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

Optimal. Leaf size=85 \[ \frac{2}{9} A b^3 x^{9/2}+\frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{6}{13} b c x^{13/2} (A c+b B)+\frac{2}{17} B c^3 x^{17/2} \]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (6*b*c*(b*B + A*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 = 0.118949, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{9} A b^3 x^{9/2}+\frac{2}{11} b^2 x^{11/2} (3 A c+b B)+\frac{2}{15} c^2 x^{15/2} (A c+3 b B)+\frac{6}{13} b c x^{13/2} (A c+b B)+\frac{2}{17} B c^3 x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (6*b*c*(b*B + A*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 = 13.7061, size = 85, normalized size = 1. \[ \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} x^{\frac{11}{2}} \left (3 A c + B b\right )}{11} + \frac{6 b c x^{\frac{13}{2}} \left (A c + B b\right )}{13} + \frac{2 c^{2} x^{\frac{15}{2}} \left (A c + 3 B b\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0449186, size = 69, normalized size = 0.81 \[ \frac{2 x^{9/2} \left (12155 A b^3+9945 b^2 x (3 A c+b B)+7293 c^2 x^3 (A c+3 b B)+25245 b c x^2 (A c+b B)+6435 B c^3 x^4\right )}{109395} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(12155*A*b^3 + 9945*b^2*(b*B + 3*A*c)*x + 25245*b*c*(b*B + A*c)*x^2 +
7293*c^2*(3*b*B + A*c)*x^3 + 6435*B*c^3*x^4))/109395

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \[{\frac{12870\,B{c}^{3}{x}^{4}+14586\,A{c}^{3}{x}^{3}+43758\,B{x}^{3}b{c}^{2}+50490\,Ab{c}^{2}{x}^{2}+50490\,B{x}^{2}{b}^{2}c+59670\,A{b}^{2}cx+19890\,Bx{b}^{3}+24310\,A{b}^{3}}{109395}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/109395*x^(9/2)*(6435*B*c^3*x^4+7293*A*c^3*x^3+21879*B*b*c^2*x^3+25245*A*b*c^2*
x^2+25245*B*b^2*c*x^2+29835*A*b^2*c*x+9945*B*b^3*x+12155*A*b^3)

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Maxima [A]  time = 0.678404, size = 99, normalized size = 1.16 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287978, size = 105, normalized size = 1.24 \[ \frac{2}{109395} \,{\left (6435 \, B c^{3} x^{8} + 12155 \, A b^{3} x^{4} + 7293 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 25245 \,{\left (B b^{2} c + A b c^{2}\right )} x^{6} + 9945 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/109395*(6435*B*c^3*x^8 + 12155*A*b^3*x^4 + 7293*(3*B*b*c^2 + A*c^3)*x^7 + 2524
5*(B*b^2*c + A*b*c^2)*x^6 + 9945*(B*b^3 + 3*A*b^2*c)*x^5)*sqrt(x)

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Sympy [A]  time = 5.64248, size = 95, normalized size = 1.12 \[ \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (3 A b^{2} c + B b^{3}\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*b**3*x**(9/2)/9 + 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*b**2*c)/13 + 2*x**(11/2)*(3*A*b**2*c + B*b**3
)/11

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GIAC/XCAS [A]  time = 0.267162, size = 104, normalized size = 1.22 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B b c^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b^{2} c x^{\frac{13}{2}} + \frac{6}{13} \, A b c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A b^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^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*A*b*c^2*x^(13/2) + 2/11*B*b^3*x^(11/2) + 6/11*A*b^2*c*x^(11/2) +
2/9*A*b^3*x^(9/2)