3.807 \(\int (a+b x)^3 \sqrt{-\frac{a^2 c}{b^2}+c x^2} \, dx\)

Optimal. Leaf size=167 \[ \frac{7 a^2 b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2}+\frac{7}{8} a^3 x \sqrt{c x^2-\frac{a^2 c}{b^2}} \]

[Out]

(7*a^3*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (7*a^2*b*(-((a^2*c)/b^2) + c*x^2)^(3/
2))/(12*c) + (7*a*b*(a + b*x)*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(20*c) + (b*(a + b
*x)^2*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(5*c) - (7*a^5*Sqrt[c]*ArcTanh[(Sqrt[c]*x)
/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

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Rubi [A]  time = 0.260387, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{7 a^2 b \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{12 c}+\frac{7 a b (a+b x) \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{20 c}+\frac{b (a+b x)^2 \left (c x^2-\frac{a^2 c}{b^2}\right )^{3/2}}{5 c}-\frac{7 a^5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2-\frac{a^2 c}{b^2}}}\right )}{8 b^2}+\frac{7}{8} a^3 x \sqrt{c x^2-\frac{a^2 c}{b^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*Sqrt[-((a^2*c)/b^2) + c*x^2],x]

[Out]

(7*a^3*x*Sqrt[-((a^2*c)/b^2) + c*x^2])/8 + (7*a^2*b*(-((a^2*c)/b^2) + c*x^2)^(3/
2))/(12*c) + (7*a*b*(a + b*x)*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(20*c) + (b*(a + b
*x)^2*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(5*c) - (7*a^5*Sqrt[c]*ArcTanh[(Sqrt[c]*x)
/Sqrt[-((a^2*c)/b^2) + c*x^2]])/(8*b^2)

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Rubi in Sympy [A]  time = 29.2076, size = 151, normalized size = 0.9 \[ - \frac{7 a^{5} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}} \right )}}{8 b^{2}} + \frac{7 a^{3} x \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{8} + \frac{7 a^{2} b \left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{12 c} + \frac{7 a b \left (a + b x\right ) \left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{20 c} + \frac{b \left (a + b x\right )^{2} \left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(-a**2*c/b**2+c*x**2)**(1/2),x)

[Out]

-7*a**5*sqrt(c)*atanh(sqrt(c)*x/sqrt(-a**2*c/b**2 + c*x**2))/(8*b**2) + 7*a**3*x
*sqrt(-a**2*c/b**2 + c*x**2)/8 + 7*a**2*b*(-a**2*c/b**2 + c*x**2)**(3/2)/(12*c)
+ 7*a*b*(a + b*x)*(-a**2*c/b**2 + c*x**2)**(3/2)/(20*c) + b*(a + b*x)**2*(-a**2*
c/b**2 + c*x**2)**(3/2)/(5*c)

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Mathematica [A]  time = 0.184279, size = 125, normalized size = 0.75 \[ \frac{\sqrt{c \left (x^2-\frac{a^2}{b^2}\right )} \left (b \sqrt{x^2-\frac{a^2}{b^2}} \left (-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right )-105 a^5 \log \left (\sqrt{x^2-\frac{a^2}{b^2}}+x\right )\right )}{120 b^2 \sqrt{x^2-\frac{a^2}{b^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*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]*(-136*a^4 + 15*a^3*b*x + 1
12*a^2*b^2*x^2 + 90*a*b^3*x^3 + 24*b^4*x^4) - 105*a^5*Log[x + Sqrt[-(a^2/b^2) +
x^2]]))/(120*b^2*Sqrt[-(a^2/b^2) + x^2])

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Maple [A]  time = 0.01, size = 169, normalized size = 1. \[{\frac{7\,{a}^{3}x}{8}\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}}}-{\frac{7\,{a}^{5}}{8\,{b}^{2}}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{-{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2}} \right ) }+{\frac{{b}^{3}{x}^{2}}{5\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{a}^{2}b}{15\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a{b}^{2}x}{4\,c} \left ( -{\frac{{a}^{2}c}{{b}^{2}}}+c{x}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}b}{c} \left ({\frac{c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{{b}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(-a^2*c/b^2+c*x^2)^(1/2),x)

[Out]

7/8*a^3*x*(-a^2*c/b^2+c*x^2)^(1/2)-7/8*a^5*c^(1/2)/b^2*ln(c^(1/2)*x+(-a^2*c/b^2+
c*x^2)^(1/2))+1/5*b^3*x^2*(-a^2*c/b^2+c*x^2)^(3/2)/c+2/15*a^2*b*(-a^2*c/b^2+c*x^
2)^(3/2)/c+3/4*a*b^2*x*(-a^2*c/b^2+c*x^2)^(3/2)/c+a^2*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238529, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, a^{5} \sqrt{c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt{c} x \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \,{\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}{240 \, b^{2}}, -\frac{105 \, a^{5} \sqrt{-c} \arctan \left (\frac{c x}{\sqrt{-c} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}\right ) -{\left (24 \, b^{5} x^{4} + 90 \, a b^{4} x^{3} + 112 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x - 136 \, a^{4} b\right )} \sqrt{\frac{b^{2} c x^{2} - a^{2} c}{b^{2}}}}{120 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 - a^2*c/b^2)*(b*x + a)^3,x, algorithm="fricas")

[Out]

[1/240*(105*a^5*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*(24*b^5*x^4 + 90*a*b^4*x^3 + 112*a^2*b^3*x^2 + 15*a^3*b^2*x
- 136*a^4*b)*sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2, -1/120*(105*a^5*sqrt(-c)*arcta
n(c*x/(sqrt(-c)*sqrt((b^2*c*x^2 - a^2*c)/b^2))) - (24*b^5*x^4 + 90*a*b^4*x^3 + 1
12*a^2*b^3*x^2 + 15*a^3*b^2*x - 136*a^4*b)*sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2]

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Sympy [A]  time = 19.7681, size = 491, normalized size = 2.94 \[ - \frac{2 a^{4} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{15 b} + a^{3} \left (\begin{cases} - \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b^{2}} - \frac{a \sqrt{c} x}{2 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b^{2}} + \frac{i a \sqrt{c} x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2 b} & \text{otherwise} \end{cases}\right ) - \frac{a^{2} b x^{2} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{15} + 3 a^{2} b \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\left (- \frac{a^{2} c}{b^{2}} + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 3 a b^{2} \left (\begin{cases} - \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{4}} + \frac{a^{3} \sqrt{c} x}{8 b^{3} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 a \sqrt{c} x^{3}}{8 b \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{b \sqrt{c} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{i a^{4} \sqrt{c} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{4}} - \frac{i a^{3} \sqrt{c} x}{8 b^{3} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 i a \sqrt{c} x^{3}}{8 b \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{i b \sqrt{c} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) + \frac{b^{3} x^{4} \sqrt{- \frac{a^{2} c}{b^{2}} + c x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(-a**2*c/b**2+c*x**2)**(1/2),x)

[Out]

-2*a**4*sqrt(-a**2*c/b**2 + c*x**2)/(15*b) + a**3*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)) - a**2*b*
x**2*sqrt(-a**2*c/b**2 + c*x**2)/15 + 3*a**2*b*Piecewise((0, Eq(c, 0)), ((-a**2*
c/b**2 + c*x**2)**(3/2)/(3*c), True)) + 3*a*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)) + b**
3*x**4*sqrt(-a**2*c/b**2 + c*x**2)/5

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GIAC/XCAS [A]  time = 0.236735, size = 153, normalized size = 0.92 \[ \frac{{\left (\frac{105 \, a^{5} \sqrt{c}{\rm ln}\left ({\left | -\sqrt{b^{2} c} x + \sqrt{b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} - \sqrt{b^{2} c x^{2} - a^{2} c}{\left (\frac{136 \, a^{4}}{b} -{\left (15 \, a^{3} + 2 \,{\left (56 \, a^{2} b + 3 \,{\left (4 \, b^{3} x + 15 \, a b^{2}\right )} x\right )} x\right )} x\right )}\right )}{\left | b \right |}}{120 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 - a^2*c/b^2)*(b*x + a)^3,x, algorithm="giac")

[Out]

1/120*(105*a^5*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)*(136*a^4/b - (15*a^3 + 2*(56*a^2*b + 3*(4*b^3*x + 15*a
*b^2)*x)*x)*x))*abs(b)/b^2