3.63 \(\int \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx\)

Optimal. Leaf size=247 \[ \frac{b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac{5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac{a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac{10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5} \]

[Out]

(a^5*x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a^4*b*x^4*(a^2 + 2*
a*b*x^3 + b^2*x^6)^(5/2))/(4*(a + b*x^3)^5) + (10*a^3*b^2*x^7*(a^2 + 2*a*b*x^3 +
b^2*x^6)^(5/2))/(7*(a + b*x^3)^5) + (a^2*b^3*x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(
5/2))/(a + b*x^3)^5 + (5*a*b^4*x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(13*(a +
b*x^3)^5) + (b^5*x^16*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(16*(a + b*x^3)^5)

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Rubi [A]  time = 0.12218, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b^5 x^{16} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{16 \left (a+b x^3\right )^5}+\frac{5 a b^4 x^{13} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{13 \left (a+b x^3\right )^5}+\frac{a^2 b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{a^5 x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{\left (a+b x^3\right )^5}+\frac{5 a^4 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{4 \left (a+b x^3\right )^5}+\frac{10 a^3 b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{7 \left (a+b x^3\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^5*x*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(a + b*x^3)^5 + (5*a^4*b*x^4*(a^2 + 2*
a*b*x^3 + b^2*x^6)^(5/2))/(4*(a + b*x^3)^5) + (10*a^3*b^2*x^7*(a^2 + 2*a*b*x^3 +
b^2*x^6)^(5/2))/(7*(a + b*x^3)^5) + (a^2*b^3*x^10*(a^2 + 2*a*b*x^3 + b^2*x^6)^(
5/2))/(a + b*x^3)^5 + (5*a*b^4*x^13*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(13*(a +
b*x^3)^5) + (b^5*x^16*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(16*(a + b*x^3)^5)

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Rubi in Sympy [A]  time = 9.86023, size = 197, normalized size = 0.8 \[ \frac{729 a^{5} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{1456 \left (a + b x^{3}\right )} + \frac{243 a^{4} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{1456} + \frac{81 a^{3} x \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{728} + \frac{9 a^{2} x \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{104} + \frac{15 a x \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{208} + \frac{x \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

729*a**5*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(1456*(a + b*x**3)) + 243*a**4*x*
sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/1456 + 81*a**3*x*(a + b*x**3)*sqrt(a**2 + 2*
a*b*x**3 + b**2*x**6)/728 + 9*a**2*x*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/104
+ 15*a*x*(a + b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/208 + x*(a**2 + 2*a
*b*x**3 + b**2*x**6)**(5/2)/16

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Mathematica [A]  time = 0.0367961, size = 81, normalized size = 0.33 \[ \frac{x \sqrt{\left (a+b x^3\right )^2} \left (1456 a^5+1820 a^4 b x^3+2080 a^3 b^2 x^6+1456 a^2 b^3 x^9+560 a b^4 x^{12}+91 b^5 x^{15}\right )}{1456 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x^3)^2]*(1456*a^5 + 1820*a^4*b*x^3 + 2080*a^3*b^2*x^6 + 1456*a^2*
b^3*x^9 + 560*a*b^4*x^12 + 91*b^5*x^15))/(1456*(a + b*x^3))

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Maple [A]  time = 0.006, size = 78, normalized size = 0.3 \[{\frac{x \left ( 91\,{b}^{5}{x}^{15}+560\,a{b}^{4}{x}^{12}+1456\,{a}^{2}{b}^{3}{x}^{9}+2080\,{a}^{3}{b}^{2}{x}^{6}+1820\,{a}^{4}b{x}^{3}+1456\,{a}^{5} \right ) }{1456\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/1456*x*(91*b^5*x^15+560*a*b^4*x^12+1456*a^2*b^3*x^9+2080*a^3*b^2*x^6+1820*a^4*
b*x^3+1456*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^3+a)^5

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Maxima [A]  time = 0.811495, size = 72, normalized size = 0.29 \[ \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/16*b^5*x^16 + 5/13*a*b^4*x^13 + a^2*b^3*x^10 + 10/7*a^3*b^2*x^7 + 5/4*a^4*b*x^
4 + a^5*x

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Fricas [A]  time = 0.258865, size = 72, normalized size = 0.29 \[ \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*b^5*x^16 + 5/13*a*b^4*x^13 + a^2*b^3*x^10 + 10/7*a^3*b^2*x^7 + 5/4*a^4*b*x^
4 + a^5*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2), x)

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GIAC/XCAS [A]  time = 0.271573, size = 136, normalized size = 0.55 \[ \frac{1}{16} \, b^{5} x^{16}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{13} \, a b^{4} x^{13}{\rm sign}\left (b x^{3} + a\right ) + a^{2} b^{3} x^{10}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{7} \, a^{3} b^{2} x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{4} \, a^{4} b x^{4}{\rm sign}\left (b x^{3} + a\right ) + a^{5} x{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="giac")

[Out]

1/16*b^5*x^16*sign(b*x^3 + a) + 5/13*a*b^4*x^13*sign(b*x^3 + a) + a^2*b^3*x^10*s
ign(b*x^3 + a) + 10/7*a^3*b^2*x^7*sign(b*x^3 + a) + 5/4*a^4*b*x^4*sign(b*x^3 + a
) + a^5*x*sign(b*x^3 + a)