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

Optimal. Leaf size=144 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac{3 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4} \]

[Out]

-(a^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^4) + (3*a^2*(a + b*x)^6*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) - (3*a*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(8*b^4) + ((a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^4)

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Rubi [A]  time = 0.155343, antiderivative size = 144, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8}{9 b^4}-\frac{3 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^4) + (3*a^2*(a + b*x)^6*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^4) - (3*a*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(8*b^4) + ((a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^4)

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Rubi in Sympy [A]  time = 19.8362, size = 139, normalized size = 0.97 \[ - \frac{a^{3} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{144 b^{4}} + \frac{a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{84 b^{4}} - \frac{a x^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 b^{2}} + \frac{x^{3} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 b} \]

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)**(5/2),x)

[Out]

-a**3*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(144*b**4) + a**2*(a**2
+ 2*a*b*x + b**2*x**2)**(7/2)/(84*b**4) - a*x**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x +
b**2*x**2)**(5/2)/(48*b**2) + x**3*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**
(5/2)/(18*b)

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Mathematica [A]  time = 0.0318111, size = 77, normalized size = 0.53 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (126 a^5+504 a^4 b x+840 a^3 b^2 x^2+720 a^2 b^3 x^3+315 a b^4 x^4+56 b^5 x^5\right )}{504 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[(a + b*x)^2]*(126*a^5 + 504*a^4*b*x + 840*a^3*b^2*x^2 + 720*a^2*b^3*x^
3 + 315*a*b^4*x^4 + 56*b^5*x^5))/(504*(a + b*x))

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Maple [A]  time = 0.009, size = 74, normalized size = 0.5 \[{\frac{{x}^{4} \left ( 56\,{b}^{5}{x}^{5}+315\,a{b}^{4}{x}^{4}+720\,{a}^{2}{b}^{3}{x}^{3}+840\,{a}^{3}{b}^{2}{x}^{2}+504\,xb{a}^{4}+126\,{a}^{5} \right ) }{504\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/504*x^4*(56*b^5*x^5+315*a*b^4*x^4+720*a^2*b^3*x^3+840*a^3*b^2*x^2+504*a^4*b*x+
126*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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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)^(5/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21954, size = 76, normalized size = 0.53 \[ \frac{1}{9} \, b^{5} x^{9} + \frac{5}{8} \, a b^{4} x^{8} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{3} \, a^{3} b^{2} x^{6} + a^{4} b x^{5} + \frac{1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*b^5*x^9 + 5/8*a*b^4*x^8 + 10/7*a^2*b^3*x^7 + 5/3*a^3*b^2*x^6 + a^4*b*x^5 + 1
/4*a^5*x^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212933, size = 143, normalized size = 0.99 \[ \frac{1}{9} \, b^{5} x^{9}{\rm sign}\left (b x + a\right ) + \frac{5}{8} \, a b^{4} x^{8}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, a^{3} b^{2} x^{6}{\rm sign}\left (b x + a\right ) + a^{4} b x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a^{5} x^{4}{\rm sign}\left (b x + a\right ) - \frac{a^{9}{\rm sign}\left (b x + a\right )}{504 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*b^5*x^9*sign(b*x + a) + 5/8*a*b^4*x^8*sign(b*x + a) + 10/7*a^2*b^3*x^7*sign(
b*x + a) + 5/3*a^3*b^2*x^6*sign(b*x + a) + a^4*b*x^5*sign(b*x + a) + 1/4*a^5*x^4
*sign(b*x + a) - 1/504*a^9*sign(b*x + a)/b^4