3.207 \(\int \frac{1}{x^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=278 \[ \frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(10*b^2)/(a^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(4*a^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(a^
4*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2)/(a^5*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a + b*
x)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b*(a + b*x))/(a^6*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*b^2
*(a + b*x)*Log[x])/(a^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*b^2*(a + b*x)*Log[a + b*x])/(a^7*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])

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Rubi [A]  time = 0.114067, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 44} \[ \frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(10*b^2)/(a^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(4*a^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(a^
4*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2)/(a^5*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (a + b*
x)/(2*a^5*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b*(a + b*x))/(a^6*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (15*b^2
*(a + b*x)*Log[x])/(a^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*b^2*(a + b*x)*Log[a + b*x])/(a^7*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
 c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{1}{a^5 b^5 x^3}-\frac{5}{a^6 b^4 x^2}+\frac{15}{a^7 b^3 x}-\frac{1}{a^3 b^2 (a+b x)^5}-\frac{3}{a^4 b^2 (a+b x)^4}-\frac{6}{a^5 b^2 (a+b x)^3}-\frac{10}{a^6 b^2 (a+b x)^2}-\frac{15}{a^7 b^2 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 b^2 (a+b x) \log (x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0433473, size = 121, normalized size = 0.44 \[ \frac{a \left (125 a^3 b^2 x^2+260 a^2 b^3 x^3+12 a^4 b x-2 a^5+210 a b^4 x^4+60 b^5 x^5\right )+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(a*(-2*a^5 + 12*a^4*b*x + 125*a^3*b^2*x^2 + 260*a^2*b^3*x^3 + 210*a*b^4*x^4 + 60*b^5*x^5) + 60*b^2*x^2*(a + b*
x)^4*Log[x] - 60*b^2*x^2*(a + b*x)^4*Log[a + b*x])/(4*a^7*x^2*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.231, size = 218, normalized size = 0.8 \begin{align*}{\frac{ \left ( 60\,\ln \left ( x \right ){x}^{6}{b}^{6}-60\,\ln \left ( bx+a \right ){x}^{6}{b}^{6}+240\,\ln \left ( x \right ){x}^{5}a{b}^{5}-240\,\ln \left ( bx+a \right ){x}^{5}a{b}^{5}+360\,\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{4}-360\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}+60\,{x}^{5}a{b}^{5}+240\,\ln \left ( x \right ){x}^{3}{a}^{3}{b}^{3}-240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}+210\,{a}^{2}{x}^{4}{b}^{4}+60\,\ln \left ( x \right ){x}^{2}{a}^{4}{b}^{2}-60\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}+260\,{a}^{3}{x}^{3}{b}^{3}+125\,{a}^{4}{x}^{2}{b}^{2}+12\,{a}^{5}xb-2\,{a}^{6} \right ) \left ( bx+a \right ) }{4\,{x}^{2}{a}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/4*(60*ln(x)*x^6*b^6-60*ln(b*x+a)*x^6*b^6+240*ln(x)*x^5*a*b^5-240*ln(b*x+a)*x^5*a*b^5+360*ln(x)*x^4*a^2*b^4-3
60*ln(b*x+a)*x^4*a^2*b^4+60*x^5*a*b^5+240*ln(x)*x^3*a^3*b^3-240*ln(b*x+a)*x^3*a^3*b^3+210*a^2*x^4*b^4+60*ln(x)
*x^2*a^4*b^2-60*ln(b*x+a)*x^2*a^4*b^2+260*a^3*x^3*b^3+125*a^4*x^2*b^2+12*a^5*x*b-2*a^6)*(b*x+a)/x^2/a^7/((b*x+
a)^2)^(5/2)

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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(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74119, size = 455, normalized size = 1.64 \begin{align*} \frac{60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6} - 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4*(60*a*b^5*x^5 + 210*a^2*b^4*x^4 + 260*a^3*b^3*x^3 + 125*a^4*b^2*x^2 + 12*a^5*b*x - 2*a^6 - 60*(b^6*x^6 + 4
*a*b^5*x^5 + 6*a^2*b^4*x^4 + 4*a^3*b^3*x^3 + a^4*b^2*x^2)*log(b*x + a) + 60*(b^6*x^6 + 4*a*b^5*x^5 + 6*a^2*b^4
*x^4 + 4*a^3*b^3*x^3 + a^4*b^2*x^2)*log(x))/(a^7*b^4*x^6 + 4*a^8*b^3*x^5 + 6*a^9*b^2*x^4 + 4*a^10*b*x^3 + a^11
*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(1/(x**3*((a + b*x)**2)**(5/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x