3.706 \(\int \frac{x^2 (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{x^2 (a+b x) (A b-a B)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (a+b x) (A b-a B) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^3 (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-((a*(A*b - a*B)*x*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((A*b - a*B
)*x^2*(a + b*x))/(2*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^3*(a + b*x))/(3*b*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (a^2*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(b^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.270503, antiderivative size = 166, 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 \[ \frac{x^2 (a+b x) (A b-a B)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (a+b x) (A b-a B) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^3 (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(A*b - a*B)*x*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + ((A*b - a*B
)*x^2*(a + b*x))/(2*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*x^3*(a + b*x))/(3*b*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (a^2*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(b^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 27.3032, size = 158, normalized size = 0.95 \[ \frac{B x^{3} \left (2 a + 2 b x\right )}{6 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{a^{2} \left (a + b x\right ) \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{4}} + \frac{x^{2} \left (2 a + 2 b x\right ) \left (A b - B a\right )}{4 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

B*x**3*(2*a + 2*b*x)/(6*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + a**2*(a + b*x)*(A*
b - B*a)*log(a + b*x)/(b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - a*(A*b - B*a)*sq
rt(a**2 + 2*a*b*x + b**2*x**2)/b**4 + x**2*(2*a + 2*b*x)*(A*b - B*a)/(4*b**2*sqr
t(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0734073, size = 77, normalized size = 0.46 \[ \frac{(a+b x) \left (b x \left (6 a^2 B-3 a b (2 A+B x)+b^2 x (3 A+2 B x)\right )+6 a^2 (A b-a B) \log (a+b x)\right )}{6 b^4 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 90, normalized size = 0.5 \[{\frac{ \left ( bx+a \right ) \left ( 2\,B{b}^{3}{x}^{3}+3\,A{b}^{3}{x}^{2}-3\,B{x}^{2}a{b}^{2}+6\,A\ln \left ( bx+a \right ){a}^{2}b-6\,Axa{b}^{2}-6\,B\ln \left ( bx+a \right ){a}^{3}+6\,Bx{a}^{2}b \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.695047, size = 225, normalized size = 1.36 \[ -\frac{5 \, B a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{A a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, B a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{A a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{A x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \, B a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{2 \, B a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B x^{2}}{3 \, b^{2}} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^2/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

-5/3*B*a^3*b*log(x + a/b)/(b^2)^(5/2) + A*a^2*b^2*log(x + a/b)/(b^2)^(5/2) + 5/3
*B*a^2*x/(b^2)^(3/2) - A*a*b*x/(b^2)^(3/2) + 1/2*A*x^2/sqrt(b^2) - 5/6*B*a*x^2/(
sqrt(b^2)*b) + 2/3*B*a^3*sqrt(b^(-2))*log(x + a/b)/b^3 + 1/3*sqrt(b^2*x^2 + 2*a*
b*x + a^2)*B*x^2/b^2 - 2/3*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^2/b^4

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Fricas [A]  time = 0.305425, size = 96, normalized size = 0.58 \[ \frac{2 \, B b^{3} x^{3} - 3 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 6 \,{\left (B a^{2} b - A a b^{2}\right )} x - 6 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^2/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.53694, size = 58, normalized size = 0.35 \[ \frac{B x^{3}}{3 b} - \frac{a^{2} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{4}} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} + \frac{x \left (- A a b + B a^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271425, size = 153, normalized size = 0.92 \[ \frac{2 \, B b^{2} x^{3}{\rm sign}\left (b x + a\right ) - 3 \, B a b x^{2}{\rm sign}\left (b x + a\right ) + 3 \, A b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} x{\rm sign}\left (b x + a\right ) - 6 \, A a b x{\rm sign}\left (b x + a\right )}{6 \, b^{3}} - \frac{{\left (B a^{3}{\rm sign}\left (b x + a\right ) - A a^{2} b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^2/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

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