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

Optimal. Leaf size=113 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{11} x^{11/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{9} x^{9/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{15} B c^2 x^{15/2} \]

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*(2*a*b*B + A*(b^2 + 2*a*c))*x^(9/2))/9 + (2*(b^2*B +
2*A*b*c + 2*a*B*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(15/2))/15

________________________________________________________________________________________

Rubi [A]  time = 0.0590619, antiderivative size = 113, 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} \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{11} x^{11/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{9} x^{9/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{15} B c^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.123023, size = 102, normalized size = 0.9 \[ \frac{2 x^{5/2} \left (1287 a^2 (7 A+5 B x)+130 a x (11 A (9 b+7 c x)+7 B x (11 b+9 c x))+7 x^2 \left (5 A \left (143 b^2+234 b c x+99 c^2 x^2\right )+3 B x \left (195 b^2+330 b c x+143 c^2 x^2\right )\right )\right )}{45045} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(1287*a^2*(7*A + 5*B*x) + 130*a*x*(11*A*(9*b + 7*c*x) + 7*B*x*(11*b + 9*c*x)) + 7*x^2*(5*A*(143*b^2
+ 234*b*c*x + 99*c^2*x^2) + 3*B*x*(195*b^2 + 330*b*c*x + 143*c^2*x^2))))/45045

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 102, normalized size = 0.9 \begin{align*}{\frac{6006\,B{c}^{2}{x}^{5}+6930\,A{c}^{2}{x}^{4}+13860\,B{x}^{4}bc+16380\,A{x}^{3}bc+16380\,aBc{x}^{3}+8190\,{b}^{2}B{x}^{3}+20020\,aAc{x}^{2}+10010\,A{b}^{2}{x}^{2}+20020\,B{x}^{2}ab+25740\,aAbx+12870\,{a}^{2}Bx+18018\,A{a}^{2}}{45045}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*x^(5/2)*(3003*B*c^2*x^5+3465*A*c^2*x^4+6930*B*b*c*x^4+8190*A*b*c*x^3+8190*B*a*c*x^3+4095*B*b^2*x^3+100
10*A*a*c*x^2+5005*A*b^2*x^2+10010*B*a*b*x^2+12870*A*a*b*x+6435*B*a^2*x+9009*A*a^2)

________________________________________________________________________________________

Maxima [A]  time = 1.04265, size = 126, normalized size = 1.12 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{13} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{11}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*B*c^2*x^(15/2) + 2/13*(2*B*b*c + A*c^2)*x^(13/2) + 2/11*(B*b^2 + 2*(B*a + A*b)*c)*x^(11/2) + 2/5*A*a^2*x^
(5/2) + 2/9*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(9/2) + 2/7*(B*a^2 + 2*A*a*b)*x^(7/2)

________________________________________________________________________________________

Fricas [A]  time = 1.02175, size = 250, normalized size = 2.21 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B c^{2} x^{7} + 3465 \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + 4095 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{5} + 9009 \, A a^{2} x^{2} + 5005 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{4} + 6435 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*B*c^2*x^7 + 3465*(2*B*b*c + A*c^2)*x^6 + 4095*(B*b^2 + 2*(B*a + A*b)*c)*x^5 + 9009*A*a^2*x^2 + 5
005*(2*B*a*b + A*b^2 + 2*A*a*c)*x^4 + 6435*(B*a^2 + 2*A*a*b)*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 7.29826, size = 162, normalized size = 1.43 \begin{align*} \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a b x^{\frac{7}{2}}}{7} + \frac{4 A a c x^{\frac{9}{2}}}{9} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{4 B a c x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(5/2)/5 + 4*A*a*b*x**(7/2)/7 + 4*A*a*c*x**(9/2)/9 + 2*A*b**2*x**(9/2)/9 + 4*A*b*c*x**(11/2)/11 + 2
*A*c**2*x**(13/2)/13 + 2*B*a**2*x**(7/2)/7 + 4*B*a*b*x**(9/2)/9 + 4*B*a*c*x**(11/2)/11 + 2*B*b**2*x**(11/2)/11
+ 4*B*b*c*x**(13/2)/13 + 2*B*c**2*x**(15/2)/15

________________________________________________________________________________________

Giac [A]  time = 1.1868, size = 139, normalized size = 1.23 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B b c x^{\frac{13}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, B a c x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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