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

Optimal. Leaf size=85 \[ \frac{2}{7} A b^3 x^{7/2}+\frac{2}{9} b^2 x^{9/2} (3 A c+b B)+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{6}{11} b c x^{11/2} (A c+b B)+\frac{2}{15} B c^3 x^{15/2} \]

[Out]

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

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Rubi [A]  time = 0.121031, 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 \[ \frac{2}{7} A b^3 x^{7/2}+\frac{2}{9} b^2 x^{9/2} (3 A c+b B)+\frac{2}{13} c^2 x^{13/2} (A c+3 b B)+\frac{6}{11} b c x^{11/2} (A c+b B)+\frac{2}{15} B c^3 x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0927, size = 85, normalized size = 1. \[ \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{15}{2}}}{15} + \frac{2 b^{2} x^{\frac{9}{2}} \left (3 A c + B b\right )}{9} + \frac{6 b c x^{\frac{11}{2}} \left (A c + B b\right )}{11} + \frac{2 c^{2} x^{\frac{13}{2}} \left (A c + 3 B b\right )}{13} \]

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

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Mathematica [A]  time = 0.042344, size = 69, normalized size = 0.81 \[ \frac{2 x^{7/2} \left (6435 A b^3+5005 b^2 x (3 A c+b B)+3465 c^2 x^3 (A c+3 b B)+12285 b c x^2 (A c+b B)+3003 B c^3 x^4\right )}{45045} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*(6435*A*b^3 + 5005*b^2*(b*B + 3*A*c)*x + 12285*b*c*(b*B + A*c)*x^2 +
3465*c^2*(3*b*B + A*c)*x^3 + 3003*B*c^3*x^4))/45045

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \[{\frac{6006\,B{c}^{3}{x}^{4}+6930\,A{c}^{3}{x}^{3}+20790\,B{x}^{3}b{c}^{2}+24570\,Ab{c}^{2}{x}^{2}+24570\,B{x}^{2}{b}^{2}c+30030\,A{b}^{2}cx+10010\,Bx{b}^{3}+12870\,A{b}^{3}}{45045}{x}^{{\frac{7}{2}}}} \]

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]

2/45045*x^(7/2)*(3003*B*c^3*x^4+3465*A*c^3*x^3+10395*B*b*c^2*x^3+12285*A*b*c^2*x
^2+12285*B*b^2*c*x^2+15015*A*b^2*c*x+5005*B*b^3*x+6435*A*b^3)

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Maxima [A]  time = 0.687442, size = 99, normalized size = 1.16 \[ \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{9}{2}} \]

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="maxima")

[Out]

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

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Fricas [A]  time = 0.280278, size = 105, normalized size = 1.24 \[ \frac{2}{45045} \,{\left (3003 \, B c^{3} x^{7} + 6435 \, A b^{3} x^{3} + 3465 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 12285 \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + 5005 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4}\right )} \sqrt{x} \]

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]

2/45045*(3003*B*c^3*x^7 + 6435*A*b^3*x^3 + 3465*(3*B*b*c^2 + A*c^3)*x^6 + 12285*
(B*b^2*c + A*b*c^2)*x^5 + 5005*(B*b^3 + 3*A*b^2*c)*x^4)*sqrt(x)

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Sympy [A]  time = 5.40621, size = 114, normalized size = 1.34 \[ \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 A b^{2} c x^{\frac{9}{2}}}{3} + \frac{6 A b c^{2} x^{\frac{11}{2}}}{11} + \frac{2 A c^{3} x^{\frac{13}{2}}}{13} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} + \frac{6 B b^{2} c x^{\frac{11}{2}}}{11} + \frac{6 B b c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{3} x^{\frac{15}{2}}}{15} \]

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**(7/2)/7 + 2*A*b**2*c*x**(9/2)/3 + 6*A*b*c**2*x**(11/2)/11 + 2*A*c**3
*x**(13/2)/13 + 2*B*b**3*x**(9/2)/9 + 6*B*b**2*c*x**(11/2)/11 + 6*B*b*c**2*x**(1
3/2)/13 + 2*B*c**3*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.269046, size = 104, normalized size = 1.22 \[ \frac{2}{15} \, B c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B b c^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B b^{2} c x^{\frac{11}{2}} + \frac{6}{11} \, A b c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A b^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} \]

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]

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