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

Optimal. Leaf size=157 \[ 2 a^6 A \sqrt{x}+\frac{2}{3} a^5 x^{3/2} (a B+6 A b)+\frac{6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac{10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac{10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac{2}{13} b^5 x^{13/2} (6 a B+A b)+\frac{6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac{2}{15} b^6 B x^{15/2} \]

[Out]

2*a^6*A*Sqrt[x] + (2*a^5*(6*A*b + a*B)*x^(3/2))/3 + (6*a^4*b*(5*A*b + 2*a*B)*x^(
5/2))/5 + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(7/2))/7 + (10*a^2*b^3*(3*A*b + 4*a*B)*x
^(9/2))/9 + (6*a*b^4*(2*A*b + 5*a*B)*x^(11/2))/11 + (2*b^5*(A*b + 6*a*B)*x^(13/2
))/13 + (2*b^6*B*x^(15/2))/15

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Rubi [A]  time = 0.194519, antiderivative size = 157, 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 \[ 2 a^6 A \sqrt{x}+\frac{2}{3} a^5 x^{3/2} (a B+6 A b)+\frac{6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac{10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac{10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac{2}{13} b^5 x^{13/2} (6 a B+A b)+\frac{6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac{2}{15} b^6 B x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

2*a^6*A*Sqrt[x] + (2*a^5*(6*A*b + a*B)*x^(3/2))/3 + (6*a^4*b*(5*A*b + 2*a*B)*x^(
5/2))/5 + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(7/2))/7 + (10*a^2*b^3*(3*A*b + 4*a*B)*x
^(9/2))/9 + (6*a*b^4*(2*A*b + 5*a*B)*x^(11/2))/11 + (2*b^5*(A*b + 6*a*B)*x^(13/2
))/13 + (2*b^6*B*x^(15/2))/15

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Rubi in Sympy [A]  time = 42.2759, size = 163, normalized size = 1.04 \[ 2 A a^{6} \sqrt{x} + \frac{2 B b^{6} x^{\frac{15}{2}}}{15} + \frac{2 a^{5} x^{\frac{3}{2}} \left (6 A b + B a\right )}{3} + \frac{6 a^{4} b x^{\frac{5}{2}} \left (5 A b + 2 B a\right )}{5} + \frac{10 a^{3} b^{2} x^{\frac{7}{2}} \left (4 A b + 3 B a\right )}{7} + \frac{10 a^{2} b^{3} x^{\frac{9}{2}} \left (3 A b + 4 B a\right )}{9} + \frac{6 a b^{4} x^{\frac{11}{2}} \left (2 A b + 5 B a\right )}{11} + \frac{2 b^{5} x^{\frac{13}{2}} \left (A b + 6 B a\right )}{13} \]

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**(1/2),x)

[Out]

2*A*a**6*sqrt(x) + 2*B*b**6*x**(15/2)/15 + 2*a**5*x**(3/2)*(6*A*b + B*a)/3 + 6*a
**4*b*x**(5/2)*(5*A*b + 2*B*a)/5 + 10*a**3*b**2*x**(7/2)*(4*A*b + 3*B*a)/7 + 10*
a**2*b**3*x**(9/2)*(3*A*b + 4*B*a)/9 + 6*a*b**4*x**(11/2)*(2*A*b + 5*B*a)/11 + 2
*b**5*x**(13/2)*(A*b + 6*B*a)/13

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Mathematica [A]  time = 0.0671468, size = 157, normalized size = 1. \[ 2 a^6 A \sqrt{x}+\frac{2}{3} a^5 x^{3/2} (a B+6 A b)+\frac{6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac{10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac{10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac{2}{13} b^5 x^{13/2} (6 a B+A b)+\frac{6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac{2}{15} b^6 B x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

2*a^6*A*Sqrt[x] + (2*a^5*(6*A*b + a*B)*x^(3/2))/3 + (6*a^4*b*(5*A*b + 2*a*B)*x^(
5/2))/5 + (10*a^3*b^2*(4*A*b + 3*a*B)*x^(7/2))/7 + (10*a^2*b^3*(3*A*b + 4*a*B)*x
^(9/2))/9 + (6*a*b^4*(2*A*b + 5*a*B)*x^(11/2))/11 + (2*b^5*(A*b + 6*a*B)*x^(13/2
))/13 + (2*b^6*B*x^(15/2))/15

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Maple [A]  time = 0.011, size = 148, normalized size = 0.9 \[{\frac{6006\,B{b}^{6}{x}^{7}+6930\,A{b}^{6}{x}^{6}+41580\,B{x}^{6}a{b}^{5}+49140\,aA{b}^{5}{x}^{5}+122850\,B{x}^{5}{a}^{2}{b}^{4}+150150\,{a}^{2}A{b}^{4}{x}^{4}+200200\,B{x}^{4}{a}^{3}{b}^{3}+257400\,{a}^{3}A{b}^{3}{x}^{3}+193050\,B{x}^{3}{a}^{4}{b}^{2}+270270\,{a}^{4}A{b}^{2}{x}^{2}+108108\,B{x}^{2}{a}^{5}b+180180\,{a}^{5}Abx+30030\,B{a}^{6}x+90090\,A{a}^{6}}{45045}\sqrt{x}} \]

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)^3/x^(1/2),x)

[Out]

2/45045*x^(1/2)*(3003*B*b^6*x^7+3465*A*b^6*x^6+20790*B*a*b^5*x^6+24570*A*a*b^5*x
^5+61425*B*a^2*b^4*x^5+75075*A*a^2*b^4*x^4+100100*B*a^3*b^3*x^4+128700*A*a^3*b^3
*x^3+96525*B*a^4*b^2*x^3+135135*A*a^4*b^2*x^2+54054*B*a^5*b*x^2+90090*A*a^5*b*x+
15015*B*a^6*x+45045*A*a^6)

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Maxima [A]  time = 0.697891, size = 198, normalized size = 1.26 \[ \frac{2}{15} \, B b^{6} x^{\frac{15}{2}} + 2 \, A a^{6} \sqrt{x} + \frac{2}{13} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{11}{2}} + \frac{10}{9} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{9}{2}} + \frac{10}{7} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*b^6*x^(15/2) + 2*A*a^6*sqrt(x) + 2/13*(6*B*a*b^5 + A*b^6)*x^(13/2) + 6/11
*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(11/2) + 10/9*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(9/2) +
10/7*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(7/2) + 6/5*(2*B*a^5*b + 5*A*a^4*b^2)*x^(5/2
) + 2/3*(B*a^6 + 6*A*a^5*b)*x^(3/2)

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Fricas [A]  time = 0.300154, size = 198, normalized size = 1.26 \[ \frac{2}{45045} \,{\left (3003 \, B b^{6} x^{7} + 45045 \, A a^{6} + 3465 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 12285 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 25025 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 32175 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 27027 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 15015 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*B*b^6*x^7 + 45045*A*a^6 + 3465*(6*B*a*b^5 + A*b^6)*x^6 + 12285*(5*
B*a^2*b^4 + 2*A*a*b^5)*x^5 + 25025*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 32175*(3*B*
a^4*b^2 + 4*A*a^3*b^3)*x^3 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 15015*(B*a^6
+ 6*A*a^5*b)*x)*sqrt(x)

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Sympy [A]  time = 20.4286, size = 211, normalized size = 1.34 \[ 2 A a^{6} \sqrt{x} + 4 A a^{5} b x^{\frac{3}{2}} + 6 A a^{4} b^{2} x^{\frac{5}{2}} + \frac{40 A a^{3} b^{3} x^{\frac{7}{2}}}{7} + \frac{10 A a^{2} b^{4} x^{\frac{9}{2}}}{3} + \frac{12 A a b^{5} x^{\frac{11}{2}}}{11} + \frac{2 A b^{6} x^{\frac{13}{2}}}{13} + \frac{2 B a^{6} x^{\frac{3}{2}}}{3} + \frac{12 B a^{5} b x^{\frac{5}{2}}}{5} + \frac{30 B a^{4} b^{2} x^{\frac{7}{2}}}{7} + \frac{40 B a^{3} b^{3} x^{\frac{9}{2}}}{9} + \frac{30 B a^{2} b^{4} x^{\frac{11}{2}}}{11} + \frac{12 B a b^{5} x^{\frac{13}{2}}}{13} + \frac{2 B b^{6} x^{\frac{15}{2}}}{15} \]

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**(1/2),x)

[Out]

2*A*a**6*sqrt(x) + 4*A*a**5*b*x**(3/2) + 6*A*a**4*b**2*x**(5/2) + 40*A*a**3*b**3
*x**(7/2)/7 + 10*A*a**2*b**4*x**(9/2)/3 + 12*A*a*b**5*x**(11/2)/11 + 2*A*b**6*x*
*(13/2)/13 + 2*B*a**6*x**(3/2)/3 + 12*B*a**5*b*x**(5/2)/5 + 30*B*a**4*b**2*x**(7
/2)/7 + 40*B*a**3*b**3*x**(9/2)/9 + 30*B*a**2*b**4*x**(11/2)/11 + 12*B*a*b**5*x*
*(13/2)/13 + 2*B*b**6*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.268496, size = 201, normalized size = 1.28 \[ \frac{2}{15} \, B b^{6} x^{\frac{15}{2}} + \frac{12}{13} \, B a b^{5} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{6} x^{\frac{13}{2}} + \frac{30}{11} \, B a^{2} b^{4} x^{\frac{11}{2}} + \frac{12}{11} \, A a b^{5} x^{\frac{11}{2}} + \frac{40}{9} \, B a^{3} b^{3} x^{\frac{9}{2}} + \frac{10}{3} \, A a^{2} b^{4} x^{\frac{9}{2}} + \frac{30}{7} \, B a^{4} b^{2} x^{\frac{7}{2}} + \frac{40}{7} \, A a^{3} b^{3} x^{\frac{7}{2}} + \frac{12}{5} \, B a^{5} b x^{\frac{5}{2}} + 6 \, A a^{4} b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{6} x^{\frac{3}{2}} + 4 \, A a^{5} b x^{\frac{3}{2}} + 2 \, A a^{6} \sqrt{x} \]

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)/sqrt(x),x, algorithm="giac")

[Out]

2/15*B*b^6*x^(15/2) + 12/13*B*a*b^5*x^(13/2) + 2/13*A*b^6*x^(13/2) + 30/11*B*a^2
*b^4*x^(11/2) + 12/11*A*a*b^5*x^(11/2) + 40/9*B*a^3*b^3*x^(9/2) + 10/3*A*a^2*b^4
*x^(9/2) + 30/7*B*a^4*b^2*x^(7/2) + 40/7*A*a^3*b^3*x^(7/2) + 12/5*B*a^5*b*x^(5/2
) + 6*A*a^4*b^2*x^(5/2) + 2/3*B*a^6*x^(3/2) + 4*A*a^5*b*x^(3/2) + 2*A*a^6*sqrt(x
)