3.165 \(\int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=83 \[ 2 A b^3 \sqrt{x}+\frac{2}{3} b^2 x^{3/2} (3 A c+b B)+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{6}{5} b c x^{5/2} (A c+b B)+\frac{2}{9} B c^3 x^{9/2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.120045, antiderivative size = 83, 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 \[ 2 A b^3 \sqrt{x}+\frac{2}{3} b^2 x^{3/2} (3 A c+b B)+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{6}{5} b c x^{5/2} (A c+b B)+\frac{2}{9} B c^3 x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7748, size = 82, normalized size = 0.99 \[ 2 A b^{3} \sqrt{x} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} + 2 b^{2} x^{\frac{3}{2}} \left (A c + \frac{B b}{3}\right ) + \frac{6 b c x^{\frac{5}{2}} \left (A c + B b\right )}{5} + \frac{2 c^{2} x^{\frac{7}{2}} \left (A c + 3 B b\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**3/x**(7/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0401191, size = 69, normalized size = 0.83 \[ \frac{2}{315} \sqrt{x} \left (315 A b^3+105 b^2 x (3 A c+b B)+45 c^2 x^3 (A c+3 b B)+189 b c x^2 (A c+b B)+35 B c^3 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(315*A*b^3 + 105*b^2*(b*B + 3*A*c)*x + 189*b*c*(b*B + A*c)*x^2 + 45*c
^2*(3*b*B + A*c)*x^3 + 35*B*c^3*x^4))/315

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 76, normalized size = 0.9 \[{\frac{70\,B{c}^{3}{x}^{4}+90\,A{c}^{3}{x}^{3}+270\,B{x}^{3}b{c}^{2}+378\,Ab{c}^{2}{x}^{2}+378\,B{x}^{2}{b}^{2}c+630\,A{b}^{2}cx+210\,Bx{b}^{3}+630\,A{b}^{3}}{315}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*x^(1/2)*(35*B*c^3*x^4+45*A*c^3*x^3+135*B*b*c^2*x^3+189*A*b*c^2*x^2+189*B*b
^2*c*x^2+315*A*b^2*c*x+105*B*b^3*x+315*A*b^3)

_______________________________________________________________________________________

Maxima [A]  time = 0.683821, size = 99, normalized size = 1.19 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + 2 \, A b^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.269291, size = 99, normalized size = 1.19 \[ \frac{2}{315} \,{\left (35 \, B c^{3} x^{4} + 315 \, A b^{3} + 45 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 189 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 105 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*c^3*x^4 + 315*A*b^3 + 45*(3*B*b*c^2 + A*c^3)*x^3 + 189*(B*b^2*c + A*
b*c^2)*x^2 + 105*(B*b^3 + 3*A*b^2*c)*x)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 17.3269, size = 110, normalized size = 1.33 \[ 2 A b^{3} \sqrt{x} + 2 A b^{2} c x^{\frac{3}{2}} + \frac{6 A b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A c^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{3}{2}}}{3} + \frac{6 B b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**3/x**(7/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268607, size = 104, normalized size = 1.25 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b c^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b^{2} c x^{\frac{5}{2}} + \frac{6}{5} \, A b c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 2 \, A b^{2} c x^{\frac{3}{2}} + 2 \, A b^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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