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

Optimal. Leaf size=296 \[ -\frac{10 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}+\frac{5 a b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{b^5 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^3 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^2 (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (10*a^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^4*(A*b + 5*a*B)*x*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(a + b*x) + (b^5*B*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (5*a
*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi [A]  time = 0.374003, antiderivative size = 296, 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 \[ -\frac{10 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}+\frac{5 a b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{b^5 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^3 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (10*a^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^4*(A*b + 5*a*B)*x*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(a + b*x) + (b^5*B*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (5*a
*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi in Sympy [A]  time = 34.5215, size = 275, normalized size = 0.93 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{8 a x^{4}} + \frac{5 a b^{3} \left (A b + 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 5 b^{3} \left (A b + 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{5 b^{3} \left (a + b x\right ) \left (A b + 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 a} - \frac{5 b^{2} \left (A b + 2 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a x} - \frac{5 b \left (a + b x\right ) \left (A b + 2 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 a x^{2}} - \frac{\left (A b + 2 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{6 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.096184, size = 126, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 (3 A+4 B x)+10 a^4 b x (2 A+3 B x)+60 a^3 b^2 x^2 (A+2 B x)+120 a^2 A b^3 x^3-60 a b^3 x^4 \log (x) (2 a B+A b)-60 a b^4 B x^5-6 b^5 x^5 (2 A+B x)\right )}{12 x^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(120*a^2*A*b^3*x^3 - 60*a*b^4*B*x^5 - 6*b^5*x^5*(2*A + B*x)
+ 60*a^3*b^2*x^2*(A + 2*B*x) + 10*a^4*b*x*(2*A + 3*B*x) + a^5*(3*A + 4*B*x) - 60
*a*b^3*(A*b + 2*a*B)*x^4*Log[x]))/(12*x^4*(a + b*x))

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Maple [A]  time = 0.021, size = 144, normalized size = 0.5 \[{\frac{6\,B{b}^{5}{x}^{6}+60\,A\ln \left ( x \right ){x}^{4}a{b}^{4}+12\,A{x}^{5}{b}^{5}+120\,B\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{3}+60\,B{x}^{5}a{b}^{4}-120\,A{x}^{3}{a}^{2}{b}^{3}-120\,B{x}^{3}{a}^{3}{b}^{2}-60\,A{x}^{2}{a}^{3}{b}^{2}-30\,B{x}^{2}{a}^{4}b-20\,Ax{a}^{4}b-4\,Bx{a}^{5}-3\,A{a}^{5}}{12\, \left ( bx+a \right ) ^{5}{x}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((b*x+a)^2)^(5/2)*(6*B*b^5*x^6+60*A*ln(x)*x^4*a*b^4+12*A*x^5*b^5+120*B*ln(x
)*x^4*a^2*b^3+60*B*x^5*a*b^4-120*A*x^3*a^2*b^3-120*B*x^3*a^3*b^2-60*A*x^2*a^3*b^
2-30*B*x^2*a^4*b-20*A*x*a^4*b-4*B*x*a^5-3*A*a^5)/(b*x+a)^5/x^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320187, size = 163, normalized size = 0.55 \[ \frac{6 \, B b^{5} x^{6} - 3 \, A a^{5} + 12 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 60 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} \log \left (x\right ) - 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(6*B*b^5*x^6 - 3*A*a^5 + 12*(5*B*a*b^4 + A*b^5)*x^5 + 60*(2*B*a^2*b^3 + A*a
*b^4)*x^4*log(x) - 120*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 30*(B*a^4*b + 2*A*a^3*b^2)*
x^2 - 4*(B*a^5 + 5*A*a^4*b)*x)/x^4

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

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**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272795, size = 254, normalized size = 0.86 \[ \frac{1}{2} \, B b^{5} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a b^{4} x{\rm sign}\left (b x + a\right ) + A b^{5} x{\rm sign}\left (b x + a\right ) + 5 \,{\left (2 \, B a^{2} b^{3}{\rm sign}\left (b x + a\right ) + A a b^{4}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, A a^{5}{\rm sign}\left (b x + a\right ) + 120 \,{\left (B a^{3} b^{2}{\rm sign}\left (b x + a\right ) + A a^{2} b^{3}{\rm sign}\left (b x + a\right )\right )} x^{3} + 30 \,{\left (B a^{4} b{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 4 \,{\left (B a^{5}{\rm sign}\left (b x + a\right ) + 5 \, A a^{4} b{\rm sign}\left (b x + a\right )\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*b^5*x^2*sign(b*x + a) + 5*B*a*b^4*x*sign(b*x + a) + A*b^5*x*sign(b*x + a)
+ 5*(2*B*a^2*b^3*sign(b*x + a) + A*a*b^4*sign(b*x + a))*ln(abs(x)) - 1/12*(3*A*a
^5*sign(b*x + a) + 120*(B*a^3*b^2*sign(b*x + a) + A*a^2*b^3*sign(b*x + a))*x^3 +
30*(B*a^4*b*sign(b*x + a) + 2*A*a^3*b^2*sign(b*x + a))*x^2 + 4*(B*a^5*sign(b*x
+ a) + 5*A*a^4*b*sign(b*x + a))*x)/x^4