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

Optimal. Leaf size=282 \[ -\frac{A b-a B}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\log (x) (a+b x) (5 A b-a B)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (5 A b-a B) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 A b-a B}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 A b-a B}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A b-a B}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.487894, antiderivative size = 282, 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 \[ -\frac{A b-a B}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\log (x) (a+b x) (5 A b-a B)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (5 A b-a B) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 A b-a B}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 A b-a B}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A b-a B}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.4222, size = 270, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right )}{2 a x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (5 A b - B a\right )}{8 a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{5 A b - B a}{3 a^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (5 A b - B a\right )}{4 a^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 A b - B a}{a^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{\left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{6} \left (a + b x\right )} + \frac{\left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{6} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.183086, size = 148, normalized size = 0.52 \[ \frac{a \left (a^4 (25 B x-12 A)+a^3 b x (52 B x-125 A)+2 a^2 b^2 x^2 (21 B x-130 A)+6 a b^3 x^3 (2 B x-35 A)-60 A b^4 x^4\right )+12 x \log (x) (a+b x)^4 (a B-5 A b)+12 x (a+b x)^4 (5 A b-a B) \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(-60*A*b^4*x^4 + 6*a*b^3*x^3*(-35*A + 2*B*x) + 2*a^2*b^2*x^2*(-130*A + 21*B*x
) + a^4*(-12*A + 25*B*x) + a^3*b*x*(-125*A + 52*B*x)) + 12*(-5*A*b + a*B)*x*(a +
b*x)^4*Log[x] + 12*(5*A*b - a*B)*x*(a + b*x)^4*Log[a + b*x])/(12*a^6*x*(a + b*x
)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.028, size = 397, normalized size = 1.4 \[ -{\frac{ \left ( 240\,A\ln \left ( x \right ){x}^{2}{a}^{3}{b}^{2}+12\,B\ln \left ( bx+a \right ){x}^{5}a{b}^{4}-240\,A\ln \left ( bx+a \right ){x}^{4}a{b}^{4}+48\,B\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{3}-360\,A\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}-25\,Bx{a}^{5}+360\,A\ln \left ( x \right ){x}^{3}{a}^{2}{b}^{3}-52\,B{x}^{2}{a}^{4}b+125\,Ax{a}^{4}b-48\,B\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{3}-12\,B\ln \left ( x \right ) x{a}^{5}-48\,B\ln \left ( x \right ){x}^{2}{a}^{4}b+60\,A{x}^{4}a{b}^{4}-12\,B{x}^{4}{a}^{2}{b}^{3}+210\,A{x}^{3}{a}^{2}{b}^{3}-42\,B{x}^{3}{a}^{3}{b}^{2}+260\,A{x}^{2}{a}^{3}{b}^{2}-12\,B\ln \left ( x \right ){x}^{5}a{b}^{4}+48\,B\ln \left ( bx+a \right ){x}^{2}{a}^{4}b+72\,B\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{2}-240\,A\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}-60\,A\ln \left ( bx+a \right ) x{a}^{4}b+60\,A\ln \left ( x \right ){x}^{5}{b}^{5}+12\,A{a}^{5}+60\,A\ln \left ( x \right ) x{a}^{4}b-72\,B\ln \left ( x \right ){x}^{3}{a}^{3}{b}^{2}-60\,A\ln \left ( bx+a \right ){x}^{5}{b}^{5}+12\,B\ln \left ( bx+a \right ) x{a}^{5}+240\,A\ln \left ( x \right ){x}^{4}a{b}^{4} \right ) \left ( bx+a \right ) }{12\,{a}^{6}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(240*A*ln(x)*x^2*a^3*b^2+12*B*ln(b*x+a)*x^5*a*b^4-240*A*ln(b*x+a)*x^4*a*b^
4+48*B*ln(b*x+a)*x^4*a^2*b^3-360*A*ln(b*x+a)*x^3*a^2*b^3-25*B*x*a^5+360*A*ln(x)*
x^3*a^2*b^3-52*B*x^2*a^4*b+125*A*x*a^4*b-48*B*ln(x)*x^4*a^2*b^3-12*B*ln(x)*x*a^5
-48*B*ln(x)*x^2*a^4*b+60*A*x^4*a*b^4-12*B*x^4*a^2*b^3+210*A*x^3*a^2*b^3-42*B*x^3
*a^3*b^2+260*A*x^2*a^3*b^2-12*B*ln(x)*x^5*a*b^4+48*B*ln(b*x+a)*x^2*a^4*b+72*B*ln
(b*x+a)*x^3*a^3*b^2-240*A*ln(b*x+a)*x^2*a^3*b^2-60*A*ln(b*x+a)*x*a^4*b+60*A*ln(x
)*x^5*b^5+12*A*a^5+60*A*ln(x)*x*a^4*b-72*B*ln(x)*x^3*a^3*b^2-60*A*ln(b*x+a)*x^5*
b^5+12*B*ln(b*x+a)*x*a^5+240*A*ln(x)*x^4*a*b^4)*(b*x+a)/a^6/x/((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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317214, size = 468, normalized size = 1.66 \[ -\frac{12 \, A a^{5} - 12 \,{\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 42 \,{\left (B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 52 \,{\left (B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} - 25 \,{\left (B a^{5} - 5 \, A a^{4} b\right )} x + 12 \,{\left ({\left (B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 4 \,{\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \,{\left (B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} - 5 \, A a^{4} b\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 4 \,{\left (B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \,{\left (B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} - 5 \, A a^{4} b\right )} x\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )}} \]

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^2),x, algorithm="fricas")

[Out]

-1/12*(12*A*a^5 - 12*(B*a^2*b^3 - 5*A*a*b^4)*x^4 - 42*(B*a^3*b^2 - 5*A*a^2*b^3)*
x^3 - 52*(B*a^4*b - 5*A*a^3*b^2)*x^2 - 25*(B*a^5 - 5*A*a^4*b)*x + 12*((B*a*b^4 -
5*A*b^5)*x^5 + 4*(B*a^2*b^3 - 5*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 5*A*a^2*b^3)*x^3
+ 4*(B*a^4*b - 5*A*a^3*b^2)*x^2 + (B*a^5 - 5*A*a^4*b)*x)*log(b*x + a) - 12*((B*a
*b^4 - 5*A*b^5)*x^5 + 4*(B*a^2*b^3 - 5*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 5*A*a^2*b^3
)*x^3 + 4*(B*a^4*b - 5*A*a^3*b^2)*x^2 + (B*a^5 - 5*A*a^4*b)*x)*log(x))/(a^6*b^4*
x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.573969, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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^2),x, algorithm="giac")

[Out]

sage0*x