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

Optimal. Leaf size=113 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{11} x^{11/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{17} B c^2 x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.0596836, 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} \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{11} x^{11/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{17} B c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0990822, size = 102, normalized size = 0.9 \[ \frac{2 x^{7/2} \left (12155 a^2 (9 A+7 B x)+1190 a x (13 A (11 b+9 c x)+9 B x (13 b+11 c x))+21 x^2 \left (17 A \left (195 b^2+330 b c x+143 c^2 x^2\right )+11 B x \left (255 b^2+442 b c x+195 c^2 x^2\right )\right )\right )}{765765} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(12155*a^2*(9*A + 7*B*x) + 1190*a*x*(13*A*(11*b + 9*c*x) + 9*B*x*(13*b + 11*c*x)) + 21*x^2*(17*A*(1
95*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))))/765765

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Maple [A]  time = 0.006, size = 102, normalized size = 0.9 \begin{align*}{\frac{90090\,B{c}^{2}{x}^{5}+102102\,A{c}^{2}{x}^{4}+204204\,B{x}^{4}bc+235620\,A{x}^{3}bc+235620\,aBc{x}^{3}+117810\,{b}^{2}B{x}^{3}+278460\,aAc{x}^{2}+139230\,A{b}^{2}{x}^{2}+278460\,B{x}^{2}ab+340340\,aAbx+170170\,{a}^{2}Bx+218790\,A{a}^{2}}{765765}{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)^2,x)

[Out]

2/765765*x^(7/2)*(45045*B*c^2*x^5+51051*A*c^2*x^4+102102*B*b*c*x^4+117810*A*b*c*x^3+117810*B*a*c*x^3+58905*B*b
^2*x^3+139230*A*a*c*x^2+69615*A*b^2*x^2+139230*B*a*b*x^2+170170*A*a*b*x+85085*B*a^2*x+109395*A*a^2)

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Maxima [A]  time = 1.10861, size = 126, normalized size = 1.12 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{13} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{13}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.940941, size = 261, normalized size = 2.31 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B c^{2} x^{8} + 51051 \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + 58905 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{6} + 109395 \, A a^{2} x^{3} + 69615 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + 85085 \,{\left (B a^{2} + 2 \, A a 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)^2,x, algorithm="fricas")

[Out]

2/765765*(45045*B*c^2*x^8 + 51051*(2*B*b*c + A*c^2)*x^7 + 58905*(B*b^2 + 2*(B*a + A*b)*c)*x^6 + 109395*A*a^2*x
^3 + 69615*(2*B*a*b + A*b^2 + 2*A*a*c)*x^5 + 85085*(B*a^2 + 2*A*a*b)*x^4)*sqrt(x)

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Sympy [A]  time = 15.9874, size = 162, normalized size = 1.43 \begin{align*} \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{4 A b c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B b c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \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)**2,x)

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*b*x**(9/2)/9 + 4*A*a*c*x**(11/2)/11 + 2*A*b**2*x**(11/2)/11 + 4*A*b*c*x**(13/2)/13
+ 2*A*c**2*x**(15/2)/15 + 2*B*a**2*x**(9/2)/9 + 4*B*a*b*x**(11/2)/11 + 4*B*a*c*x**(13/2)/13 + 2*B*b**2*x**(13
/2)/13 + 4*B*b*c*x**(15/2)/15 + 2*B*c**2*x**(17/2)/17

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Giac [A]  time = 1.21022, size = 139, normalized size = 1.23 \begin{align*} \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B b c x^{\frac{15}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{13} \, A b c x^{\frac{13}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} 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)^2,x, algorithm="giac")

[Out]

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