3.806 \(\int (a+b x)^2 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx\)

Optimal. Leaf size=130 \[ \frac{5}{8} a^2 x \sqrt{c x^2-\frac{a^2 c}{b^2}}+\frac{5 a b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{4 c}-\frac{5 a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2} \]

[Out]

(5*a^2*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (5*a*b*(-((a^2*c)/b^2) + c*x^2)^(3/2)
)/(12*c) + (b*(a + b*x)*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(4*c) - (5*a^4*Sqrt[c]*A
rcTanh[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

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Rubi [A]  time = 0.181774, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{5}{8} a^2 x \sqrt{c x^2-\frac{a^2 c}{b^2}}+\frac{5 a b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{4 c}-\frac{5 a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2*Sqrt[-((a^2*c)/b^2) + c*x^2],x]

[Out]

(5*a^2*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (5*a*b*(-((a^2*c)/b^2) + c*x^2)^(3/2)
)/(12*c) + (b*(a + b*x)*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(4*c) - (5*a^4*Sqrt[c]*A
rcTanh[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

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Rubi in Sympy [A]  time = 22.4028, size = 117, normalized size = 0.9 \[ - \frac{5 a^{4} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}} \right )}}{8 b^{2}} + \frac{5 a^{2} x \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{8} + \frac{5 a b \left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{12 c} + \frac{b \left (a + b x\right ) \left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(-a**2*c/b**2+c*x**2)**(1/2),x)

[Out]

-5*a**4*sqrt(c)*atanh(sqrt(c)*x/sqrt(-a**2*c/b**2 + c*x**2))/(8*b**2) + 5*a**2*x
*sqrt(-a**2*c/b**2 + c*x**2)/8 + 5*a*b*(-a**2*c/b**2 + c*x**2)**(3/2)/(12*c) + b
*(a + b*x)*(-a**2*c/b**2 + c*x**2)**(3/2)/(4*c)

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Mathematica [A]  time = 0.181677, size = 114, normalized size = 0.88 \[ \frac{\sqrt{c \left (x^2-\frac{a^2}{b^2}\right )} \left (b \sqrt{x^2-\frac{a^2}{b^2}} \left (-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right )-15 a^4 \log \left (\sqrt{x^2-\frac{a^2}{b^2}}+x\right )\right )}{24 b^2 \sqrt{x^2-\frac{a^2}{b^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2*Sqrt[-((a^2*c)/b^2) + c*x^2],x]

[Out]

(Sqrt[c*(-(a^2/b^2) + x^2)]*(b*Sqrt[-(a^2/b^2) + x^2]*(-16*a^3 + 9*a^2*b*x + 16*
a*b^2*x^2 + 6*b^3*x^3) - 15*a^4*Log[x + Sqrt[-(a^2/b^2) + x^2]]))/(24*b^2*Sqrt[-
(a^2/b^2) + x^2])

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Maple [A]  time = 0.026, size = 113, normalized size = 0.9 \[{\frac{5\,{a}^{2}x}{8}\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}}}-{\frac{5\,{a}^{4}}{8\,{b}^{2}}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}} \right ) }+{\frac{{b}^{2}x}{4\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{2\,ab}{3\,c} \left ({\frac{c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{{b}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(-a^2*c/b^2+c*x^2)^(1/2),x)

[Out]

5/8*a^2*x*(-a^2*c/b^2+c*x^2)^(1/2)-5/8*a^4*c^(1/2)/b^2*ln(c^(1/2)*x+(-a^2*c/b^2+
c*x^2)^(1/2))+1/4*b^2*x*(-a^2*c/b^2+c*x^2)^(3/2)/c+2/3*a*b/c*(c*(b^2*x^2-a^2)/b^
2)^(3/2)

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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(sqrt(c*x^2 - a^2*c/b^2)*(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242017, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \sqrt{c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt{c} x \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \,{\left (6 \, b^{4} x^{3} + 16 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x - 16 \, a^{3} b\right )} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}{48 \, b^{2}}, -\frac{15 \, a^{4} \sqrt{-c} \arctan \left (\frac{c x}{\sqrt{-c} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}\right ) -{\left (6 \, b^{4} x^{3} + 16 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x - 16 \, a^{3} b\right )} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}{24 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 - a^2*c/b^2)*(b*x + a)^2,x, algorithm="fricas")

[Out]

[1/48*(15*a^4*sqrt(c)*log(2*b^2*c*x^2 - 2*b^2*sqrt(c)*x*sqrt((b^2*c*x^2 - a^2*c)
/b^2) - a^2*c) + 2*(6*b^4*x^3 + 16*a*b^3*x^2 + 9*a^2*b^2*x - 16*a^3*b)*sqrt((b^2
*c*x^2 - a^2*c)/b^2))/b^2, -1/24*(15*a^4*sqrt(-c)*arctan(c*x/(sqrt(-c)*sqrt((b^2
*c*x^2 - a^2*c)/b^2))) - (6*b^4*x^3 + 16*a*b^3*x^2 + 9*a^2*b^2*x - 16*a^3*b)*sqr
t((b^2*c*x^2 - a^2*c)/b^2))/b^2]

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Sympy [A]  time = 19.1444, size = 408, normalized size = 3.14 \[ a^{2} \left (\begin{cases} - \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b^{2}} - \frac{a \sqrt{c} x}{2 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b^{2}} + \frac{i a \sqrt{c} x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2 b} & \text{otherwise} \end{cases}\right ) + 2 a b \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + b^{2} \left (\begin{cases} - \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{4}} + \frac{a^{3} \sqrt{c} x}{8 b^{3} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 a \sqrt{c} x^{3}}{8 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{i a^{4} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{4}} - \frac{i a^{3} \sqrt{c} x}{8 b^{3} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 i a \sqrt{c} x^{3}}{8 b \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{i b \sqrt{c} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(-a**2*c/b**2+c*x**2)**(1/2),x)

[Out]

a**2*Piecewise((-a**2*sqrt(c)*acosh(b*x/a)/(2*b**2) - a*sqrt(c)*x/(2*b*sqrt(-1 +
b**2*x**2/a**2)) + b*sqrt(c)*x**3/(2*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**
2/a**2) > 1), (I*a**2*sqrt(c)*asin(b*x/a)/(2*b**2) + I*a*sqrt(c)*x*sqrt(1 - b**2
*x**2/a**2)/(2*b), True)) + 2*a*b*Piecewise((0, Eq(c, 0)), ((-a**2*c/b**2 + c*x*
*2)**(3/2)/(3*c), True)) + b**2*Piecewise((-a**4*sqrt(c)*acosh(b*x/a)/(8*b**4) +
a**3*sqrt(c)*x/(8*b**3*sqrt(-1 + b**2*x**2/a**2)) - 3*a*sqrt(c)*x**3/(8*b*sqrt(
-1 + b**2*x**2/a**2)) + b*sqrt(c)*x**5/(4*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2
*x**2/a**2) > 1), (I*a**4*sqrt(c)*asin(b*x/a)/(8*b**4) - I*a**3*sqrt(c)*x/(8*b**
3*sqrt(1 - b**2*x**2/a**2)) + 3*I*a*sqrt(c)*x**3/(8*b*sqrt(1 - b**2*x**2/a**2))
- I*b*sqrt(c)*x**5/(4*a*sqrt(1 - b**2*x**2/a**2)), True))

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GIAC/XCAS [A]  time = 0.236494, size = 136, normalized size = 1.05 \[ \frac{{\left (\frac{15 \, a^{4} \sqrt{c}{\rm ln}\left ({\left | -\sqrt{b^{2} c} x + \sqrt{b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} - \sqrt{b^{2} c x^{2} - a^{2} c}{\left (\frac{16 \, a^{3}}{b} -{\left (9 \, a^{2} + 2 \,{\left (3 \, b^{2} x + 8 \, a b\right )} x\right )} x\right )}\right )}{\left | b \right |}}{24 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 - a^2*c/b^2)*(b*x + a)^2,x, algorithm="giac")

[Out]

1/24*(15*a^4*sqrt(c)*ln(abs(-sqrt(b^2*c)*x + sqrt(b^2*c*x^2 - a^2*c)))/abs(b) -
sqrt(b^2*c*x^2 - a^2*c)*(16*a^3/b - (9*a^2 + 2*(3*b^2*x + 8*a*b)*x)*x))*abs(b)/b
^2