3.705 \(\int \frac{x^3 (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=212 \[ \frac{x^3 (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^2 (a+b x) (A b-a B)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^4 (a+b x)}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 (a+b x) (A b-a B) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.323933, antiderivative size = 212, 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 \[ \frac{x^3 (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 x (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a x^2 (a+b x) (A b-a B)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^4 (a+b x)}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^3 (a+b x) (A b-a B) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.5191, size = 206, normalized size = 0.97 \[ \frac{B x^{4} \left (2 a + 2 b x\right )}{8 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a^{3} \left (a + b x\right ) \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{a^{2} \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{5}} - \frac{a x^{2} \left (2 a + 2 b x\right ) \left (A b - B a\right )}{4 b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{x^{3} \left (2 a + 2 b x\right ) \left (A b - B a\right )}{6 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

B*x**4*(2*a + 2*b*x)/(8*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - a**3*(a + b*x)*(A*
b - B*a)*log(a + b*x)/(b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + a**2*(A*b - B*a)
*sqrt(a**2 + 2*a*b*x + b**2*x**2)/b**5 - a*x**2*(2*a + 2*b*x)*(A*b - B*a)/(4*b**
3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + x**3*(2*a + 2*b*x)*(A*b - B*a)/(6*b**2*sqr
t(a**2 + 2*a*b*x + b**2*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0873419, size = 96, normalized size = 0.45 \[ \frac{(a+b x) \left (12 a^3 (a B-A b) \log (a+b x)+b x \left (-12 a^3 B+6 a^2 b (2 A+B x)-2 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )\right )}{12 b^5 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*(b*x*(-12*a^3*B + 6*a^2*b*(2*A + B*x) - 2*a*b^2*x*(3*A + 2*B*x) + b^3
*x^2*(4*A + 3*B*x)) + 12*a^3*(-(A*b) + a*B)*Log[a + b*x]))/(12*b^5*Sqrt[(a + b*x
)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 114, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( -3\,{b}^{4}B{x}^{4}-4\,A{x}^{3}{b}^{4}+4\,B{x}^{3}a{b}^{3}+6\,A{x}^{2}a{b}^{3}-6\,B{x}^{2}{a}^{2}{b}^{2}+12\,A\ln \left ( bx+a \right ){a}^{3}b-12\,Ax{a}^{2}{b}^{2}-12\,B\ln \left ( bx+a \right ){a}^{4}+12\,Bx{a}^{3}b \right ) }{12\,{b}^{5}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

-1/12*(b*x+a)*(-3*b^4*B*x^4-4*A*x^3*b^4+4*B*x^3*a*b^3+6*A*x^2*a*b^3-6*B*x^2*a^2*
b^2+12*A*ln(b*x+a)*a^3*b-12*A*x*a^2*b^2-12*B*ln(b*x+a)*a^4+12*B*x*a^3*b)/((b*x+a
)^2)^(1/2)/b^5

_______________________________________________________________________________________

Maxima [A]  time = 0.692588, size = 377, normalized size = 1.78 \[ \frac{13 \, B a^{4} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{5 \, A a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, A a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{13 \, B a^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{13 \, B a^{2} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} - \frac{5 \, A a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B x^{3}}{4 \, b^{2}} - \frac{7 \, B a^{4} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} + \frac{2 \, A a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a x^{2}}{12 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A x^{2}}{3 \, b^{2}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3}}{6 \, b^{5}} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{2}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

13/6*B*a^4*log(x + a/b)/(b^2)^(5/2) - 5/3*A*a^3*b*log(x + a/b)/(b^2)^(5/2) + 5/3
*A*a^2*x/(b^2)^(3/2) - 13/6*B*a^3*x/((b^2)^(3/2)*b) + 13/12*B*a^2*x^2/(sqrt(b^2)
*b^2) - 5/6*A*a*x^2/(sqrt(b^2)*b) + 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*x^3/b^2
- 7/6*B*a^4*sqrt(b^(-2))*log(x + a/b)/b^4 + 2/3*A*a^3*sqrt(b^(-2))*log(x + a/b)/
b^3 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*x^2/b^3 + 1/3*sqrt(b^2*x^2 + 2*a*b*
x + a^2)*A*x^2/b^2 + 7/6*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^3/b^5 - 2/3*sqrt(b^2*
x^2 + 2*a*b*x + a^2)*A*a^2/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.299461, size = 127, normalized size = 0.6 \[ \frac{3 \, B b^{4} x^{4} - 4 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 6 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 12 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x + 12 \,{\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/sqrt((b*x + a)^2),x, algorithm="fricas")

[Out]

1/12*(3*B*b^4*x^4 - 4*(B*a*b^3 - A*b^4)*x^3 + 6*(B*a^2*b^2 - A*a*b^3)*x^2 - 12*(
B*a^3*b - A*a^2*b^2)*x + 12*(B*a^4 - A*a^3*b)*log(b*x + a))/b^5

_______________________________________________________________________________________

Sympy [A]  time = 1.57843, size = 78, normalized size = 0.37 \[ \frac{B x^{4}}{4 b} + \frac{a^{3} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} + \frac{x^{2} \left (- A a b + B a^{2}\right )}{2 b^{3}} - \frac{x \left (- A a^{2} b + B a^{3}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269235, size = 200, normalized size = 0.94 \[ \frac{3 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) - 4 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 4 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) - 6 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) - 12 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 12 \, A a^{2} b x{\rm sign}\left (b x + a\right )}{12 \, b^{4}} + \frac{{\left (B a^{4}{\rm sign}\left (b x + a\right ) - A a^{3} b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^3/sqrt((b*x + a)^2),x, algorithm="giac")

[Out]

1/12*(3*B*b^3*x^4*sign(b*x + a) - 4*B*a*b^2*x^3*sign(b*x + a) + 4*A*b^3*x^3*sign
(b*x + a) + 6*B*a^2*b*x^2*sign(b*x + a) - 6*A*a*b^2*x^2*sign(b*x + a) - 12*B*a^3
*x*sign(b*x + a) + 12*A*a^2*b*x*sign(b*x + a))/b^4 + (B*a^4*sign(b*x + a) - A*a^
3*b*sign(b*x + a))*ln(abs(b*x + a))/b^5