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

Optimal. Leaf size=151 \[ \frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

(a^3*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (3*a^2*b*x^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a*b^2*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2
*(a + b*x)) + (b^3*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x))

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Rubi [A]  time = 0.127411, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{a b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{b^3 x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^3 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (3*a^2*b*x^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a*b^2*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2
*(a + b*x)) + (b^3*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x))

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Rubi in Sympy [A]  time = 14.6516, size = 124, normalized size = 0.82 \[ \frac{a^{3} x^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 \left (a + b x\right )} + \frac{a^{2} x^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35} + \frac{a x^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{42} + \frac{x^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(140*(a + b*x)) + a**2*x**4*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/35 + a*x**4*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/42 + x**4*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/7

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Mathematica [A]  time = 0.0231604, size = 55, normalized size = 0.36 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (35 a^3+84 a^2 b x+70 a b^2 x^2+20 b^3 x^3\right )}{140 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[(a + b*x)^2]*(35*a^3 + 84*a^2*b*x + 70*a*b^2*x^2 + 20*b^3*x^3))/(140*(
a + b*x))

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Maple [A]  time = 0.006, size = 52, normalized size = 0.3 \[{\frac{{x}^{4} \left ( 20\,{b}^{3}{x}^{3}+70\,a{b}^{2}{x}^{2}+84\,{a}^{2}bx+35\,{a}^{3} \right ) }{140\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/140*x^4*(20*b^3*x^3+70*a*b^2*x^2+84*a^2*b*x+35*a^3)*((b*x+a)^2)^(3/2)/(b*x+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^2 + 2*a*b*x + a^2)^(3/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243706, size = 47, normalized size = 0.31 \[ \frac{1}{7} \, b^{3} x^{7} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{4} \, a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^3*x^7 + 1/2*a*b^2*x^6 + 3/5*a^2*b*x^5 + 1/4*a^3*x^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209191, size = 99, normalized size = 0.66 \[ \frac{1}{7} \, b^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a b^{2} x^{6}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, a^{2} b x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a^{3} x^{4}{\rm sign}\left (b x + a\right ) - \frac{a^{7}{\rm sign}\left (b x + a\right )}{140 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^3*x^7*sign(b*x + a) + 1/2*a*b^2*x^6*sign(b*x + a) + 3/5*a^2*b*x^5*sign(b*x
+ a) + 1/4*a^3*x^4*sign(b*x + a) - 1/140*a^7*sign(b*x + a)/b^4