3.742 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=109 \[ 2 a^4 A \sqrt{x}+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+\frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.132234, antiderivative size = 109, 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 \[ 2 a^4 A \sqrt{x}+\frac{2}{3} a^3 x^{3/2} (a B+4 A b)+\frac{4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac{2}{9} b^3 x^{9/2} (4 a B+A b)+\frac{4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac{2}{11} b^4 B x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.8236, size = 112, normalized size = 1.03 \[ 2 A a^{4} \sqrt{x} + \frac{2 B b^{4} x^{\frac{11}{2}}}{11} + \frac{2 a^{3} x^{\frac{3}{2}} \left (4 A b + B a\right )}{3} + \frac{4 a^{2} b x^{\frac{5}{2}} \left (3 A b + 2 B a\right )}{5} + \frac{4 a b^{2} x^{\frac{7}{2}} \left (2 A b + 3 B a\right )}{7} + \frac{2 b^{3} x^{\frac{9}{2}} \left (A b + 4 B a\right )}{9} \]

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)**2/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0498812, size = 89, normalized size = 0.82 \[ \frac{2 \sqrt{x} \left (1155 a^4 (3 A+B x)+924 a^3 b x (5 A+3 B x)+594 a^2 b^2 x^2 (7 A+5 B x)+220 a b^3 x^3 (9 A+7 B x)+35 b^4 x^4 (11 A+9 B x)\right )}{3465} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(1155*a^4*(3*A + B*x) + 924*a^3*b*x*(5*A + 3*B*x) + 594*a^2*b^2*x^2*(
7*A + 5*B*x) + 220*a*b^3*x^3*(9*A + 7*B*x) + 35*b^4*x^4*(11*A + 9*B*x)))/3465

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Maple [A]  time = 0.01, size = 100, normalized size = 0.9 \[{\frac{630\,{b}^{4}B{x}^{5}+770\,A{b}^{4}{x}^{4}+3080\,B{x}^{4}a{b}^{3}+3960\,aA{b}^{3}{x}^{3}+5940\,B{x}^{3}{a}^{2}{b}^{2}+8316\,{a}^{2}A{b}^{2}{x}^{2}+5544\,B{x}^{2}{a}^{3}b+9240\,{a}^{3}Abx+2310\,{a}^{4}Bx+6930\,A{a}^{4}}{3465}\sqrt{x}} \]

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^(1/2),x)

[Out]

2/3465*x^(1/2)*(315*B*b^4*x^5+385*A*b^4*x^4+1540*B*a*b^3*x^4+1980*A*a*b^3*x^3+29
70*B*a^2*b^2*x^3+4158*A*a^2*b^2*x^2+2772*B*a^3*b*x^2+4620*A*a^3*b*x+1155*B*a^4*x
+3465*A*a^4)

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Maxima [A]  time = 0.682923, size = 134, normalized size = 1.23 \[ \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + 2 \, A a^{4} \sqrt{x} + \frac{2}{9} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{9}{2}} + \frac{4}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{7}{2}} + \frac{4}{5} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289646, size = 134, normalized size = 1.23 \[ \frac{2}{3465} \,{\left (315 \, B b^{4} x^{5} + 3465 \, A a^{4} + 385 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 990 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 1386 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 1155 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} \sqrt{x} \]

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)/sqrt(x),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^4*x^5 + 3465*A*a^4 + 385*(4*B*a*b^3 + A*b^4)*x^4 + 990*(3*B*a^2*
b^2 + 2*A*a*b^3)*x^3 + 1386*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 1155*(B*a^4 + 4*A*a^
3*b)*x)*sqrt(x)

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Sympy [A]  time = 10.807, size = 146, normalized size = 1.34 \[ 2 A a^{4} \sqrt{x} + \frac{8 A a^{3} b x^{\frac{3}{2}}}{3} + \frac{12 A a^{2} b^{2} x^{\frac{5}{2}}}{5} + \frac{8 A a b^{3} x^{\frac{7}{2}}}{7} + \frac{2 A b^{4} x^{\frac{9}{2}}}{9} + \frac{2 B a^{4} x^{\frac{3}{2}}}{3} + \frac{8 B a^{3} b x^{\frac{5}{2}}}{5} + \frac{12 B a^{2} b^{2} x^{\frac{7}{2}}}{7} + \frac{8 B a b^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{4} x^{\frac{11}{2}}}{11} \]

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)**2/x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269203, size = 136, normalized size = 1.25 \[ \frac{2}{11} \, B b^{4} x^{\frac{11}{2}} + \frac{8}{9} \, B a b^{3} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{4} x^{\frac{9}{2}} + \frac{12}{7} \, B a^{2} b^{2} x^{\frac{7}{2}} + \frac{8}{7} \, A a b^{3} x^{\frac{7}{2}} + \frac{8}{5} \, B a^{3} b x^{\frac{5}{2}} + \frac{12}{5} \, A a^{2} b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{4} x^{\frac{3}{2}} + \frac{8}{3} \, A a^{3} b x^{\frac{3}{2}} + 2 \, A a^{4} \sqrt{x} \]

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)/sqrt(x),x, algorithm="giac")

[Out]

2/11*B*b^4*x^(11/2) + 8/9*B*a*b^3*x^(9/2) + 2/9*A*b^4*x^(9/2) + 12/7*B*a^2*b^2*x
^(7/2) + 8/7*A*a*b^3*x^(7/2) + 8/5*B*a^3*b*x^(5/2) + 12/5*A*a^2*b^2*x^(5/2) + 2/
3*B*a^4*x^(3/2) + 8/3*A*a^3*b*x^(3/2) + 2*A*a^4*sqrt(x)