3.675 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=187 \[ -\frac{A (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 a x^4}-\frac{3 a^2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.197772, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{A (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 a x^4}-\frac{3 a^2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 B \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^5,x]

[Out]

-(a^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (3*a^2*b*B*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (3*a*b^2*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(x*(a + b*x)) - (A*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*a*x^4) + (b^
3*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 = 28.8702, size = 165, normalized size = 0.88 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 a x^{4}} - \frac{B a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{B b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} - \frac{B b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x^{2}} - \frac{B \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

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)**(3/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0600371, size = 88, normalized size = 0.47 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 (3 A+4 B x)+6 a^2 b x (2 A+3 B x)+18 a b^2 x^2 (A+2 B x)+12 A b^3 x^3-12 b^3 B x^4 \log (x)\right )}{12 x^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^5,x]

[Out]

-(Sqrt[(a + b*x)^2]*(12*A*b^3*x^3 + 18*a*b^2*x^2*(A + 2*B*x) + 6*a^2*b*x*(2*A +
3*B*x) + a^3*(3*A + 4*B*x) - 12*b^3*B*x^4*Log[x]))/(12*x^4*(a + b*x))

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Maple [A]  time = 0.019, size = 94, normalized size = 0.5 \[ -{\frac{-12\,B{b}^{3}\ln \left ( x \right ){x}^{4}+12\,A{b}^{3}{x}^{3}+36\,B{x}^{3}a{b}^{2}+18\,A{x}^{2}a{b}^{2}+18\,B{x}^{2}{a}^{2}b+12\,A{a}^{2}bx+4\,{a}^{3}Bx+3\,A{a}^{3}}{12\, \left ( bx+a \right ) ^{3}{x}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/x^5,x)

[Out]

-1/12*((b*x+a)^2)^(3/2)*(-12*B*b^3*ln(x)*x^4+12*A*b^3*x^3+36*B*x^3*a*b^2+18*A*x^
2*a*b^2+18*B*x^2*a^2*b+12*A*a^2*b*x+4*a^3*B*x+3*A*a^3)/(b*x+a)^3/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)^(3/2)*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285931, size = 101, normalized size = 0.54 \[ \frac{12 \, B b^{3} x^{4} \log \left (x\right ) - 3 \, A a^{3} - 12 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} - 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/x^5,x, algorithm="fricas")

[Out]

1/12*(12*B*b^3*x^4*log(x) - 3*A*a^3 - 12*(3*B*a*b^2 + A*b^3)*x^3 - 18*(B*a^2*b +
A*a*b^2)*x^2 - 4*(B*a^3 + 3*A*a^2*b)*x)/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

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)**(3/2)/x**5,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**5, x)

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GIAC/XCAS [A]  time = 0.273089, size = 163, normalized size = 0.87 \[ B b^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{3 \, A a^{3}{\rm sign}\left (b x + a\right ) + 12 \,{\left (3 \, B a b^{2}{\rm sign}\left (b x + a\right ) + A b^{3}{\rm sign}\left (b x + a\right )\right )} x^{3} + 18 \,{\left (B a^{2} b{\rm sign}\left (b x + a\right ) + A a b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 4 \,{\left (B a^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b{\rm sign}\left (b x + a\right )\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/x^5,x, algorithm="giac")

[Out]

B*b^3*ln(abs(x))*sign(b*x + a) - 1/12*(3*A*a^3*sign(b*x + a) + 12*(3*B*a*b^2*sig
n(b*x + a) + A*b^3*sign(b*x + a))*x^3 + 18*(B*a^2*b*sign(b*x + a) + A*a*b^2*sign
(b*x + a))*x^2 + 4*(B*a^3*sign(b*x + a) + 3*A*a^2*b*sign(b*x + a))*x)/x^4