3.713 \(\int \frac{A+B x}{x^5 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=256 \[ \frac{b^2 (a+b x) (A b-a B)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{4 a x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.120323, antiderivative size = 256, 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, Rules used = {770, 77} \[ \frac{b^2 (a+b x) (A b-a B)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{4 a x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{x^5 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{A+B x}{x^5 \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{A}{a b x^5}+\frac{-A b+a B}{a^2 b x^4}+\frac{A b-a B}{a^3 x^3}+\frac{b (-A b+a B)}{a^4 x^2}-\frac{b^2 (-A b+a B)}{a^5 x}+\frac{b^3 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A (a+b x)}{4 a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (a+b x)}{3 a^2 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (A b-a B) (a+b x)}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (A b-a B) (a+b x)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (A b-a B) (a+b x) \log (x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 (A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0609364, size = 121, normalized size = 0.47 \[ -\frac{(a+b x) \left (a \left (-2 a^2 b x (2 A+3 B x)+a^3 (3 A+4 B x)+6 a b^2 x^2 (A+2 B x)-12 A b^3 x^3\right )-12 b^3 x^4 \log (x) (A b-a B)+12 b^3 x^4 (A b-a B) \log (a+b x)\right )}{12 a^5 x^4 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.015, size = 143, normalized size = 0.6 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( 12\,A\ln \left ( x \right ){x}^{4}{b}^{4}-12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{4}-12\,B\ln \left ( x \right ){x}^{4}a{b}^{3}+12\,B\ln \left ( bx+a \right ){x}^{4}a{b}^{3}+12\,aA{b}^{3}{x}^{3}-12\,B{x}^{3}{a}^{2}{b}^{2}-6\,{a}^{2}A{b}^{2}{x}^{2}+6\,B{x}^{2}{a}^{3}b+4\,{a}^{3}Abx-4\,{a}^{4}Bx-3\,A{a}^{4} \right ) }{12\,{a}^{5}{x}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^5/((b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65311, size = 250, normalized size = 0.98 \begin{align*} \frac{12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.821217, size = 189, normalized size = 0.74 \begin{align*} - \frac{3 A a^{3} + x^{3} \left (- 12 A b^{3} + 12 B a b^{2}\right ) + x^{2} \left (6 A a b^{2} - 6 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**5/((b*x+a)**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.21414, size = 254, normalized size = 0.99 \begin{align*} -\frac{{\left (B a b^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{4} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (B a b^{4} \mathrm{sgn}\left (b x + a\right ) - A b^{5} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{3 \, A a^{4} \mathrm{sgn}\left (b x + a\right ) + 12 \,{\left (B a^{2} b^{2} \mathrm{sgn}\left (b x + a\right ) - A a b^{3} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} - 6 \,{\left (B a^{3} b \mathrm{sgn}\left (b x + a\right ) - A a^{2} b^{2} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 4 \,{\left (B a^{4} \mathrm{sgn}\left (b x + a\right ) - A a^{3} b \mathrm{sgn}\left (b x + a\right )\right )} x}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

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