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

Optimal. Leaf size=294 \[ \frac{5 a^2 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{4 (a+b x)}+\frac{5 a b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{b^5 B x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{a^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{a+b x}+\frac{5 a^3 b x \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

[Out]

-((a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (5*a^3*b*(2*A*b + a*B)*
x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^2*b^2*(A*b + a*B)*x^2*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a*b^3*(A*b + 2*a*B)*x^3*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*(a + b*x)) + (b^4*(A*b + 5*a*B)*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(4*(a + b*x)) + (b^5*B*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^
4*(5*A*b + 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 = 0.375203, antiderivative size = 294, 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 \[ \frac{5 a^2 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{4 (a+b x)}+\frac{5 a b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{b^5 B x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{a^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{a+b x}+\frac{5 a^3 b x \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

-((a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (5*a^3*b*(2*A*b + a*B)*
x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^2*b^2*(A*b + a*B)*x^2*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a*b^3*(A*b + 2*a*B)*x^3*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(3*(a + b*x)) + (b^4*(A*b + 5*a*B)*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(4*(a + b*x)) + (b^5*B*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^
4*(5*A*b + 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 = 33.7558, size = 253, normalized size = 0.86 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{2 a x} + \frac{a^{4} \left (5 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + a^{3} \left (5 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{a^{2} \left (3 a + 3 b x\right ) \left (5 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{a \left (5 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} + \left (5 a + 5 b x\right ) \left (\frac{A b}{4} + \frac{B a}{20}\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} + \frac{\left (5 A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 a} \]

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**2,x)

[Out]

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

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Mathematica [A]  time = 0.0873333, size = 128, normalized size = 0.44 \[ \frac{\sqrt{(a+b x)^2} \left (-60 a^5 A+60 a^4 x \log (x) (a B+5 A b)+300 a^4 b B x^2+300 a^3 b^2 x^2 (2 A+B x)+100 a^2 b^3 x^3 (3 A+2 B x)+25 a b^4 x^4 (4 A+3 B x)+3 b^5 x^5 (5 A+4 B x)\right )}{60 x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279017, size = 163, normalized size = 0.55 \[ \frac{12 \, B b^{5} x^{6} - 60 \, A a^{5} + 15 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 300 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 60 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x \log \left (x\right )}{60 \, x} \]

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^2,x, algorithm="fricas")

[Out]

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

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273578, size = 258, normalized size = 0.88 \[ \frac{1}{5} \, B b^{5} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, B a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A b^{5} x^{4}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, B a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, A a b^{4} x^{3}{\rm sign}\left (b x + a\right ) + 5 \, B a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, A a^{2} b^{3} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a^{4} b x{\rm sign}\left (b x + a\right ) + 10 \, A a^{3} b^{2} x{\rm sign}\left (b x + a\right ) - \frac{A a^{5}{\rm sign}\left (b x + a\right )}{x} +{\left (B a^{5}{\rm sign}\left (b x + a\right ) + 5 \, A a^{4} b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) \]

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^2,x, algorithm="giac")

[Out]

1/5*B*b^5*x^5*sign(b*x + a) + 5/4*B*a*b^4*x^4*sign(b*x + a) + 1/4*A*b^5*x^4*sign
(b*x + a) + 10/3*B*a^2*b^3*x^3*sign(b*x + a) + 5/3*A*a*b^4*x^3*sign(b*x + a) + 5
*B*a^3*b^2*x^2*sign(b*x + a) + 5*A*a^2*b^3*x^2*sign(b*x + a) + 5*B*a^4*b*x*sign(
b*x + a) + 10*A*a^3*b^2*x*sign(b*x + a) - A*a^5*sign(b*x + a)/x + (B*a^5*sign(b*
x + a) + 5*A*a^4*b*sign(b*x + a))*ln(abs(x))