3.727 \(\int \frac{A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{A b-a B}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

A/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + A/(3*a^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + A/
(2*a^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*(a + b*x)*Log[x])/(a^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]) - (A*(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 = 0.317069, antiderivative size = 210, 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 \[ \frac{A b-a B}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

A/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + A/(3*a^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + A/
(2*a^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*(a + b*x)*Log[x])/(a^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2]) - (A*(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 in Sympy [A]  time = 48.583, size = 199, normalized size = 0.95 \[ \frac{A}{3 a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{A \left (2 a + 2 b x\right )}{4 a^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{A}{a^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{A \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{5} \left (a + b x\right )} - \frac{A \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{5} \left (a + b x\right )} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{8 a b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

A/(3*a**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + A*(2*a + 2*b*x)/(4*a**3*(a**2 +
2*a*b*x + b**2*x**2)**(3/2)) + A/(a**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + A*sq
rt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a**5*(a + b*x)) - A*sqrt(a**2 + 2*a*b*x +
b**2*x**2)*log(a + b*x)/(a**5*(a + b*x)) + (2*a + 2*b*x)*(A*b - B*a)/(8*a*b*(a*
*2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.105655, size = 104, normalized size = 0.5 \[ \frac{a \left (-3 a^4 B+25 a^3 A b+52 a^2 A b^2 x+42 a A b^3 x^2+12 A b^4 x^3\right )+12 A b \log (x) (a+b x)^4-12 A b (a+b x)^4 \log (a+b x)}{12 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(25*a^3*A*b - 3*a^4*B + 52*a^2*A*b^2*x + 42*a*A*b^3*x^2 + 12*A*b^4*x^3) + 12*
A*b*(a + b*x)^4*Log[x] - 12*A*b*(a + b*x)^4*Log[a + b*x])/(12*a^5*b*(a + b*x)^3*
Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.023, size = 205, normalized size = 1. \[{\frac{ \left ( 12\,A\ln \left ( x \right ){x}^{4}{b}^{5}-12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}+48\,A\ln \left ( x \right ){x}^{3}a{b}^{4}-48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}+72\,A\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{3}-72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}+12\,A{x}^{3}a{b}^{4}+48\,A\ln \left ( x \right ) x{a}^{3}{b}^{2}-48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}+42\,A{x}^{2}{a}^{2}{b}^{3}+12\,A\ln \left ( x \right ){a}^{4}b-12\,A\ln \left ( bx+a \right ){a}^{4}b+52\,Ax{a}^{3}{b}^{2}+25\,A{a}^{4}b-3\,B{a}^{5} \right ) \left ( bx+a \right ) }{12\,b{a}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*A*ln(x)*x^4*b^5-12*A*ln(b*x+a)*x^4*b^5+48*A*ln(x)*x^3*a*b^4-48*A*ln(b*x
+a)*x^3*a*b^4+72*A*ln(x)*x^2*a^2*b^3-72*A*ln(b*x+a)*x^2*a^2*b^3+12*A*x^3*a*b^4+4
8*A*ln(x)*x*a^3*b^2-48*A*ln(b*x+a)*x*a^3*b^2+42*A*x^2*a^2*b^3+12*A*ln(x)*a^4*b-1
2*A*ln(b*x+a)*a^4*b+52*A*x*a^3*b^2+25*A*a^4*b-3*B*a^5)*(b*x+a)/b/a^5/((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((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330617, size = 274, normalized size = 1.3 \[ \frac{12 \, A a b^{4} x^{3} + 42 \, A a^{2} b^{3} x^{2} + 52 \, A a^{3} b^{2} x - 3 \, B a^{5} + 25 \, A a^{4} b - 12 \,{\left (A b^{5} x^{4} + 4 \, A a b^{4} x^{3} + 6 \, A a^{2} b^{3} x^{2} + 4 \, A a^{3} b^{2} x + A a^{4} b\right )} \log \left (b x + a\right ) + 12 \,{\left (A b^{5} x^{4} + 4 \, A a b^{4} x^{3} + 6 \, A a^{2} b^{3} x^{2} + 4 \, A a^{3} b^{2} x + A a^{4} b\right )} \log \left (x\right )}{12 \,{\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*A*a*b^4*x^3 + 42*A*a^2*b^3*x^2 + 52*A*a^3*b^2*x - 3*B*a^5 + 25*A*a^4*b
- 12*(A*b^5*x^4 + 4*A*a*b^4*x^3 + 6*A*a^2*b^3*x^2 + 4*A*a^3*b^2*x + A*a^4*b)*log
(b*x + a) + 12*(A*b^5*x^4 + 4*A*a*b^4*x^3 + 6*A*a^2*b^3*x^2 + 4*A*a^3*b^2*x + A*
a^4*b)*log(x))/(a^5*b^5*x^4 + 4*a^6*b^4*x^3 + 6*a^7*b^3*x^2 + 4*a^8*b^2*x + a^9*
b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.593037, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x