} x^{3}+77805 A b \,c^{2} x^{2}+77805 B \,b^{2} c \,x^{2}+88179 A \,b^{2} c x +29393 B \,b^{3} x +33915 A \,b^{3}\right ) x^{\frac {13}{2}}}{440895} \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)^3,x)

[Out]

2/440895*x^(13/2)*(20995*B*c^3*x^4+23205*A*c^3*x^3+69615*B*b*c^2*x^3+77805*A*b*c^2*x^2+77805*B*b^2*c*x^2+88179
*A*b^2*c*x+29393*B*b^3*x+33915*A*b^3)

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maxima [A]  time = 0.49, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{21} \, B c^{3} x^{\frac {21}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {2}{19} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {19}{2}} + \frac {6}{17} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {15}{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)^3,x, algorithm="maxima")

[Out]

2/21*B*c^3*x^(21/2) + 2/13*A*b^3*x^(13/2) + 2/19*(3*B*b*c^2 + A*c^3)*x^(19/2) + 6/17*(B*b^2*c + A*b*c^2)*x^(17
/2) + 2/15*(B*b^3 + 3*A*b^2*c)*x^(15/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{15/2}\,\left (\frac {2\,B\,b^3}{15}+\frac {2\,A\,c\,b^2}{5}\right )+x^{19/2}\,\left (\frac {2\,A\,c^3}{19}+\frac {6\,B\,b\,c^2}{19}\right )+\frac {2\,A\,b^3\,x^{13/2}}{13}+\frac {2\,B\,c^3\,x^{21/2}}{21}+\frac {6\,b\,c\,x^{17/2}\,\left (A\,c+B\,b\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x + c*x^2)^3*(A + B*x),x)

[Out]

x^(15/2)*((2*B*b^3)/15 + (2*A*b^2*c)/5) + x^(19/2)*((2*A*c^3)/19 + (6*B*b*c^2)/19) + (2*A*b^3*x^(13/2))/13 + (
2*B*c^3*x^(21/2))/21 + (6*b*c*x^(17/2)*(A*c + B*b))/17

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sympy [A]  time = 15.92, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} + \frac {6 B b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 B b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {21}{2}}}{21} \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)**3,x)

[Out]

2*A*b**3*x**(13/2)/13 + 2*A*b**2*c*x**(15/2)/5 + 6*A*b*c**2*x**(17/2)/17 + 2*A*c**3*x**(19/2)/19 + 2*B*b**3*x*
*(15/2)/15 + 6*B*b**2*c*x**(17/2)/17 + 6*B*b*c**2*x**(19/2)/19 + 2*B*c**3*x**(21/2)/21

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