3.164 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{3} A b^3 x^{3/2}+\frac{2}{5} b^2 x^{5/2} (3 A c+b B)+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{6}{7} b c x^{7/2} (A c+b B)+\frac{2}{11} B c^3 x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.118401, 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 \[ \frac{2}{3} A b^3 x^{3/2}+\frac{2}{5} b^2 x^{5/2} (3 A c+b B)+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{6}{7} b c x^{7/2} (A c+b B)+\frac{2}{11} B c^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.7772, size = 85, normalized size = 1. \[ \frac{2 A b^{3} x^{\frac{3}{2}}}{3} + \frac{2 B c^{3} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} x^{\frac{5}{2}} \left (3 A c + B b\right )}{5} + \frac{6 b c x^{\frac{7}{2}} \left (A c + B b\right )}{7} + \frac{2 c^{2} x^{\frac{9}{2}} \left (A c + 3 B b\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0409805, size = 69, normalized size = 0.81 \[ \frac{2 x^{3/2} \left (1155 A b^3+693 b^2 x (3 A c+b B)+385 c^2 x^3 (A c+3 b B)+1485 b c x^2 (A c+b B)+315 B c^3 x^4\right )}{3465} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]

[Out]

(2*x^(3/2)*(1155*A*b^3 + 693*b^2*(b*B + 3*A*c)*x + 1485*b*c*(b*B + A*c)*x^2 + 38
5*c^2*(3*b*B + A*c)*x^3 + 315*B*c^3*x^4))/3465

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Maple [A]  time = 0.009, size = 76, normalized size = 0.9 \[{\frac{630\,B{c}^{3}{x}^{4}+770\,A{c}^{3}{x}^{3}+2310\,B{x}^{3}b{c}^{2}+2970\,Ab{c}^{2}{x}^{2}+2970\,B{x}^{2}{b}^{2}c+4158\,A{b}^{2}cx+1386\,Bx{b}^{3}+2310\,A{b}^{3}}{3465}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x)

[Out]

2/3465*x^(3/2)*(315*B*c^3*x^4+385*A*c^3*x^3+1155*B*b*c^2*x^3+1485*A*b*c^2*x^2+14
85*B*b^2*c*x^2+2079*A*b^2*c*x+693*B*b^3*x+1155*A*b^3)

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Maxima [A]  time = 0.677002, size = 99, normalized size = 1.16 \[ \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.267138, size = 103, normalized size = 1.21 \[ \frac{2}{3465} \,{\left (315 \, B c^{3} x^{5} + 1155 \, A b^{3} x + 385 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 1485 \,{\left (B b^{2} c + A b c^{2}\right )} x^{3} + 693 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^(5/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^3*x^5 + 1155*A*b^3*x + 385*(3*B*b*c^2 + A*c^3)*x^4 + 1485*(B*b^2
*c + A*b*c^2)*x^3 + 693*(B*b^3 + 3*A*b^2*c)*x^2)*sqrt(x)

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Sympy [A]  time = 12.3489, size = 114, normalized size = 1.34 \[ \frac{2 A b^{3} x^{\frac{3}{2}}}{3} + \frac{6 A b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 A b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A c^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} + \frac{6 B b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 B b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B c^{3} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.266769, size = 104, normalized size = 1.22 \[ \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b c^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, A b c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A b^{2} c x^{\frac{5}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)/x^(5/2),x, algorithm="giac")

[Out]

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