3.695 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=293 \[ -\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^2 (a+b x)}+\frac{b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x (a+b x)}+\frac{b^5 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{4 x^4 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^3 (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (5*a^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(x^2*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(x*(a + b*x)) + (b^5*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (b^
4*(A*b + 5*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 = 0.358091, antiderivative size = 293, 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 \[ -\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^2 (a+b x)}+\frac{b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x (a+b x)}+\frac{b^5 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{4 x^4 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^6,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (5*a^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(x^2*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(x*(a + b*x)) + (b^5*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (b^
4*(A*b + 5*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 = 35.0235, size = 270, normalized size = 0.92 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{10 a x^{5}} + \frac{b^{4} \left (A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + \frac{b^{4} \left (A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a} - \frac{b^{3} \left (a + b x\right ) \left (A b + 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 a x} - \frac{b^{2} \left (A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 a x^{2}} - \frac{b \left (a + b x\right ) \left (A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 a x^{3}} - \frac{\left (A b + 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{20 a x^{4}} \]

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**6,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(10*a*x**5) + b**4*(A*b + 5
*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + b**4*(A*b + 5*B*a)*sqr
t(a**2 + 2*a*b*x + b**2*x**2)/a - b**3*(a + b*x)*(A*b + 5*B*a)*sqrt(a**2 + 2*a*b
*x + b**2*x**2)/(2*a*x) - b**2*(A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)
/(6*a*x**2) - b*(a + b*x)*(A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*
a*x**3) - (A*b + 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(20*a*x**4)

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Mathematica [A]  time = 0.112792, size = 127, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^5 (4 A+5 B x)+25 a^4 b x (3 A+4 B x)+100 a^3 b^2 x^2 (2 A+3 B x)+300 a^2 b^3 x^3 (A+2 B x)-60 b^4 x^5 \log (x) (5 a B+A b)+300 a A b^4 x^4-60 b^5 B x^6\right )}{60 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(300*a*A*b^4*x^4 - 60*b^5*B*x^6 + 300*a^2*b^3*x^3*(A + 2*B*x
) + 100*a^3*b^2*x^2*(2*A + 3*B*x) + 25*a^4*b*x*(3*A + 4*B*x) + 3*a^5*(4*A + 5*B*
x) - 60*b^4*(A*b + 5*a*B)*x^5*Log[x]))/(60*x^5*(a + b*x))

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Maple [A]  time = 0.02, size = 144, normalized size = 0.5 \[{\frac{60\,A\ln \left ( x \right ){x}^{5}{b}^{5}+300\,B\ln \left ( x \right ){x}^{5}a{b}^{4}+60\,B{b}^{5}{x}^{6}-300\,A{x}^{4}a{b}^{4}-600\,B{x}^{4}{a}^{2}{b}^{3}-300\,A{x}^{3}{a}^{2}{b}^{3}-300\,B{x}^{3}{a}^{3}{b}^{2}-200\,A{x}^{2}{a}^{3}{b}^{2}-100\,B{x}^{2}{a}^{4}b-75\,Ax{a}^{4}b-15\,Bx{a}^{5}-12\,A{a}^{5}}{60\,{x}^{5} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(60*A*ln(x)*x^5*b^5+300*B*ln(x)*x^5*a*b^4+60*B*b^5*x^6-30
0*A*x^4*a*b^4-600*B*x^4*a^2*b^3-300*A*x^3*a^2*b^3-300*B*x^3*a^3*b^2-200*A*x^2*a^
3*b^2-100*B*x^2*a^4*b-75*A*x*a^4*b-15*B*x*a^5-12*A*a^5)/x^5/(b*x+a)^5

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319649, size = 163, normalized size = 0.56 \[ \frac{60 \, B b^{5} x^{6} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} \log \left (x\right ) - 12 \, A a^{5} - 300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6 + 60*(5*B*a*b^4 + A*b^5)*x^5*log(x) - 12*A*a^5 - 300*(2*B*a^2
*b^3 + A*a*b^4)*x^4 - 300*(B*a^3*b^2 + A*a^2*b^3)*x^3 - 100*(B*a^4*b + 2*A*a^3*b
^2)*x^2 - 15*(B*a^5 + 5*A*a^4*b)*x)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.273528, size = 254, normalized size = 0.87 \[ B b^{5} x{\rm sign}\left (b x + a\right ) +{\left (5 \, B a b^{4}{\rm sign}\left (b x + a\right ) + A b^{5}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{12 \, A a^{5}{\rm sign}\left (b x + a\right ) + 300 \,{\left (2 \, B a^{2} b^{3}{\rm sign}\left (b x + a\right ) + A a b^{4}{\rm sign}\left (b x + a\right )\right )} x^{4} + 300 \,{\left (B a^{3} b^{2}{\rm sign}\left (b x + a\right ) + A a^{2} b^{3}{\rm sign}\left (b x + a\right )\right )} x^{3} + 100 \,{\left (B a^{4} b{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 15 \,{\left (B a^{5}{\rm sign}\left (b x + a\right ) + 5 \, A a^{4} b{\rm sign}\left (b x + a\right )\right )} x}{60 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

B*b^5*x*sign(b*x + a) + (5*B*a*b^4*sign(b*x + a) + A*b^5*sign(b*x + a))*ln(abs(x
)) - 1/60*(12*A*a^5*sign(b*x + a) + 300*(2*B*a^2*b^3*sign(b*x + a) + A*a*b^4*sig
n(b*x + a))*x^4 + 300*(B*a^3*b^2*sign(b*x + a) + A*a^2*b^3*sign(b*x + a))*x^3 +
100*(B*a^4*b*sign(b*x + a) + 2*A*a^3*b^2*sign(b*x + a))*x^2 + 15*(B*a^5*sign(b*x
+ a) + 5*A*a^4*b*sign(b*x + a))*x)/x^5