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

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

[Out]

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

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Rubi [A]  time = 0.197197, 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 \[ -\frac{2 a^6 A}{5 x^{5/2}}-\frac{2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac{6 a^4 b (2 a B+5 A b)}{\sqrt{x}}+10 a^3 b^2 \sqrt{x} (3 a B+4 A b)+\frac{10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+\frac{2}{7} b^5 x^{7/2} (6 a B+A b)+\frac{6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac{2}{9} b^6 B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.6824, size = 160, normalized size = 1.03 \[ - \frac{2 A a^{6}}{5 x^{\frac{5}{2}}} + \frac{2 B b^{6} x^{\frac{9}{2}}}{9} - \frac{2 a^{5} \left (6 A b + B a\right )}{3 x^{\frac{3}{2}}} - \frac{6 a^{4} b \left (5 A b + 2 B a\right )}{\sqrt{x}} + 10 a^{3} b^{2} \sqrt{x} \left (4 A b + 3 B a\right ) + 10 a^{2} b^{3} x^{\frac{3}{2}} \left (A b + \frac{4 B a}{3}\right ) + \frac{6 a b^{4} x^{\frac{5}{2}} \left (2 A b + 5 B a\right )}{5} + \frac{2 b^{5} x^{\frac{7}{2}} \left (A b + 6 B a\right )}{7} \]

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**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0640398, size = 124, normalized size = 0.8 \[ \frac{2 \left (-21 a^6 (3 A+5 B x)-630 a^5 b x (A+3 B x)+4725 a^4 b^2 x^2 (B x-A)+2100 a^3 b^3 x^3 (3 A+B x)+315 a^2 b^4 x^4 (5 A+3 B x)+54 a b^5 x^5 (7 A+5 B x)+5 b^6 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 148, normalized size = 1. \[ -{\frac{-70\,B{b}^{6}{x}^{7}-90\,A{b}^{6}{x}^{6}-540\,B{x}^{6}a{b}^{5}-756\,aA{b}^{5}{x}^{5}-1890\,B{x}^{5}{a}^{2}{b}^{4}-3150\,{a}^{2}A{b}^{4}{x}^{4}-4200\,B{x}^{4}{a}^{3}{b}^{3}-12600\,{a}^{3}A{b}^{3}{x}^{3}-9450\,B{x}^{3}{a}^{4}{b}^{2}+9450\,{a}^{4}A{b}^{2}{x}^{2}+3780\,B{x}^{2}{a}^{5}b+1260\,{a}^{5}Abx+210\,B{a}^{6}x+126\,A{a}^{6}}{315}{x}^{-{\frac{5}{2}}}} \]

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^(7/2),x)

[Out]

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

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Maxima [A]  time = 0.698487, size = 200, normalized size = 1.29 \[ \frac{2}{9} \, B b^{6} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{5}{2}} + \frac{10}{3} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{3}{2}} + 10 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{6} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

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^(7/2),x, algorithm="maxima")

[Out]

2/9*B*b^6*x^(9/2) + 2/7*(6*B*a*b^5 + A*b^6)*x^(7/2) + 6/5*(5*B*a^2*b^4 + 2*A*a*b
^5)*x^(5/2) + 10/3*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(3/2) + 10*(3*B*a^4*b^2 + 4*A*a
^3*b^3)*sqrt(x) - 2/15*(3*A*a^6 + 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 5*(B*a^6 +
6*A*a^5*b)*x)/x^(5/2)

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Fricas [A]  time = 0.29666, size = 198, normalized size = 1.28 \[ \frac{2 \,{\left (35 \, B b^{6} x^{7} - 63 \, A a^{6} + 45 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 189 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 525 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1575 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 945 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 105 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{5}{2}}} \]

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^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.3438, size = 204, normalized size = 1.32 \[ - \frac{2 A a^{6}}{5 x^{\frac{5}{2}}} - \frac{4 A a^{5} b}{x^{\frac{3}{2}}} - \frac{30 A a^{4} b^{2}}{\sqrt{x}} + 40 A a^{3} b^{3} \sqrt{x} + 10 A a^{2} b^{4} x^{\frac{3}{2}} + \frac{12 A a b^{5} x^{\frac{5}{2}}}{5} + \frac{2 A b^{6} x^{\frac{7}{2}}}{7} - \frac{2 B a^{6}}{3 x^{\frac{3}{2}}} - \frac{12 B a^{5} b}{\sqrt{x}} + 30 B a^{4} b^{2} \sqrt{x} + \frac{40 B a^{3} b^{3} x^{\frac{3}{2}}}{3} + 6 B a^{2} b^{4} x^{\frac{5}{2}} + \frac{12 B a b^{5} x^{\frac{7}{2}}}{7} + \frac{2 B b^{6} x^{\frac{9}{2}}}{9} \]

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**(7/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27142, size = 200, normalized size = 1.29 \[ \frac{2}{9} \, B b^{6} x^{\frac{9}{2}} + \frac{12}{7} \, B a b^{5} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{6} x^{\frac{7}{2}} + 6 \, B a^{2} b^{4} x^{\frac{5}{2}} + \frac{12}{5} \, A a b^{5} x^{\frac{5}{2}} + \frac{40}{3} \, B a^{3} b^{3} x^{\frac{3}{2}} + 10 \, A a^{2} b^{4} x^{\frac{3}{2}} + 30 \, B a^{4} b^{2} \sqrt{x} + 40 \, A a^{3} b^{3} \sqrt{x} - \frac{2 \,{\left (90 \, B a^{5} b x^{2} + 225 \, A a^{4} b^{2} x^{2} + 5 \, B a^{6} x + 30 \, A a^{5} b x + 3 \, A a^{6}\right )}}{15 \, x^{\frac{5}{2}}} \]

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^(7/2),x, algorithm="giac")

[Out]

2/9*B*b^6*x^(9/2) + 12/7*B*a*b^5*x^(7/2) + 2/7*A*b^6*x^(7/2) + 6*B*a^2*b^4*x^(5/
2) + 12/5*A*a*b^5*x^(5/2) + 40/3*B*a^3*b^3*x^(3/2) + 10*A*a^2*b^4*x^(3/2) + 30*B
*a^4*b^2*sqrt(x) + 40*A*a^3*b^3*sqrt(x) - 2/15*(90*B*a^5*b*x^2 + 225*A*a^4*b^2*x
^2 + 5*B*a^6*x + 30*A*a^5*b*x + 3*A*a^6)/x^(5/2)