3.674 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=199 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{2 x^2 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac{b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}+\frac{b^3 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^3*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a +
b*x) + (b^2*(A*b + 3*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 = 0.246001, antiderivative size = 199, 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 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{2 x^2 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac{b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{a+b x}+\frac{b^3 B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^4,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(x*(a + b*x)) + (b^3*B*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a +
b*x) + (b^2*(A*b + 3*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 = 24.9494, size = 187, normalized size = 0.94 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 a x^{3}} + \frac{b^{2} \left (A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + \frac{b^{2} \left (A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a} - \frac{b \left (a + b x\right ) \left (A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 a x} - \frac{\left (A b + 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 a x^{2}} \]

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**4,x)

[Out]

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

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Mathematica [A]  time = 0.066201, size = 88, normalized size = 0.44 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 (2 A+3 B x)+9 a^2 b x (A+2 B x)-6 b^2 x^3 \log (x) (3 a B+A b)+18 a A b^2 x^2-6 b^3 B x^4\right )}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^4,x]

[Out]

-(Sqrt[(a + b*x)^2]*(18*a*A*b^2*x^2 - 6*b^3*B*x^4 + 9*a^2*b*x*(A + 2*B*x) + a^3*
(2*A + 3*B*x) - 6*b^2*(A*b + 3*a*B)*x^3*Log[x]))/(6*x^3*(a + b*x))

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Maple [A]  time = 0.02, size = 96, normalized size = 0.5 \[{\frac{6\,A\ln \left ( x \right ){x}^{3}{b}^{3}+18\,B\ln \left ( x \right ){x}^{3}a{b}^{2}+6\,B{x}^{4}{b}^{3}-18\,A{x}^{2}a{b}^{2}-18\,B{x}^{2}{a}^{2}b-9\,A{a}^{2}bx-3\,{a}^{3}Bx-2\,A{a}^{3}}{6\, \left ( bx+a \right ) ^{3}{x}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^4,x)

[Out]

1/6*((b*x+a)^2)^(3/2)*(6*A*ln(x)*x^3*b^3+18*B*ln(x)*x^3*a*b^2+6*B*x^4*b^3-18*A*x
^2*a*b^2-18*B*x^2*a^2*b-9*A*a^2*b*x-3*a^3*B*x-2*A*a^3)/(b*x+a)^3/x^3

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302323, size = 101, normalized size = 0.51 \[ \frac{6 \, B b^{3} x^{4} + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} \log \left (x\right ) - 2 \, A a^{3} - 18 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/6*(6*B*b^3*x^4 + 6*(3*B*a*b^2 + A*b^3)*x^3*log(x) - 2*A*a^3 - 18*(B*a^2*b + A*
a*b^2)*x^2 - 3*(B*a^3 + 3*A*a^2*b)*x)/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

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**4,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.273398, size = 159, normalized size = 0.8 \[ B b^{3} x{\rm sign}\left (b x + a\right ) +{\left (3 \, B a b^{2}{\rm sign}\left (b x + a\right ) + A b^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{3}{\rm sign}\left (b x + a\right ) + 18 \,{\left (B a^{2} b{\rm sign}\left (b x + a\right ) + A a b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (B a^{3}{\rm sign}\left (b x + a\right ) + 3 \, A a^{2} b{\rm sign}\left (b x + a\right )\right )} x}{6 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

B*b^3*x*sign(b*x + a) + (3*B*a*b^2*sign(b*x + a) + A*b^3*sign(b*x + a))*ln(abs(x
)) - 1/6*(2*A*a^3*sign(b*x + a) + 18*(B*a^2*b*sign(b*x + a) + A*a*b^2*sign(b*x +
a))*x^2 + 3*(B*a^3*sign(b*x + a) + 3*A*a^2*b*sign(b*x + a))*x)/x^3