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

Optimal. Leaf size=162 \[ \frac{3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac{3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac{b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}+\frac{a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.0801765, antiderivative size = 162, 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 \[ \frac{3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac{3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac{b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}+\frac{a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.60486, size = 129, normalized size = 0.8 \[ \frac{81 a^{3} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{140 \left (a + b x^{3}\right )} + \frac{27 a^{2} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{140} + \frac{9 a x \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{70} + \frac{x \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*a**3*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(140*(a + b*x**3)) + 27*a**2*x*sqr
t(a**2 + 2*a*b*x**3 + b**2*x**6)/140 + 9*a*x*(a + b*x**3)*sqrt(a**2 + 2*a*b*x**3
+ b**2*x**6)/70 + x*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/10

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Mathematica [A]  time = 0.0311651, size = 59, normalized size = 0.36 \[ \frac{x \sqrt{\left (a+b x^3\right )^2} \left (140 a^3+105 a^2 b x^3+60 a b^2 x^6+14 b^3 x^9\right )}{140 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x^3)^2]*(140*a^3 + 105*a^2*b*x^3 + 60*a*b^2*x^6 + 14*b^3*x^9))/(1
40*(a + b*x^3))

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Maple [A]  time = 0.005, size = 56, normalized size = 0.4 \[{\frac{x \left ( 14\,{b}^{3}{x}^{9}+60\,a{b}^{2}{x}^{6}+105\,{a}^{2}b{x}^{3}+140\,{a}^{3} \right ) }{140\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/140*x*(14*b^3*x^9+60*a*b^2*x^6+105*a^2*b*x^3+140*a^3)*((b*x^3+a)^2)^(3/2)/(b*x
^3+a)^3

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Maxima [A]  time = 0.78541, size = 43, normalized size = 0.27 \[ \frac{1}{10} \, b^{3} x^{10} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{4} \, a^{2} b x^{4} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.250921, size = 43, normalized size = 0.27 \[ \frac{1}{10} \, b^{3} x^{10} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{4} \, a^{2} b x^{4} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.259844, size = 86, normalized size = 0.53 \[ \frac{1}{10} \, b^{3} x^{10}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{7} \, a b^{2} x^{7}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{4} \, a^{2} b x^{4}{\rm sign}\left (b x^{3} + a\right ) + a^{3} x{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*x^10*sign(b*x^3 + a) + 3/7*a*b^2*x^7*sign(b*x^3 + a) + 3/4*a^2*b*x^4*si
gn(b*x^3 + a) + a^3*x*sign(b*x^3 + a)