3.559 \(\int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]

[Out]

(a^3*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (3*a^2*b*x^10*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a*b^2*x^12*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(4*(a + b*x^2)) + (b^3*x^14*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a +
b*x^2))

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Rubi [A]  time = 0.270207, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{a b^2 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^{10} \sqrt{a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac{b^3 x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{a^3 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(a^3*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (3*a^2*b*x^10*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(10*(a + b*x^2)) + (a*b^2*x^12*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(4*(a + b*x^2)) + (b^3*x^14*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a +
b*x^2))

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Rubi in Sympy [A]  time = 17.1272, size = 131, normalized size = 0.78 \[ \frac{a^{3} x^{8} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{280 \left (a + b x^{2}\right )} + \frac{a^{2} x^{8} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{70} + \frac{a x^{8} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{28} + \frac{x^{8} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

a**3*x**8*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(280*(a + b*x**2)) + a**2*x**8*sqr
t(a**2 + 2*a*b*x**2 + b**2*x**4)/70 + a*x**8*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2
+ b**2*x**4)/28 + x**8*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/14

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Mathematica [A]  time = 0.0269624, size = 61, normalized size = 0.37 \[ \frac{x^8 \sqrt{\left (a+b x^2\right )^2} \left (35 a^3+84 a^2 b x^2+70 a b^2 x^4+20 b^3 x^6\right )}{280 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(x^8*Sqrt[(a + b*x^2)^2]*(35*a^3 + 84*a^2*b*x^2 + 70*a*b^2*x^4 + 20*b^3*x^6))/(2
80*(a + b*x^2))

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Maple [A]  time = 0.011, size = 58, normalized size = 0.4 \[{\frac{{x}^{8} \left ( 20\,{b}^{3}{x}^{6}+70\,a{b}^{2}{x}^{4}+84\,{a}^{2}b{x}^{2}+35\,{a}^{3} \right ) }{280\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/280*x^8*(20*b^3*x^6+70*a*b^2*x^4+84*a^2*b*x^2+35*a^3)*((b*x^2+a)^2)^(3/2)/(b*x
^2+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267843, size = 47, normalized size = 0.28 \[ \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^7,x, algorithm="fricas")

[Out]

1/14*b^3*x^14 + 1/4*a*b^2*x^12 + 3/10*a^2*b*x^10 + 1/8*a^3*x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{7} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(x**7*((a + b*x**2)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.270687, size = 90, normalized size = 0.54 \[ \frac{1}{14} \, b^{3} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{4} \, a b^{2} x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{10} \, a^{2} b x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{8} \, a^{3} x^{8}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^7,x, algorithm="giac")

[Out]

1/14*b^3*x^14*sign(b*x^2 + a) + 1/4*a*b^2*x^12*sign(b*x^2 + a) + 3/10*a^2*b*x^10
*sign(b*x^2 + a) + 1/8*a^3*x^8*sign(b*x^2 + a)