3.756 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx\)

Optimal. Leaf size=155 \[ -\frac{2 a^6 A}{9 x^{9/2}}-\frac{2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac{6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac{10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac{10 a^2 b^3 (4 a B+3 A b)}{\sqrt{x}}+\frac{2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt{x} (5 a B+2 A b)+\frac{2}{5} b^6 B x^{5/2} \]

[Out]

(-2*a^6*A)/(9*x^(9/2)) - (2*a^5*(6*A*b + a*B))/(7*x^(7/2)) - (6*a^4*b*(5*A*b + 2
*a*B))/(5*x^(5/2)) - (10*a^3*b^2*(4*A*b + 3*a*B))/(3*x^(3/2)) - (10*a^2*b^3*(3*A
*b + 4*a*B))/Sqrt[x] + 6*a*b^4*(2*A*b + 5*a*B)*Sqrt[x] + (2*b^5*(A*b + 6*a*B)*x^
(3/2))/3 + (2*b^6*B*x^(5/2))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.196571, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 a^6 A}{9 x^{9/2}}-\frac{2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac{6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac{10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac{10 a^2 b^3 (4 a B+3 A b)}{\sqrt{x}}+\frac{2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt{x} (5 a B+2 A b)+\frac{2}{5} b^6 B x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(11/2),x]

[Out]

(-2*a^6*A)/(9*x^(9/2)) - (2*a^5*(6*A*b + a*B))/(7*x^(7/2)) - (6*a^4*b*(5*A*b + 2
*a*B))/(5*x^(5/2)) - (10*a^3*b^2*(4*A*b + 3*a*B))/(3*x^(3/2)) - (10*a^2*b^3*(3*A
*b + 4*a*B))/Sqrt[x] + 6*a*b^4*(2*A*b + 5*a*B)*Sqrt[x] + (2*b^5*(A*b + 6*a*B)*x^
(3/2))/3 + (2*b^6*B*x^(5/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.3172, size = 160, normalized size = 1.03 \[ - \frac{2 A a^{6}}{9 x^{\frac{9}{2}}} + \frac{2 B b^{6} x^{\frac{5}{2}}}{5} - \frac{2 a^{5} \left (6 A b + B a\right )}{7 x^{\frac{7}{2}}} - \frac{6 a^{4} b \left (5 A b + 2 B a\right )}{5 x^{\frac{5}{2}}} - \frac{10 a^{3} b^{2} \left (\frac{4 A b}{3} + B a\right )}{x^{\frac{3}{2}}} - \frac{10 a^{2} b^{3} \left (3 A b + 4 B a\right )}{\sqrt{x}} + 6 a b^{4} \sqrt{x} \left (2 A b + 5 B a\right ) + \frac{2 b^{5} x^{\frac{3}{2}} \left (A b + 6 B a\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**(11/2),x)

[Out]

-2*A*a**6/(9*x**(9/2)) + 2*B*b**6*x**(5/2)/5 - 2*a**5*(6*A*b + B*a)/(7*x**(7/2))
- 6*a**4*b*(5*A*b + 2*B*a)/(5*x**(5/2)) - 10*a**3*b**2*(4*A*b/3 + B*a)/x**(3/2)
- 10*a**2*b**3*(3*A*b + 4*B*a)/sqrt(x) + 6*a*b**4*sqrt(x)*(2*A*b + 5*B*a) + 2*b
**5*x**(3/2)*(A*b + 6*B*a)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0608342, size = 123, normalized size = 0.79 \[ -\frac{2 \left (5 a^6 (7 A+9 B x)+54 a^5 b x (5 A+7 B x)+315 a^4 b^2 x^2 (3 A+5 B x)+2100 a^3 b^3 x^3 (A+3 B x)+4725 a^2 b^4 x^4 (A-B x)-630 a b^5 x^5 (3 A+B x)-21 b^6 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(11/2),x]

[Out]

(-2*(4725*a^2*b^4*x^4*(A - B*x) - 630*a*b^5*x^5*(3*A + B*x) + 2100*a^3*b^3*x^3*(
A + 3*B*x) - 21*b^6*x^6*(5*A + 3*B*x) + 315*a^4*b^2*x^2*(3*A + 5*B*x) + 54*a^5*b
*x*(5*A + 7*B*x) + 5*a^6*(7*A + 9*B*x)))/(315*x^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 148, normalized size = 1. \[ -{\frac{-126\,B{b}^{6}{x}^{7}-210\,A{b}^{6}{x}^{6}-1260\,B{x}^{6}a{b}^{5}-3780\,aA{b}^{5}{x}^{5}-9450\,B{x}^{5}{a}^{2}{b}^{4}+9450\,{a}^{2}A{b}^{4}{x}^{4}+12600\,B{x}^{4}{a}^{3}{b}^{3}+4200\,{a}^{3}A{b}^{3}{x}^{3}+3150\,B{x}^{3}{a}^{4}{b}^{2}+1890\,{a}^{4}A{b}^{2}{x}^{2}+756\,B{x}^{2}{a}^{5}b+540\,{a}^{5}Abx+90\,B{a}^{6}x+70\,A{a}^{6}}{315}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(11/2),x)

[Out]

-2/315*(-63*B*b^6*x^7-105*A*b^6*x^6-630*B*a*b^5*x^6-1890*A*a*b^5*x^5-4725*B*a^2*
b^4*x^5+4725*A*a^2*b^4*x^4+6300*B*a^3*b^3*x^4+2100*A*a^3*b^3*x^3+1575*B*a^4*b^2*
x^3+945*A*a^4*b^2*x^2+378*B*a^5*b*x^2+270*A*a^5*b*x+45*B*a^6*x+35*A*a^6)/x^(9/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.69485, size = 200, normalized size = 1.29 \[ \frac{2}{5} \, B b^{6} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{3}{2}} + 6 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \sqrt{x} - \frac{2 \,{\left (35 \, A a^{6} + 1575 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 189 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 45 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^(11/2),x, algorithm="maxima")

[Out]

2/5*B*b^6*x^(5/2) + 2/3*(6*B*a*b^5 + A*b^6)*x^(3/2) + 6*(5*B*a^2*b^4 + 2*A*a*b^5
)*sqrt(x) - 2/315*(35*A*a^6 + 1575*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 525*(3*B*a^
4*b^2 + 4*A*a^3*b^3)*x^3 + 189*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 45*(B*a^6 + 6*A*a
^5*b)*x)/x^(9/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.309171, size = 198, normalized size = 1.28 \[ \frac{2 \,{\left (63 \, B b^{6} x^{7} - 35 \, A a^{6} + 105 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 945 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 1575 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 189 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 45 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*b^6*x^7 - 35*A*a^6 + 105*(6*B*a*b^5 + A*b^6)*x^6 + 945*(5*B*a^2*b^4
+ 2*A*a*b^5)*x^5 - 1575*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 525*(3*B*a^4*b^2 + 4*A
*a^3*b^3)*x^3 - 189*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 45*(B*a^6 + 6*A*a^5*b)*x)/x^
(9/2)

_______________________________________________________________________________________

Sympy [A]  time = 36.6896, size = 204, normalized size = 1.32 \[ - \frac{2 A a^{6}}{9 x^{\frac{9}{2}}} - \frac{12 A a^{5} b}{7 x^{\frac{7}{2}}} - \frac{6 A a^{4} b^{2}}{x^{\frac{5}{2}}} - \frac{40 A a^{3} b^{3}}{3 x^{\frac{3}{2}}} - \frac{30 A a^{2} b^{4}}{\sqrt{x}} + 12 A a b^{5} \sqrt{x} + \frac{2 A b^{6} x^{\frac{3}{2}}}{3} - \frac{2 B a^{6}}{7 x^{\frac{7}{2}}} - \frac{12 B a^{5} b}{5 x^{\frac{5}{2}}} - \frac{10 B a^{4} b^{2}}{x^{\frac{3}{2}}} - \frac{40 B a^{3} b^{3}}{\sqrt{x}} + 30 B a^{2} b^{4} \sqrt{x} + 4 B a b^{5} x^{\frac{3}{2}} + \frac{2 B b^{6} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**(11/2),x)

[Out]

-2*A*a**6/(9*x**(9/2)) - 12*A*a**5*b/(7*x**(7/2)) - 6*A*a**4*b**2/x**(5/2) - 40*
A*a**3*b**3/(3*x**(3/2)) - 30*A*a**2*b**4/sqrt(x) + 12*A*a*b**5*sqrt(x) + 2*A*b*
*6*x**(3/2)/3 - 2*B*a**6/(7*x**(7/2)) - 12*B*a**5*b/(5*x**(5/2)) - 10*B*a**4*b**
2/x**(3/2) - 40*B*a**3*b**3/sqrt(x) + 30*B*a**2*b**4*sqrt(x) + 4*B*a*b**5*x**(3/
2) + 2*B*b**6*x**(5/2)/5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269177, size = 200, normalized size = 1.29 \[ \frac{2}{5} \, B b^{6} x^{\frac{5}{2}} + 4 \, B a b^{5} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{6} x^{\frac{3}{2}} + 30 \, B a^{2} b^{4} \sqrt{x} + 12 \, A a b^{5} \sqrt{x} - \frac{2 \,{\left (6300 \, B a^{3} b^{3} x^{4} + 4725 \, A a^{2} b^{4} x^{4} + 1575 \, B a^{4} b^{2} x^{3} + 2100 \, A a^{3} b^{3} x^{3} + 378 \, B a^{5} b x^{2} + 945 \, A a^{4} b^{2} x^{2} + 45 \, B a^{6} x + 270 \, A a^{5} b x + 35 \, A a^{6}\right )}}{315 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^(11/2),x, algorithm="giac")

[Out]

2/5*B*b^6*x^(5/2) + 4*B*a*b^5*x^(3/2) + 2/3*A*b^6*x^(3/2) + 30*B*a^2*b^4*sqrt(x)
+ 12*A*a*b^5*sqrt(x) - 2/315*(6300*B*a^3*b^3*x^4 + 4725*A*a^2*b^4*x^4 + 1575*B*
a^4*b^2*x^3 + 2100*A*a^3*b^3*x^3 + 378*B*a^5*b*x^2 + 945*A*a^4*b^2*x^2 + 45*B*a^
6*x + 270*A*a^5*b*x + 35*A*a^6)/x^(9/2)