3.666 \(\int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=210 \[ \frac{b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac{3 a b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{a^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac{b^3 B x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^3 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

[Out]

(a^3*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^2*(3*A*b + a*B)*x^6
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (3*a*b*(A*b + a*B)*x^7*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (b^2*(A*b + 3*a*B)*x^8*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(8*(a + b*x)) + (b^3*B*x^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*
x))

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Rubi [A]  time = 0.325335, 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 \[ \frac{b^2 x^8 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 (a+b x)}+\frac{3 a b x^7 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{a^2 x^6 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 (a+b x)}+\frac{b^3 B x^9 \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{a^3 A x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^2*(3*A*b + a*B)*x^6
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (3*a*b*(A*b + a*B)*x^7*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (b^2*(A*b + 3*a*B)*x^8*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(8*(a + b*x)) + (b^3*B*x^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*
x))

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Rubi in Sympy [A]  time = 26.0687, size = 207, normalized size = 0.99 \[ \frac{B x^{5} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{18 b} + \frac{a^{3} x^{5} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2520 b \left (a + b x\right )} + \frac{a^{2} x^{5} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{504 b} + \frac{a x^{5} \left (3 a + 3 b x\right ) \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{504 b} + \frac{x^{5} \left (9 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{72 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**5*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(18*b) + a**3*x**5*(9*A
*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2520*b*(a + b*x)) + a**2*x**5*(9*A
*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(504*b) + a*x**5*(3*a + 3*b*x)*(9*A
*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(504*b) + x**5*(9*A*b - 5*B*a)*(a**
2 + 2*a*b*x + b**2*x**2)**(3/2)/(72*b)

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Mathematica [A]  time = 0.0614569, size = 87, normalized size = 0.41 \[ \frac{x^5 \sqrt{(a+b x)^2} \left (84 a^3 (6 A+5 B x)+180 a^2 b x (7 A+6 B x)+135 a b^2 x^2 (8 A+7 B x)+35 b^3 x^3 (9 A+8 B x)\right )}{2520 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^5*Sqrt[(a + b*x)^2]*(84*a^3*(6*A + 5*B*x) + 180*a^2*b*x*(7*A + 6*B*x) + 135*a
*b^2*x^2*(8*A + 7*B*x) + 35*b^3*x^3*(9*A + 8*B*x)))/(2520*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[{\frac{{x}^{5} \left ( 280\,B{b}^{3}{x}^{4}+315\,A{b}^{3}{x}^{3}+945\,{x}^{3}a{b}^{2}B+1080\,{x}^{2}a{b}^{2}A+1080\,{x}^{2}B{a}^{2}b+1260\,xA{a}^{2}b+420\,{a}^{3}Bx+504\,A{a}^{3} \right ) }{2520\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2520*x^5*(280*B*b^3*x^4+315*A*b^3*x^3+945*B*a*b^2*x^3+1080*A*a*b^2*x^2+1080*B*
a^2*b*x^2+1260*A*a^2*b*x+420*B*a^3*x+504*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29064, size = 99, normalized size = 0.47 \[ \frac{1}{9} \, B b^{3} x^{9} + \frac{1}{5} \, A a^{3} x^{5} + \frac{1}{8} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + \frac{3}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6} \]

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^4,x, algorithm="fricas")

[Out]

1/9*B*b^3*x^9 + 1/5*A*a^3*x^5 + 1/8*(3*B*a*b^2 + A*b^3)*x^8 + 3/7*(B*a^2*b + A*a
*b^2)*x^7 + 1/6*(B*a^3 + 3*A*a^2*b)*x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(A + B*x)*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.273198, size = 203, normalized size = 0.97 \[ \frac{1}{9} \, B b^{3} x^{9}{\rm sign}\left (b x + a\right ) + \frac{3}{8} \, B a b^{2} x^{8}{\rm sign}\left (b x + a\right ) + \frac{1}{8} \, A b^{3} x^{8}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, B a^{2} b x^{7}{\rm sign}\left (b x + a\right ) + \frac{3}{7} \, A a b^{2} x^{7}{\rm sign}\left (b x + a\right ) + \frac{1}{6} \, B a^{3} x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A a^{2} b x^{6}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, A a^{3} x^{5}{\rm sign}\left (b x + a\right ) - \frac{{\left (5 \, B a^{9} - 9 \, A a^{8} b\right )}{\rm sign}\left (b x + a\right )}{2520 \, b^{6}} \]

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^4,x, algorithm="giac")

[Out]

1/9*B*b^3*x^9*sign(b*x + a) + 3/8*B*a*b^2*x^8*sign(b*x + a) + 1/8*A*b^3*x^8*sign
(b*x + a) + 3/7*B*a^2*b*x^7*sign(b*x + a) + 3/7*A*a*b^2*x^7*sign(b*x + a) + 1/6*
B*a^3*x^6*sign(b*x + a) + 1/2*A*a^2*b*x^6*sign(b*x + a) + 1/5*A*a^3*x^5*sign(b*x
+ a) - 1/2520*(5*B*a^9 - 9*A*a^8*b)*sign(b*x + a)/b^6