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

Optimal. Leaf size=210 \[ \frac{b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{a b x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 (a+b x)}+\frac{a^2 x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{b^3 B x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{a^3 A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

(a^3*A*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (a^2*(3*A*b + a*B)*x^5
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a*b*(A*b + a*B)*x^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (b^2*(A*b + 3*a*B)*x^7*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(7*(a + b*x)) + (b^3*B*x^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(a + b*x)
)

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Rubi [A]  time = 0.312047, 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 \[ \frac{b^2 x^7 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{a b x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 (a+b x)}+\frac{a^2 x^5 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{b^3 B x^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac{a^3 A x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (a^2*(3*A*b + a*B)*x^5
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a*b*(A*b + a*B)*x^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (b^2*(A*b + 3*a*B)*x^7*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(7*(a + b*x)) + (b^3*B*x^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(a + b*x)
)

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Rubi in Sympy [A]  time = 25.9573, size = 201, normalized size = 0.96 \[ \frac{B x^{4} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{16 b} + \frac{a^{3} x^{4} \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 b \left (a + b x\right )} + \frac{a^{2} x^{4} \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{70 b} + \frac{a x^{4} \left (3 a + 3 b x\right ) \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{84 b} + \frac{x^{4} \left (2 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{14 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**4*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(16*b) + a**3*x**4*(2*A
*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*b*(a + b*x)) + a**2*x**4*(2*A*b
- B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(70*b) + a*x**4*(3*a + 3*b*x)*(2*A*b - B
*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(84*b) + x**4*(2*A*b - B*a)*(a**2 + 2*a*b*x
+ b**2*x**2)**(3/2)/(14*b)

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Mathematica [A]  time = 0.0568639, size = 87, normalized size = 0.41 \[ \frac{x^4 \sqrt{(a+b x)^2} \left (14 a^3 (5 A+4 B x)+28 a^2 b x (6 A+5 B x)+20 a b^2 x^2 (7 A+6 B x)+5 b^3 x^3 (8 A+7 B x)\right )}{280 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[(a + b*x)^2]*(14*a^3*(5*A + 4*B*x) + 28*a^2*b*x*(6*A + 5*B*x) + 20*a*b
^2*x^2*(7*A + 6*B*x) + 5*b^3*x^3*(8*A + 7*B*x)))/(280*(a + b*x))

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Maple [A]  time = 0.008, size = 92, normalized size = 0.4 \[{\frac{{x}^{4} \left ( 35\,B{b}^{3}{x}^{4}+40\,A{b}^{3}{x}^{3}+120\,{x}^{3}a{b}^{2}B+140\,{x}^{2}a{b}^{2}A+140\,{x}^{2}B{a}^{2}b+168\,xA{a}^{2}b+56\,{a}^{3}Bx+70\,A{a}^{3} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/280*x^4*(35*B*b^3*x^4+40*A*b^3*x^3+120*B*a*b^2*x^3+140*A*a*b^2*x^2+140*B*a^2*b
*x^2+168*A*a^2*b*x+56*B*a^3*x+70*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296282, size = 99, normalized size = 0.47 \[ \frac{1}{8} \, B b^{3} x^{8} + \frac{1}{4} \, A a^{3} x^{4} + \frac{1}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5} \]

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^3,x, algorithm="fricas")

[Out]

1/8*B*b^3*x^8 + 1/4*A*a^3*x^4 + 1/7*(3*B*a*b^2 + A*b^3)*x^7 + 1/2*(B*a^2*b + A*a
*b^2)*x^6 + 1/5*(B*a^3 + 3*A*a^2*b)*x^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271555, size = 201, normalized size = 0.96 \[ \frac{1}{8} \, B b^{3} x^{8}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, B a b^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{7} \, A b^{3} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a^{2} b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A a b^{2} x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, B a^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, A a^{2} b x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A a^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{{\left (B a^{8} - 2 \, A a^{7} b\right )}{\rm sign}\left (b x + a\right )}{280 \, b^{5}} \]

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^3,x, algorithm="giac")

[Out]

1/8*B*b^3*x^8*sign(b*x + a) + 3/7*B*a*b^2*x^7*sign(b*x + a) + 1/7*A*b^3*x^7*sign
(b*x + a) + 1/2*B*a^2*b*x^6*sign(b*x + a) + 1/2*A*a*b^2*x^6*sign(b*x + a) + 1/5*
B*a^3*x^5*sign(b*x + a) + 3/5*A*a^2*b*x^5*sign(b*x + a) + 1/4*A*a^3*x^4*sign(b*x
+ a) + 1/280*(B*a^8 - 2*A*a^7*b)*sign(b*x + a)/b^5