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

Optimal. Leaf size=210 \[ \frac{b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 (a+b x)}+\frac{3 a b x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{a^2 x^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{4 (a+b x)}+\frac{b^3 B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^3 A x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.300273, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 (a+b x)}+\frac{3 a b x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{a^2 x^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{4 (a+b x)}+\frac{b^3 B x^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{a^3 A x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (a^2*(3*A*b + a*B)*x^4
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (3*a*b*(A*b + a*B)*x^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (b^2*(A*b + 3*a*B)*x^6*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(6*(a + b*x)) + (b^3*B*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 = 22.8605, size = 168, normalized size = 0.8 \[ \frac{B x^{3} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{14 b} + \frac{a^{2} \left (2 a + 2 b x\right ) \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{168 b^{4}} - \frac{a \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{105 b^{4}} + \frac{x^{2} \left (2 a + 2 b x\right ) \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{84 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**3*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(14*b) + a**2*(2*a + 2*
b*x)*(7*A*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(168*b**4) - a*(7*A*b -
3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(105*b**4) + x**2*(2*a + 2*b*x)*(7*A
*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(84*b**2)

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Mathematica [A]  time = 0.0556159, size = 87, normalized size = 0.41 \[ \frac{x^3 \sqrt{(a+b x)^2} \left (35 a^3 (4 A+3 B x)+63 a^2 b x (5 A+4 B x)+42 a b^2 x^2 (6 A+5 B x)+10 b^3 x^3 (7 A+6 B x)\right )}{420 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[(a + b*x)^2]*(35*a^3*(4*A + 3*B*x) + 63*a^2*b*x*(5*A + 4*B*x) + 42*a*b
^2*x^2*(6*A + 5*B*x) + 10*b^3*x^3*(7*A + 6*B*x)))/(420*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[{\frac{{x}^{3} \left ( 60\,B{b}^{3}{x}^{4}+70\,A{b}^{3}{x}^{3}+210\,{x}^{3}a{b}^{2}B+252\,{x}^{2}a{b}^{2}A+252\,{x}^{2}B{a}^{2}b+315\,xA{a}^{2}b+105\,{a}^{3}Bx+140\,A{a}^{3} \right ) }{420\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/420*x^3*(60*B*b^3*x^4+70*A*b^3*x^3+210*B*a*b^2*x^3+252*A*a*b^2*x^2+252*B*a^2*b
*x^2+315*A*a^2*b*x+105*B*a^3*x+140*A*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)*(B*x + A)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294317, size = 99, normalized size = 0.47 \[ \frac{1}{7} \, B b^{3} x^{7} + \frac{1}{3} \, A a^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b^3*x^7 + 1/3*A*a^3*x^3 + 1/6*(3*B*a*b^2 + A*b^3)*x^6 + 3/5*(B*a^2*b + A*a
*b^2)*x^5 + 1/4*(B*a^3 + 3*A*a^2*b)*x^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270935, size = 203, normalized size = 0.97 \[ \frac{1}{7} \, B b^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a b^{2} x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, A b^{3} x^{6}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, B a^{2} b x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, A a b^{2} x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, B a^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, A a^{2} b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, A a^{3} x^{3}{\rm sign}\left (b x + a\right ) - \frac{{\left (3 \, B a^{7} - 7 \, A a^{6} b\right )}{\rm sign}\left (b x + a\right )}{420 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b^3*x^7*sign(b*x + a) + 1/2*B*a*b^2*x^6*sign(b*x + a) + 1/6*A*b^3*x^6*sign
(b*x + a) + 3/5*B*a^2*b*x^5*sign(b*x + a) + 3/5*A*a*b^2*x^5*sign(b*x + a) + 1/4*
B*a^3*x^4*sign(b*x + a) + 3/4*A*a^2*b*x^4*sign(b*x + a) + 1/3*A*a^3*x^3*sign(b*x
+ a) - 1/420*(3*B*a^7 - 7*A*a^6*b)*sign(b*x + a)/b^4