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

Optimal. Leaf size=107 \[ -\frac{2 a^4 A}{\sqrt{x}}+2 a^3 \sqrt{x} (a B+4 A b)+\frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.134463, antiderivative size = 107, 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 \[ -\frac{2 a^4 A}{\sqrt{x}}+2 a^3 \sqrt{x} (a B+4 A b)+\frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.8906, size = 110, normalized size = 1.03 \[ - \frac{2 A a^{4}}{\sqrt{x}} + \frac{2 B b^{4} x^{\frac{9}{2}}}{9} + 2 a^{3} \sqrt{x} \left (4 A b + B a\right ) + \frac{4 a^{2} b x^{\frac{3}{2}} \left (3 A b + 2 B a\right )}{3} + \frac{4 a b^{2} x^{\frac{5}{2}} \left (2 A b + 3 B a\right )}{5} + \frac{2 b^{3} x^{\frac{7}{2}} \left (A b + 4 B a\right )}{7} \]

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)**2/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0398174, size = 87, normalized size = 0.81 \[ \frac{-630 a^4 (A-B x)+840 a^3 b x (3 A+B x)+252 a^2 b^2 x^2 (5 A+3 B x)+72 a b^3 x^3 (7 A+5 B x)+10 b^4 x^4 (9 A+7 B x)}{315 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-630*a^4*(A - B*x) + 840*a^3*b*x*(3*A + B*x) + 252*a^2*b^2*x^2*(5*A + 3*B*x) +
72*a*b^3*x^3*(7*A + 5*B*x) + 10*b^4*x^4*(9*A + 7*B*x))/(315*Sqrt[x])

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Maple [A]  time = 0.011, size = 100, normalized size = 0.9 \[ -{\frac{-70\,{b}^{4}B{x}^{5}-90\,A{b}^{4}{x}^{4}-360\,B{x}^{4}a{b}^{3}-504\,aA{b}^{3}{x}^{3}-756\,B{x}^{3}{a}^{2}{b}^{2}-1260\,{a}^{2}A{b}^{2}{x}^{2}-840\,B{x}^{2}{a}^{3}b-2520\,{a}^{3}Abx-630\,{a}^{4}Bx+630\,A{a}^{4}}{315}{\frac{1}{\sqrt{x}}}} \]

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)^2/x^(3/2),x)

[Out]

-2/315*(-35*B*b^4*x^5-45*A*b^4*x^4-180*B*a*b^3*x^4-252*A*a*b^3*x^3-378*B*a^2*b^2
*x^3-630*A*a^2*b^2*x^2-420*B*a^3*b*x^2-1260*A*a^3*b*x-315*B*a^4*x+315*A*a^4)/x^(
1/2)

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Maxima [A]  time = 0.679828, size = 134, normalized size = 1.25 \[ \frac{2}{9} \, B b^{4} x^{\frac{9}{2}} - \frac{2 \, A a^{4}}{\sqrt{x}} + \frac{2}{7} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{7}{2}} + \frac{4}{5} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{5}{2}} + \frac{4}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.318536, size = 134, normalized size = 1.25 \[ \frac{2 \,{\left (35 \, B b^{4} x^{5} - 315 \, A a^{4} + 45 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 126 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{315 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b^4*x^5 - 315*A*a^4 + 45*(4*B*a*b^3 + A*b^4)*x^4 + 126*(3*B*a^2*b^2
+ 2*A*a*b^3)*x^3 + 210*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 315*(B*a^4 + 4*A*a^3*b)*x
)/sqrt(x)

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Sympy [A]  time = 11.3874, size = 141, normalized size = 1.32 \[ - \frac{2 A a^{4}}{\sqrt{x}} + 8 A a^{3} b \sqrt{x} + 4 A a^{2} b^{2} x^{\frac{3}{2}} + \frac{8 A a b^{3} x^{\frac{5}{2}}}{5} + \frac{2 A b^{4} x^{\frac{7}{2}}}{7} + 2 B a^{4} \sqrt{x} + \frac{8 B a^{3} b x^{\frac{3}{2}}}{3} + \frac{12 B a^{2} b^{2} x^{\frac{5}{2}}}{5} + \frac{8 B a b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{4} x^{\frac{9}{2}}}{9} \]

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)**2/x**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.26884, size = 136, normalized size = 1.27 \[ \frac{2}{9} \, B b^{4} x^{\frac{9}{2}} + \frac{8}{7} \, B a b^{3} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{4} x^{\frac{7}{2}} + \frac{12}{5} \, B a^{2} b^{2} x^{\frac{5}{2}} + \frac{8}{5} \, A a b^{3} x^{\frac{5}{2}} + \frac{8}{3} \, B a^{3} b x^{\frac{3}{2}} + 4 \, A a^{2} b^{2} x^{\frac{3}{2}} + 2 \, B a^{4} \sqrt{x} + 8 \, A a^{3} b \sqrt{x} - \frac{2 \, A a^{4}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*b^4*x^(9/2) + 8/7*B*a*b^3*x^(7/2) + 2/7*A*b^4*x^(7/2) + 12/5*B*a^2*b^2*x^(
5/2) + 8/5*A*a*b^3*x^(5/2) + 8/3*B*a^3*b*x^(3/2) + 4*A*a^2*b^2*x^(3/2) + 2*B*a^4
*sqrt(x) + 8*A*a^3*b*sqrt(x) - 2*A*a^4/sqrt(x)