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

Optimal. Leaf size=200 \[ \frac{3 a b x \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{2 (a+b x)}+\frac{a^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}+\frac{b^3 B x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

[Out]

-((a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (3*a*b*(A*b + a*B)*x*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (b^2*(A*b + 3*a*B)*x^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(2*(a + b*x)) + (b^3*B*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a +
b*x)) + (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi [A]  time = 0.253815, antiderivative size = 200, 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 \[ \frac{3 a b x \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{2 (a+b x)}+\frac{a^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{a+b x}+\frac{b^3 B x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-((a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (3*a*b*(A*b + a*B)*x*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (b^2*(A*b + 3*a*B)*x^2*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(2*(a + b*x)) + (b^3*B*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a +
b*x)) + (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi in Sympy [A]  time = 24.1358, size = 177, normalized size = 0.88 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 a x} + \frac{a^{2} \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + a \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \left (3 a + 3 b x\right ) \left (\frac{A b}{2} + \frac{B a}{6}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{\left (3 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a} \]

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**2,x)

[Out]

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

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Mathematica [A]  time = 0.0660199, size = 89, normalized size = 0.44 \[ \frac{\sqrt{(a+b x)^2} \left (-6 a^3 A+6 a^2 x \log (x) (a B+3 A b)+18 a^2 b B x^2+9 a b^2 x^2 (2 A+B x)+b^3 x^3 (3 A+2 B x)\right )}{6 x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(-6*a^3*A + 18*a^2*b*B*x^2 + 9*a*b^2*x^2*(2*A + B*x) + b^3*x^
3*(3*A + 2*B*x) + 6*a^2*(3*A*b + a*B)*x*Log[x]))/(6*x*(a + b*x))

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Maple [A]  time = 0.019, size = 96, normalized size = 0.5 \[{\frac{2\,B{x}^{4}{b}^{3}+3\,A{b}^{3}{x}^{3}+9\,B{x}^{3}a{b}^{2}+18\,A\ln \left ( x \right ) x{a}^{2}b+18\,A{x}^{2}a{b}^{2}+6\,B\ln \left ( x \right ) x{a}^{3}+18\,B{x}^{2}{a}^{2}b-6\,A{a}^{3}}{6\, \left ( bx+a \right ) ^{3}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^2,x)

[Out]

1/6*((b*x+a)^2)^(3/2)*(2*B*x^4*b^3+3*A*b^3*x^3+9*B*x^3*a*b^2+18*A*ln(x)*x*a^2*b+
18*A*x^2*a*b^2+6*B*ln(x)*x*a^3+18*B*x^2*a^2*b-6*A*a^3)/(b*x+a)^3/x

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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 = 0.293783, size = 101, normalized size = 0.5 \[ \frac{2 \, B b^{3} x^{4} - 6 \, A a^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x \log \left (x\right )}{6 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/6*(2*B*b^3*x^4 - 6*A*a^3 + 3*(3*B*a*b^2 + A*b^3)*x^3 + 18*(B*a^2*b + A*a*b^2)*
x^2 + 6*(B*a^3 + 3*A*a^2*b)*x*log(x))/x

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272288, size = 161, normalized size = 0.8 \[ \frac{1}{3} \, B b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, B a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A b^{3} x^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a^{2} b x{\rm sign}\left (b x + a\right ) + 3 \, A a b^{2} x{\rm sign}\left (b x + a\right ) - \frac{A a^{3}{\rm sign}\left (b x + a\right )}{x} +{\left (B a^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

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^2,x, algorithm="giac")

[Out]

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