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

Optimal. Leaf size=153 \[ -\frac{2 a^6 A}{3 x^{3/2}}-\frac{2 a^5 (a B+6 A b)}{\sqrt{x}}+6 a^4 b \sqrt{x} (2 a B+5 A b)+\frac{10}{3} a^3 b^2 x^{3/2} (3 a B+4 A b)+2 a^2 b^3 x^{5/2} (4 a B+3 A b)+\frac{2}{9} b^5 x^{9/2} (6 a B+A b)+\frac{6}{7} a b^4 x^{7/2} (5 a B+2 A b)+\frac{2}{11} b^6 B x^{11/2} \]

[Out]

(-2*a^6*A)/(3*x^(3/2)) - (2*a^5*(6*A*b + a*B))/Sqrt[x] + 6*a^4*b*(5*A*b + 2*a*B)
*Sqrt[x] + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(3/2))/3 + 2*a^2*b^3*(3*A*b + 4*a*B)*x^
(5/2) + (6*a*b^4*(2*A*b + 5*a*B)*x^(7/2))/7 + (2*b^5*(A*b + 6*a*B)*x^(9/2))/9 +
(2*b^6*B*x^(11/2))/11

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Rubi [A]  time = 0.197494, antiderivative size = 153, 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 \[ -\frac{2 a^6 A}{3 x^{3/2}}-\frac{2 a^5 (a B+6 A b)}{\sqrt{x}}+6 a^4 b \sqrt{x} (2 a B+5 A b)+\frac{10}{3} a^3 b^2 x^{3/2} (3 a B+4 A b)+2 a^2 b^3 x^{5/2} (4 a B+3 A b)+\frac{2}{9} b^5 x^{9/2} (6 a B+A b)+\frac{6}{7} a b^4 x^{7/2} (5 a B+2 A b)+\frac{2}{11} b^6 B x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^6*A)/(3*x^(3/2)) - (2*a^5*(6*A*b + a*B))/Sqrt[x] + 6*a^4*b*(5*A*b + 2*a*B)
*Sqrt[x] + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(3/2))/3 + 2*a^2*b^3*(3*A*b + 4*a*B)*x^
(5/2) + (6*a*b^4*(2*A*b + 5*a*B)*x^(7/2))/7 + (2*b^5*(A*b + 6*a*B)*x^(9/2))/9 +
(2*b^6*B*x^(11/2))/11

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Rubi in Sympy [A]  time = 42.4291, size = 158, normalized size = 1.03 \[ - \frac{2 A a^{6}}{3 x^{\frac{3}{2}}} + \frac{2 B b^{6} x^{\frac{11}{2}}}{11} - \frac{2 a^{5} \left (6 A b + B a\right )}{\sqrt{x}} + 6 a^{4} b \sqrt{x} \left (5 A b + 2 B a\right ) + 10 a^{3} b^{2} x^{\frac{3}{2}} \left (\frac{4 A b}{3} + B a\right ) + 2 a^{2} b^{3} x^{\frac{5}{2}} \left (3 A b + 4 B a\right ) + \frac{6 a b^{4} x^{\frac{7}{2}} \left (2 A b + 5 B a\right )}{7} + \frac{2 b^{5} x^{\frac{9}{2}} \left (A b + 6 B a\right )}{9} \]

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**(5/2),x)

[Out]

-2*A*a**6/(3*x**(3/2)) + 2*B*b**6*x**(11/2)/11 - 2*a**5*(6*A*b + B*a)/sqrt(x) +
6*a**4*b*sqrt(x)*(5*A*b + 2*B*a) + 10*a**3*b**2*x**(3/2)*(4*A*b/3 + B*a) + 2*a**
2*b**3*x**(5/2)*(3*A*b + 4*B*a) + 6*a*b**4*x**(7/2)*(2*A*b + 5*B*a)/7 + 2*b**5*x
**(9/2)*(A*b + 6*B*a)/9

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Mathematica [A]  time = 0.0606365, size = 124, normalized size = 0.81 \[ \frac{2 \left (-231 a^6 (A+3 B x)+4158 a^5 b x (B x-A)+3465 a^4 b^2 x^2 (3 A+B x)+924 a^3 b^3 x^3 (5 A+3 B x)+297 a^2 b^4 x^4 (7 A+5 B x)+66 a b^5 x^5 (9 A+7 B x)+7 b^6 x^6 (11 A+9 B x)\right )}{693 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4158*a^5*b*x*(-A + B*x) + 3465*a^4*b^2*x^2*(3*A + B*x) - 231*a^6*(A + 3*B*x)
+ 924*a^3*b^3*x^3*(5*A + 3*B*x) + 297*a^2*b^4*x^4*(7*A + 5*B*x) + 66*a*b^5*x^5*
(9*A + 7*B*x) + 7*b^6*x^6*(11*A + 9*B*x)))/(693*x^(3/2))

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Maple [A]  time = 0.01, size = 148, normalized size = 1. \[ -{\frac{-126\,B{b}^{6}{x}^{7}-154\,A{b}^{6}{x}^{6}-924\,B{x}^{6}a{b}^{5}-1188\,aA{b}^{5}{x}^{5}-2970\,B{x}^{5}{a}^{2}{b}^{4}-4158\,{a}^{2}A{b}^{4}{x}^{4}-5544\,B{x}^{4}{a}^{3}{b}^{3}-9240\,{a}^{3}A{b}^{3}{x}^{3}-6930\,B{x}^{3}{a}^{4}{b}^{2}-20790\,{a}^{4}A{b}^{2}{x}^{2}-8316\,B{x}^{2}{a}^{5}b+8316\,{a}^{5}Abx+1386\,B{a}^{6}x+462\,A{a}^{6}}{693}{x}^{-{\frac{3}{2}}}} \]

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^(5/2),x)

[Out]

-2/693*(-63*B*b^6*x^7-77*A*b^6*x^6-462*B*a*b^5*x^6-594*A*a*b^5*x^5-1485*B*a^2*b^
4*x^5-2079*A*a^2*b^4*x^4-2772*B*a^3*b^3*x^4-4620*A*a^3*b^3*x^3-3465*B*a^4*b^2*x^
3-10395*A*a^4*b^2*x^2-4158*B*a^5*b*x^2+4158*A*a^5*b*x+693*B*a^6*x+231*A*a^6)/x^(
3/2)

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Maxima [A]  time = 0.697415, size = 198, normalized size = 1.29 \[ \frac{2}{11} \, B b^{6} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{7}{2}} + 2 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{5}{2}} + \frac{10}{3} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{6} + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

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^(5/2),x, algorithm="maxima")

[Out]

2/11*B*b^6*x^(11/2) + 2/9*(6*B*a*b^5 + A*b^6)*x^(9/2) + 6/7*(5*B*a^2*b^4 + 2*A*a
*b^5)*x^(7/2) + 2*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(5/2) + 10/3*(3*B*a^4*b^2 + 4*A*
a^3*b^3)*x^(3/2) + 6*(2*B*a^5*b + 5*A*a^4*b^2)*sqrt(x) - 2/3*(A*a^6 + 3*(B*a^6 +
6*A*a^5*b)*x)/x^(3/2)

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Fricas [A]  time = 0.286926, size = 198, normalized size = 1.29 \[ \frac{2 \,{\left (63 \, B b^{6} x^{7} - 231 \, A a^{6} + 77 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 297 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 693 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1155 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 2079 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 693 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{693 \, x^{\frac{3}{2}}} \]

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^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^6*x^7 - 231*A*a^6 + 77*(6*B*a*b^5 + A*b^6)*x^6 + 297*(5*B*a^2*b^4
+ 2*A*a*b^5)*x^5 + 693*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 1155*(3*B*a^4*b^2 + 4*A
*a^3*b^3)*x^3 + 2079*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 693*(B*a^6 + 6*A*a^5*b)*x)/
x^(3/2)

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Sympy [A]  time = 15.8837, size = 204, normalized size = 1.33 \[ - \frac{2 A a^{6}}{3 x^{\frac{3}{2}}} - \frac{12 A a^{5} b}{\sqrt{x}} + 30 A a^{4} b^{2} \sqrt{x} + \frac{40 A a^{3} b^{3} x^{\frac{3}{2}}}{3} + 6 A a^{2} b^{4} x^{\frac{5}{2}} + \frac{12 A a b^{5} x^{\frac{7}{2}}}{7} + \frac{2 A b^{6} x^{\frac{9}{2}}}{9} - \frac{2 B a^{6}}{\sqrt{x}} + 12 B a^{5} b \sqrt{x} + 10 B a^{4} b^{2} x^{\frac{3}{2}} + 8 B a^{3} b^{3} x^{\frac{5}{2}} + \frac{30 B a^{2} b^{4} x^{\frac{7}{2}}}{7} + \frac{4 B a b^{5} x^{\frac{9}{2}}}{3} + \frac{2 B b^{6} x^{\frac{11}{2}}}{11} \]

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**(5/2),x)

[Out]

-2*A*a**6/(3*x**(3/2)) - 12*A*a**5*b/sqrt(x) + 30*A*a**4*b**2*sqrt(x) + 40*A*a**
3*b**3*x**(3/2)/3 + 6*A*a**2*b**4*x**(5/2) + 12*A*a*b**5*x**(7/2)/7 + 2*A*b**6*x
**(9/2)/9 - 2*B*a**6/sqrt(x) + 12*B*a**5*b*sqrt(x) + 10*B*a**4*b**2*x**(3/2) + 8
*B*a**3*b**3*x**(5/2) + 30*B*a**2*b**4*x**(7/2)/7 + 4*B*a*b**5*x**(9/2)/3 + 2*B*
b**6*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.269914, size = 198, normalized size = 1.29 \[ \frac{2}{11} \, B b^{6} x^{\frac{11}{2}} + \frac{4}{3} \, B a b^{5} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{6} x^{\frac{9}{2}} + \frac{30}{7} \, B a^{2} b^{4} x^{\frac{7}{2}} + \frac{12}{7} \, A a b^{5} x^{\frac{7}{2}} + 8 \, B a^{3} b^{3} x^{\frac{5}{2}} + 6 \, A a^{2} b^{4} x^{\frac{5}{2}} + 10 \, B a^{4} b^{2} x^{\frac{3}{2}} + \frac{40}{3} \, A a^{3} b^{3} x^{\frac{3}{2}} + 12 \, B a^{5} b \sqrt{x} + 30 \, A a^{4} b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{6} x + 18 \, A a^{5} b x + A a^{6}\right )}}{3 \, x^{\frac{3}{2}}} \]

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^(5/2),x, algorithm="giac")

[Out]

2/11*B*b^6*x^(11/2) + 4/3*B*a*b^5*x^(9/2) + 2/9*A*b^6*x^(9/2) + 30/7*B*a^2*b^4*x
^(7/2) + 12/7*A*a*b^5*x^(7/2) + 8*B*a^3*b^3*x^(5/2) + 6*A*a^2*b^4*x^(5/2) + 10*B
*a^4*b^2*x^(3/2) + 40/3*A*a^3*b^3*x^(3/2) + 12*B*a^5*b*sqrt(x) + 30*A*a^4*b^2*sq
rt(x) - 2/3*(3*B*a^6*x + 18*A*a^5*b*x + A*a^6)/x^(3/2)