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

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

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^2*(A*b + 3*a*B)*x*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(a + b*x) + (b^3*B*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b
*x)) + (3*a*b*(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 = 0.253533, 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 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{x (a+b x)}+\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}+\frac{3 a b \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^3 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^2*(A*b + 3*a*B)*x*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(a + b*x) + (b^3*B*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b
*x)) + (3*a*b*(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 = 23.8324, size = 177, normalized size = 0.88 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 a x^{2}} + \frac{3 a b \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 3 b \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{3 b \left (a + b x\right ) \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 a} - \frac{\left (A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{a x} \]

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**3,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(4*a*x**2) + 3*a*b*(A*b + B
*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + 3*b*(A*b + B*a)*sqrt(a**
2 + 2*a*b*x + b**2*x**2) + 3*b*(a + b*x)*(A*b + B*a)*sqrt(a**2 + 2*a*b*x + b**2*
x**2)/(2*a) - (A*b + B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(a*x)

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Mathematica [A]  time = 0.0789011, size = 85, normalized size = 0.42 \[ \frac{\sqrt{(a+b x)^2} \left (a^3 (-(A+2 B x))-6 a^2 A b x+6 a b x^2 \log (x) (a B+A b)+6 a b^2 B x^3+b^3 x^3 (2 A+B x)\right )}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 95, normalized size = 0.5 \[{\frac{B{x}^{4}{b}^{3}+6\,A\ln \left ( x \right ){x}^{2}a{b}^{2}+2\,A{b}^{3}{x}^{3}+6\,B\ln \left ( x \right ){x}^{2}{a}^{2}b+6\,B{x}^{3}a{b}^{2}-6\,A{a}^{2}bx-2\,{a}^{3}Bx-A{a}^{3}}{2\, \left ( bx+a \right ) ^{3}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^3,x)

[Out]

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

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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 = 0.303621, size = 100, normalized size = 0.5 \[ \frac{B b^{3} x^{4} - A a^{3} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

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

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

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272149, size = 158, normalized size = 0.79 \[ \frac{1}{2} \, B b^{3} x^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a b^{2} x{\rm sign}\left (b x + a\right ) + A b^{3} x{\rm sign}\left (b x + a\right ) + 3 \,{\left (B a^{2} b{\rm sign}\left (b x + a\right ) + A a b^{2}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{3}{\rm sign}\left (b x + a\right ) + 2 \,{\left (B a^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b{\rm sign}\left (b x + a\right )\right )} x}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

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