3.1000 \(\int x^{5/2} (A+B x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=182 \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{21} B c^3 x^{21/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(11/2))/11 + (2*(3*a*B*
(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13/2))/13 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 +
(6*c*(b^2*B + A*b*c + a*B*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

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Rubi [A]  time = 0.115716, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {765} \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{21} B c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(11/2))/11 + (2*(3*a*B*
(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13/2))/13 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 +
(6*c*(b^2*B + A*b*c + a*B*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 (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{7/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{9/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{11/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{13/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{15/2}+c^2 (3 b B+A c) x^{17/2}+B c^3 x^{19/2}\right ) \, dx\\ &=\frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 (3 A b+a B) x^{9/2}+\frac{6}{11} a \left (a b B+A \left (b^2+a c\right )\right ) x^{11/2}+\frac{2}{13} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{13/2}+\frac{2}{15} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{15/2}+\frac{6}{17} c \left (b^2 B+A b c+a B c\right ) 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*}

Mathematica [A]  time = 0.225599, size = 178, normalized size = 0.98 \[ \frac{2 x^{7/2} \left (33915 a^2 x (13 A (11 b+9 c x)+9 B x (13 b+11 c x))+230945 a^3 (9 A+7 B x)+1197 a x^2 \left (17 A \left (195 b^2+330 b c x+143 c^2 x^2\right )+11 B x \left (255 b^2+442 b c x+195 c^2 x^2\right )\right )+33 x^3 \left (21 A \left (4199 b^2 c x+1615 b^3+3705 b c^2 x^2+1105 c^3 x^3\right )+13 B x \left (5985 b^2 c x+2261 b^3+5355 b c^2 x^2+1615 c^3 x^3\right )\right )\right )}{14549535} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(230945*a^3*(9*A + 7*B*x) + 33915*a^2*x*(13*A*(11*b + 9*c*x) + 9*B*x*(13*b + 11*c*x)) + 1197*a*x^2*
(17*A*(195*b^2 + 330*b*c*x + 143*c^2*x^2) + 11*B*x*(255*b^2 + 442*b*c*x + 195*c^2*x^2)) + 33*x^3*(21*A*(1615*b
^3 + 4199*b^2*c*x + 3705*b*c^2*x^2 + 1105*c^3*x^3) + 13*B*x*(2261*b^3 + 5985*b^2*c*x + 5355*b*c^2*x^2 + 1615*c
^3*x^3))))/14549535

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Maple [A]  time = 0.006, size = 192, normalized size = 1.1 \begin{align*}{\frac{1385670\,B{c}^{3}{x}^{7}+1531530\,A{c}^{3}{x}^{6}+4594590\,B{x}^{6}b{c}^{2}+5135130\,A{x}^{5}b{c}^{2}+5135130\,aB{c}^{2}{x}^{5}+5135130\,B{x}^{5}{b}^{2}c+5819814\,aA{c}^{2}{x}^{4}+5819814\,A{x}^{4}{b}^{2}c+11639628\,B{x}^{4}abc+1939938\,B{x}^{4}{b}^{3}[phone]\,A{x}^{3}abc+2238390\,A{b}^{3}{x}^{3}+6715170\,{a}^{2}Bc{x}^{3}+6715170\,B{x}^{3}a{b}^{2}+7936110\,{a}^{2}Ac{x}^{2}+7936110\,A{x}^{2}a{b}^{2}+7936110\,B{x}^{2}{a}^{2}b+9699690\,A{a}^{2}bx+3233230\,{a}^{3}Bx+4157010\,A{a}^{3}}{14549535}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/14549535*x^(7/2)*(692835*B*c^3*x^7+765765*A*c^3*x^6+2297295*B*b*c^2*x^6+2567565*A*b*c^2*x^5+2567565*B*a*c^2*
x^5+2567565*B*b^2*c*x^5+2909907*A*a*c^2*x^4+2909907*A*b^2*c*x^4+5819814*B*a*b*c*x^4+969969*B*b^3*x^4+6715170*A
*a*b*c*x^3+1119195*A*b^3*x^3+3357585*B*a^2*c*x^3+3357585*B*a*b^2*x^3+3968055*A*a^2*c*x^2+3968055*A*a*b^2*x^2+3
968055*B*a^2*b*x^2+4849845*A*a^2*b*x+1616615*B*a^3*x+2078505*A*a^3)

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Maxima [A]  time = 1.11218, size = 224, normalized size = 1.23 \begin{align*} \frac{2}{21} \, B c^{3} x^{\frac{21}{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 +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/21*B*c^3*x^(21/2) + 2/19*(3*B*b*c^2 + A*c^3)*x^(19/2) + 6/17*(B*b^2*c + (B*a + A*b)*c^2)*x^(17/2) + 2/15*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(15/2) + 2/7*A*a^3*x^(7/2) + 2/13*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2
*A*a*b)*c)*x^(13/2) + 6/11*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(11/2) + 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]  time = 1.05124, size = 443, normalized size = 2.43 \begin{align*} \frac{2}{14549535} \,{\left (692835 \, B c^{3} x^{10} + 765765 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 2567565 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{8} + 969969 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + 2078505 \, A a^{3} x^{3} + 1119195 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 3968055 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + 1616615 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

2/14549535*(692835*B*c^3*x^10 + 765765*(3*B*b*c^2 + A*c^3)*x^9 + 2567565*(B*b^2*c + (B*a + A*b)*c^2)*x^8 + 969
969*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^7 + 2078505*A*a^3*x^3 + 1119195*(3*B*a*b^2 + A*b^3 + 3*(B*a^
2 + 2*A*a*b)*c)*x^6 + 3968055*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + 1616615*(B*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [A]  time = 29.8345, size = 294, normalized size = 1.62 \begin{align*} \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a^{2} c x^{\frac{11}{2}}}{11} + \frac{6 A a b^{2} x^{\frac{11}{2}}}{11} + \frac{12 A a b c x^{\frac{13}{2}}}{13} + \frac{2 A a c^{2} x^{\frac{15}{2}}}{5} + \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 a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B a b c x^{\frac{15}{2}}}{5} + \frac{6 B a c^{2} x^{\frac{17}{2}}}{17} + \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x+a)**3,x)

[Out]

2*A*a**3*x**(7/2)/7 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a**2*c*x**(11/2)/11 + 6*A*a*b**2*x**(11/2)/11 + 12*A*a*b*c*x
**(13/2)/13 + 2*A*a*c**2*x**(15/2)/5 + 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)/1
7 + 2*A*c**3*x**(19/2)/19 + 2*B*a**3*x**(9/2)/9 + 6*B*a**2*b*x**(11/2)/11 + 6*B*a**2*c*x**(13/2)/13 + 6*B*a*b*
*2*x**(13/2)/13 + 4*B*a*b*c*x**(15/2)/5 + 6*B*a*c**2*x**(17/2)/17 + 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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Giac [A]  time = 1.25924, size = 261, normalized size = 1.43 \begin{align*} \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} \, B a c^{2} 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{4}{5} \, B a b c x^{\frac{15}{2}} + \frac{2}{5} \, A b^{2} c x^{\frac{15}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{12}{13} \, A a b c x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^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*B*a*c^2*x^(17
/2) + 6/17*A*b*c^2*x^(17/2) + 2/15*B*b^3*x^(15/2) + 4/5*B*a*b*c*x^(15/2) + 2/5*A*b^2*c*x^(15/2) + 2/5*A*a*c^2*
x^(15/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 6/13*B*a^2*c*x^(13/2) + 12/13*A*a*b*c*x^(13/2) + 6/11
*B*a^2*b*x^(11/2) + 6/11*A*a*b^2*x^(11/2) + 6/11*A*a^2*c*x^(11/2) + 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) +
2/7*A*a^3*x^(7/2)