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

Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]

[Out]

((A*b - a*B)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b^2) + (B*(a^2 + 2*a*
b*x + b^2*x^2)^(5/2))/(5*b^2)

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Rubi [A]  time = 0.0641828, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (A b-a B)}{4 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(4*b^2) + (B*(a^2 + 2*a*
b*x + b^2*x^2)^(5/2))/(5*b^2)

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Rubi in Sympy [A]  time = 8.93751, size = 66, normalized size = 0.96 \[ \frac{B \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*b**2) + (2*a + 2*b*x)*(A*b - B*a)*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(8*b**2)

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Mathematica [A]  time = 0.0543402, size = 83, normalized size = 1.2 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^3 (2 A+B x)+10 a^2 b x (3 A+2 B x)+5 a b^2 x^2 (4 A+3 B x)+b^3 x^3 (5 A+4 B x)\right )}{20 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(10*a^3*(2*A + B*x) + 10*a^2*b*x*(3*A + 2*B*x) + 5*a*b^2*x^
2*(4*A + 3*B*x) + b^3*x^3*(5*A + 4*B*x)))/(20*(a + b*x))

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Maple [A]  time = 0.007, size = 90, normalized size = 1.3 \[{\frac{x \left ( 4\,B{b}^{3}{x}^{4}+5\,A{b}^{3}{x}^{3}+15\,{x}^{3}a{b}^{2}B+20\,Aa{b}^{2}{x}^{2}+20\,B{a}^{2}b{x}^{2}+30\,xA{a}^{2}b+10\,{a}^{3}Bx+20\,A{a}^{3} \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*x*(4*B*b^3*x^4+5*A*b^3*x^3+15*B*a*b^2*x^3+20*A*a*b^2*x^2+20*B*a^2*b*x^2+30*
A*a^2*b*x+10*B*a^3*x+20*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303939, size = 93, normalized size = 1.35 \[ \frac{1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} +{\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]

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, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273779, size = 194, normalized size = 2.81 \[ \frac{1}{5} \, B b^{3} x^{5}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, B a b^{2} x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A b^{3} x^{4}{\rm sign}\left (b x + a\right ) + B a^{2} b x^{3}{\rm sign}\left (b x + a\right ) + A a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, A a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + A a^{3} x{\rm sign}\left (b x + a\right ) - \frac{{\left (B a^{5} - 5 \, A a^{4} b\right )}{\rm sign}\left (b x + a\right )}{20 \, b^{2}} \]

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, algorithm="giac")

[Out]

1/5*B*b^3*x^5*sign(b*x + a) + 3/4*B*a*b^2*x^4*sign(b*x + a) + 1/4*A*b^3*x^4*sign
(b*x + a) + B*a^2*b*x^3*sign(b*x + a) + A*a*b^2*x^3*sign(b*x + a) + 1/2*B*a^3*x^
2*sign(b*x + a) + 3/2*A*a^2*b*x^2*sign(b*x + a) + A*a^3*x*sign(b*x + a) - 1/20*(
B*a^5 - 5*A*a^4*b)*sign(b*x + a)/b^2