3.999 \(\int x^{7/2} (A+B x) (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=182 \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{17} x^{17/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{19} c x^{19/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{15} x^{15/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{13} a x^{13/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{2}{23} B c^3 x^{23/2} \]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(13/2))/13 + (2*(3*a*
B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(15/2))/15 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(17/2))/17
+ (6*c*(b^2*B + A*b*c + a*B*c)*x^(19/2))/19 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(23/2))/23

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Rubi [A]  time = 0.124834, 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} \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{17} x^{17/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{6}{19} c x^{19/2} \left (a B c+A b c+b^2 B\right )+\frac{2}{15} x^{15/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{6}{13} a x^{13/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{2}{23} B c^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*(a + b*x + c*x^2)^3,x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*(a*b*B + A*(b^2 + a*c))*x^(13/2))/13 + (2*(3*a*
B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(15/2))/15 + (2*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(17/2))/17
+ (6*c*(b^2*B + A*b*c + a*B*c)*x^(19/2))/19 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(23/2))/23

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^{7/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{9/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{11/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{13/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{15/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{17/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{9} a^3 A x^{9/2}+\frac{2}{11} a^2 (3 A b+a B) x^{11/2}+\frac{6}{13} a \left (a b B+A \left (b^2+a c\right )\right ) x^{13/2}+\frac{2}{15} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{15/2}+\frac{2}{17} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{17/2}+\frac{6}{19} c \left (b^2 B+A b c+a B c\right ) x^{19/2}+\frac{2}{21} c^2 (3 b B+A c) x^{21/2}+\frac{2}{23} B c^3 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.208605, size = 178, normalized size = 0.98 \[ \frac{2 x^{9/2} \left (468027 a^2 x (15 A (13 b+11 c x)+11 B x (15 b+13 c x))+3380195 a^3 (11 A+9 B x)+15939 a x^2 \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )+429 x^3 \left (23 A \left (5985 b^2 c x+2261 b^3+5355 b c^2 x^2+1615 c^3 x^3\right )+15 B x \left (8211 b^2 c x+3059 b^3+7429 b c^2 x^2+2261 c^3 x^3\right )\right )\right )}{334639305} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x)*(a + b*x + c*x^2)^3,x]

[Out]

(2*x^(9/2)*(3380195*a^3*(11*A + 9*B*x) + 468027*a^2*x*(15*A*(13*b + 11*c*x) + 11*B*x*(15*b + 13*c*x)) + 15939*
a*x^2*(19*A*(255*b^2 + 442*b*c*x + 195*c^2*x^2) + 13*B*x*(323*b^2 + 570*b*c*x + 255*c^2*x^2)) + 429*x^3*(23*A*
(2261*b^3 + 5985*b^2*c*x + 5355*b*c^2*x^2 + 1615*c^3*x^3) + 15*B*x*(3059*b^3 + 8211*b^2*c*x + 7429*b*c^2*x^2 +
2261*c^3*x^3))))/334639305

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Maple [A]  time = 0.006, size = 192, normalized size = 1.1 \begin{align*}{\frac{29099070\,B{c}^{3}{x}^{7}[phone]\,A{c}^{3}{x}^{6}[phone]\,B{x}^{6}b{c}^{2}[phone]\,A{x}^{5}b{c}^{2}[phone]\,aB{c}^{2}{x}^{5}[phone]\,B{x}^{5}{b}^{2}c+118107990\,aA{c}^{2}{x}^{4}[phone]\,A{x}^{4}{b}^{2}c+236215980\,B{x}^{4}abc+39369330\,B{x}^{4}{b}^{3}[phone]\,A{x}^{3}abc+44618574\,A{b}^{3}{x}^{3}[phone]\,{a}^{2}Bc{x}^{3}[phone]\,B{x}^{3}a{b}^{2}[phone]\,{a}^{2}Ac{x}^{2}[phone]\,A{x}^{2}a{b}^{2}[phone]\,B{x}^{2}{a}^{2}b+182530530\,A{a}^{2}bx+60843510\,{a}^{3}Bx+74364290\,A{a}^{3}}{334639305}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*(c*x^2+b*x+a)^3,x)

[Out]

2/334639305*x^(9/2)*(14549535*B*c^3*x^7+15935205*A*c^3*x^6+47805615*B*b*c^2*x^6+52837785*A*b*c^2*x^5+52837785*
B*a*c^2*x^5+52837785*B*b^2*c*x^5+59053995*A*a*c^2*x^4+59053995*A*b^2*c*x^4+118107990*B*a*b*c*x^4+19684665*B*b^
3*x^4+133855722*A*a*b*c*x^3+22309287*A*b^3*x^3+66927861*B*a^2*c*x^3+66927861*B*a*b^2*x^3+77224455*A*a^2*c*x^2+
77224455*A*a*b^2*x^2+77224455*B*a^2*b*x^2+91265265*A*a^2*b*x+30421755*B*a^3*x+37182145*A*a^3)

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Maxima [A]  time = 0.984727, size = 224, normalized size = 1.23 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{21} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{21}{2}} + \frac{6}{19} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{19}{2}} + \frac{2}{17} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/23*B*c^3*x^(23/2) + 2/21*(3*B*b*c^2 + A*c^3)*x^(21/2) + 6/19*(B*b^2*c + (B*a + A*b)*c^2)*x^(19/2) + 2/17*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(17/2) + 2/9*A*a^3*x^(9/2) + 2/15*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2
*A*a*b)*c)*x^(15/2) + 6/13*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(13/2) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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Fricas [A]  time = 1.03096, size = 460, normalized size = 2.53 \begin{align*} \frac{2}{334639305} \,{\left (14549535 \, B c^{3} x^{11} + 15935205 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 52837785 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{9} + 19684665 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 37182145 \, A a^{3} x^{4} + 22309287 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + 77224455 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{6} + 30421755 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

2/334639305*(14549535*B*c^3*x^11 + 15935205*(3*B*b*c^2 + A*c^3)*x^10 + 52837785*(B*b^2*c + (B*a + A*b)*c^2)*x^
9 + 19684665*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 37182145*A*a^3*x^4 + 22309287*(3*B*a*b^2 + A*b^
3 + 3*(B*a^2 + 2*A*a*b)*c)*x^7 + 77224455*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^6 + 30421755*(B*a^3 + 3*A*a^2*b)*x^5
)*sqrt(x)

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Sympy [A]  time = 42.575, size = 294, normalized size = 1.62 \begin{align*} \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{6 A a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 A a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A a b c x^{\frac{15}{2}}}{5} + \frac{6 A a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 A b^{3} x^{\frac{15}{2}}}{15} + \frac{6 A b^{2} c x^{\frac{17}{2}}}{17} + \frac{6 A b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 A c^{3} x^{\frac{21}{2}}}{21} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} c x^{\frac{15}{2}}}{5} + \frac{2 B a b^{2} x^{\frac{15}{2}}}{5} + \frac{12 B a b c x^{\frac{17}{2}}}{17} + \frac{6 B a c^{2} x^{\frac{19}{2}}}{19} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} + \frac{6 B b^{2} c x^{\frac{19}{2}}}{19} + \frac{2 B b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 B c^{3} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)*(c*x**2+b*x+a)**3,x)

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a**2*c*x**(13/2)/13 + 6*A*a*b**2*x**(13/2)/13 + 4*A*a*b*c*
x**(15/2)/5 + 6*A*a*c**2*x**(17/2)/17 + 2*A*b**3*x**(15/2)/15 + 6*A*b**2*c*x**(17/2)/17 + 6*A*b*c**2*x**(19/2)
/19 + 2*A*c**3*x**(21/2)/21 + 2*B*a**3*x**(11/2)/11 + 6*B*a**2*b*x**(13/2)/13 + 2*B*a**2*c*x**(15/2)/5 + 2*B*a
*b**2*x**(15/2)/5 + 12*B*a*b*c*x**(17/2)/17 + 6*B*a*c**2*x**(19/2)/19 + 2*B*b**3*x**(17/2)/17 + 6*B*b**2*c*x**
(19/2)/19 + 2*B*b*c**2*x**(21/2)/7 + 2*B*c**3*x**(23/2)/23

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Giac [A]  time = 1.18743, size = 261, normalized size = 1.43 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{7} \, B b c^{2} x^{\frac{21}{2}} + \frac{2}{21} \, A c^{3} x^{\frac{21}{2}} + \frac{6}{19} \, B b^{2} c x^{\frac{19}{2}} + \frac{6}{19} \, B a c^{2} x^{\frac{19}{2}} + \frac{6}{19} \, A b c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{12}{17} \, B a b c x^{\frac{17}{2}} + \frac{6}{17} \, A b^{2} c x^{\frac{17}{2}} + \frac{6}{17} \, A a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{2}{5} \, B a^{2} c x^{\frac{15}{2}} + \frac{4}{5} \, A a b c x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{6}{13} \, A a^{2} c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/23*B*c^3*x^(23/2) + 2/7*B*b*c^2*x^(21/2) + 2/21*A*c^3*x^(21/2) + 6/19*B*b^2*c*x^(19/2) + 6/19*B*a*c^2*x^(19/
2) + 6/19*A*b*c^2*x^(19/2) + 2/17*B*b^3*x^(17/2) + 12/17*B*a*b*c*x^(17/2) + 6/17*A*b^2*c*x^(17/2) + 6/17*A*a*c
^2*x^(17/2) + 2/5*B*a*b^2*x^(15/2) + 2/15*A*b^3*x^(15/2) + 2/5*B*a^2*c*x^(15/2) + 4/5*A*a*b*c*x^(15/2) + 6/13*
B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13/2) + 6/13*A*a^2*c*x^(13/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2)
+ 2/9*A*a^3*x^(9/2)