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

Optimal. Leaf size=182 \[ \frac{3 a^2 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{B (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b}+\frac{a^3 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

(3*a^2*A*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (3*a*A*b^2*x^2*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (A*b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(3*(a + b*x)) + (B*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b) + (a^3*A*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi [A]  time = 0.185182, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{3 a^2 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{B (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b}+\frac{a^3 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*A*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (3*a*A*b^2*x^2*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (A*b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(3*(a + b*x)) + (B*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b) + (a^3*A*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi in Sympy [A]  time = 22.7612, size = 150, normalized size = 0.82 \[ \frac{A a^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + A a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{A a \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} + \frac{B \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b} \]

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,x)

[Out]

A*a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + A*a**2*sqrt(a**2 + 2*
a*b*x + b**2*x**2) + A*a*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/6 + A*(a
**2 + 2*a*b*x + b**2*x**2)**(3/2)/3 + B*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**
2)**(3/2)/(8*b)

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Mathematica [A]  time = 0.0761185, size = 83, normalized size = 0.46 \[ \frac{\sqrt{(a+b x)^2} \left (12 a^3 A \log (x)+x \left (12 a^3 B+18 a^2 b (2 A+B x)+6 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )\right )}{12 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(x*(12*a^3*B + 18*a^2*b*(2*A + B*x) + 6*a*b^2*x*(3*A + 2*B*x)
+ b^3*x^2*(4*A + 3*B*x)) + 12*a^3*A*Log[x]))/(12*(a + b*x))

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Maple [A]  time = 0.012, size = 91, normalized size = 0.5 \[{\frac{3\,B{x}^{4}{b}^{3}+4\,A{b}^{3}{x}^{3}+12\,B{x}^{3}a{b}^{2}+18\,A{x}^{2}a{b}^{2}+18\,B{x}^{2}{a}^{2}b+12\,{a}^{3}A\ln \left ( x \right ) +36\,A{a}^{2}bx+12\,{a}^{3}Bx}{12\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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,x)

[Out]

1/12*((b*x+a)^2)^(3/2)*(3*B*x^4*b^3+4*A*b^3*x^3+12*B*x^3*a*b^2+18*A*x^2*a*b^2+18
*B*x^2*a^2*b+12*a^3*A*ln(x)+36*A*a^2*b*x+12*a^3*B*x)/(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308194, size = 92, normalized size = 0.51 \[ \frac{1}{4} \, B b^{3} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \]

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,x, algorithm="fricas")

[Out]

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

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

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,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273077, size = 159, normalized size = 0.87 \[ \frac{1}{4} \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + B a^{3} x{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + A a^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]

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,x, algorithm="giac")

[Out]

1/4*B*b^3*x^4*sign(b*x + a) + B*a*b^2*x^3*sign(b*x + a) + 1/3*A*b^3*x^3*sign(b*x
+ a) + 3/2*B*a^2*b*x^2*sign(b*x + a) + 3/2*A*a*b^2*x^2*sign(b*x + a) + B*a^3*x*
sign(b*x + a) + 3*A*a^2*b*x*sign(b*x + a) + A*a^3*ln(abs(x))*sign(b*x + a)