3.1002 \(\int \sqrt{x} (A+B x) \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=182 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{2}{17} B c^3 x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.266349, 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 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{2}{17} B c^3 x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.8264, size = 206, normalized size = 1.13 \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 a^{2} x^{\frac{5}{2}} \left (3 A b + B a\right )}{5} + \frac{6 a x^{\frac{7}{2}} \left (A a c + A b^{2} + B a b\right )}{7} + \frac{2 c^{2} x^{\frac{15}{2}} \left (A c + 3 B b\right )}{15} + \frac{6 c x^{\frac{13}{2}} \left (A b c + B a c + B b^{2}\right )}{13} + x^{\frac{11}{2}} \left (\frac{6 A a c^{2}}{11} + \frac{6 A b^{2} c}{11} + \frac{12 B a b c}{11} + \frac{2 B b^{3}}{11}\right ) + x^{\frac{9}{2}} \left (\frac{4 A a b c}{3} + \frac{2 A b^{3}}{9} + \frac{2 B a^{2} c}{3} + \frac{2 B a b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3*x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.144181, size = 182, normalized size = 1. \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{2}{17} B c^3 x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 192, normalized size = 1.1 \[{\frac{90090\,B{c}^{3}{x}^{7}+102102\,A{c}^{3}{x}^{6}+306306\,B{x}^{6}b{c}^{2}+353430\,A{x}^{5}b{c}^{2}+353430\,aB{c}^{2}{x}^{5}+353430\,B{x}^{5}{b}^{2}c+417690\,aA{c}^{2}{x}^{4}+417690\,A{x}^{4}{b}^{2}c+835380\,B{x}^{4}abc+139230\,B{x}^{4}{b}^{3}+1021020\,A{x}^{3}abc+170170\,A{b}^{3}{x}^{3}+510510\,{a}^{2}Bc{x}^{3}+510510\,B{x}^{3}a{b}^{2}+656370\,{a}^{2}Ac{x}^{2}+656370\,A{x}^{2}a{b}^{2}+656370\,B{x}^{2}{a}^{2}b+918918\,A{a}^{2}bx+306306\,{a}^{3}Bx+510510\,A{a}^{3}}{765765}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^3*x^(1/2),x)

[Out]

2/765765*x^(3/2)*(45045*B*c^3*x^7+51051*A*c^3*x^6+153153*B*b*c^2*x^6+176715*A*b*
c^2*x^5+176715*B*a*c^2*x^5+176715*B*b^2*c*x^5+208845*A*a*c^2*x^4+208845*A*b^2*c*
x^4+417690*B*a*b*c*x^4+69615*B*b^3*x^4+510510*A*a*b*c*x^3+85085*A*b^3*x^3+255255
*B*a^2*c*x^3+255255*B*a*b^2*x^3+328185*A*a^2*c*x^2+328185*A*a*b^2*x^2+328185*B*a
^2*b*x^2+459459*A*a^2*b*x+153153*B*a^3*x+255255*A*a^3)

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Maxima [A]  time = 0.721698, size = 224, normalized size = 1.23 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{11}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*sqrt(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.28844, size = 228, normalized size = 1.25 \[ \frac{2}{765765} \,{\left (45045 \, B c^{3} x^{8} + 51051 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 176715 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{6} + 69615 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + 255255 \, A a^{3} x + 85085 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + 328185 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + 153153 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*sqrt(x),x, algorithm="fricas")

[Out]

2/765765*(45045*B*c^3*x^8 + 51051*(3*B*b*c^2 + A*c^3)*x^7 + 176715*(B*b^2*c + (B
*a + A*b)*c^2)*x^6 + 69615*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^5 + 255
255*A*a^3*x + 85085*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^4 + 328185*(B*
a^2*b + A*a*b^2 + A*a^2*c)*x^3 + 153153*(B*a^3 + 3*A*a^2*b)*x^2)*sqrt(x)

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Sympy [A]  time = 6.37024, size = 216, normalized size = 1.19 \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (3 A b c^{2} + 3 B a c^{2} + 3 B b^{2} c\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (3 A a^{2} c + 3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (3 A a^{2} b + B a^{3}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**3*x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271419, size = 261, normalized size = 1.43 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B b c^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{13} \, A b c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{12}{11} \, B a b c x^{\frac{11}{2}} + \frac{6}{11} \, A b^{2} c x^{\frac{11}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{4}{3} \, A a b c x^{\frac{9}{2}} + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}} + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)*sqrt(x),x, algorithm="giac")

[Out]

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