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

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

[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 = 0.307408, 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 \[ \frac{15 b^2 \log (x) (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{10 b^2}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x)}{a^6 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^5 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{a^4 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{4 a^3 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

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])

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Rubi in Sympy [A]  time = 45.3895, size = 275, normalized size = 0.99 \[ \frac{2 a + 2 b x}{8 a x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{1}{2 a^{2} x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5 \left (2 a + 2 b x\right )}{8 a^{3} x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5}{a^{4} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{15 \left (2 a + 2 b x\right )}{4 a^{5} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{15 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{7} \left (a + b x\right )} - \frac{15 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{7} \left (a + b x\right )} + \frac{15 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{7} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

(2*a + 2*b*x)/(8*a*x**2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) + 1/(2*a**2*x**2*(a
**2 + 2*a*b*x + b**2*x**2)**(3/2)) + 5*(2*a + 2*b*x)/(8*a**3*x**2*(a**2 + 2*a*b*
x + b**2*x**2)**(3/2)) + 5/(a**4*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - 15*(2*
a + 2*b*x)/(4*a**5*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + 15*b**2*sqrt(a**2 +
2*a*b*x + b**2*x**2)*log(x)/(a**7*(a + b*x)) - 15*b**2*sqrt(a**2 + 2*a*b*x + b**
2*x**2)*log(a + b*x)/(a**7*(a + b*x)) + 15*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(a
**7*x)

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Mathematica [A]  time = 0.0844618, size = 121, normalized size = 0.44 \[ \frac{a \left (-2 a^5+12 a^4 b x+125 a^3 b^2 x^2+260 a^2 b^3 x^3+210 a b^4 x^4+60 b^5 x^5\right )+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]

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 = 0.01, size = 218, normalized size = 0.8 \[{\frac{ \left ( 60\,\ln \left ( x \right ){x}^{6}{b}^{6}-60\,\ln \left ( bx+a \right ){x}^{6}{b}^{6}+240\,\ln \left ( x \right ){x}^{5}a{b}^{5}-240\,\ln \left ( bx+a \right ){x}^{5}a{b}^{5}+360\,\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{4}-360\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}+60\,{x}^{5}a{b}^{5}+240\,\ln \left ( x \right ){x}^{3}{a}^{3}{b}^{3}-240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}+210\,{x}^{4}{a}^{2}{b}^{4}+60\,\ln \left ( x \right ){x}^{2}{a}^{4}{b}^{2}-60\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}+260\,{x}^{3}{a}^{3}{b}^{3}+125\,{x}^{2}{a}^{4}{b}^{2}+12\,x{a}^{5}b-2\,{a}^{6} \right ) \left ( bx+a \right ) }{4\,{a}^{7}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251436, size = 294, normalized size = 1.06 \[ \frac{60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6} - 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3),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. \[ \int \frac{1}{x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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/XCAS [A]  time = 0.591391, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x