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

Optimal. Leaf size=107 \[ \frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]

[Out]

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

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Rubi [A]  time = 0.103331, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{5}{8} a^2 x \sqrt{a^2-b^2 x^2}-\frac{5 a \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{4 b}+\frac{5 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0385, size = 90, normalized size = 0.84 \[ \frac{5 a^{4} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{8 b} + \frac{5 a^{2} x \sqrt{a^{2} - b^{2} x^{2}}}{8} - \frac{5 a \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b} - \frac{\left (a + b x\right ) \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0905773, size = 80, normalized size = 0.75 \[ \frac{15 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-16 a^3+9 a^2 b x+16 a b^2 x^2+6 b^3 x^3\right )}{24 b} \]

Antiderivative was successfully verified.

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

[Out]

(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*A
rcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(24*b)

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Maple [A]  time = 0.009, size = 91, normalized size = 0.9 \[{\frac{5\,{a}^{2}x}{8}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{5\,{a}^{4}}{8}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{x}{4} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{2\,a}{3\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.766432, size = 112, normalized size = 1.05 \[ \frac{5 \, a^{4} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{8 \, \sqrt{b^{2}}} + \frac{5}{8} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{2} x - \frac{1}{4} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} x - \frac{2 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21999, size = 412, normalized size = 3.85 \[ -\frac{24 \, a b^{7} x^{7} + 64 \, a^{2} b^{6} x^{6} - 36 \, a^{3} b^{5} x^{5} - 240 \, a^{4} b^{4} x^{4} - 60 \, a^{5} b^{3} x^{3} + 192 \, a^{6} b^{2} x^{2} + 72 \, a^{7} b x + 30 \,{\left (a^{4} b^{4} x^{4} - 8 \, a^{6} b^{2} x^{2} + 8 \, a^{8} + 4 \,{\left (a^{5} b^{2} x^{2} - 2 \, a^{7}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (6 \, b^{7} x^{7} + 16 \, a b^{6} x^{6} - 39 \, a^{2} b^{5} x^{5} - 144 \, a^{3} b^{4} x^{4} - 24 \, a^{4} b^{3} x^{3} + 192 \, a^{5} b^{2} x^{2} + 72 \, a^{6} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{24 \,{\left (b^{5} x^{4} - 8 \, a^{2} b^{3} x^{2} + 8 \, a^{4} b + 4 \,{\left (a b^{3} x^{2} - 2 \, a^{3} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(24*a*b^7*x^7 + 64*a^2*b^6*x^6 - 36*a^3*b^5*x^5 - 240*a^4*b^4*x^4 - 60*a^5
*b^3*x^3 + 192*a^6*b^2*x^2 + 72*a^7*b*x + 30*(a^4*b^4*x^4 - 8*a^6*b^2*x^2 + 8*a^
8 + 4*(a^5*b^2*x^2 - 2*a^7)*sqrt(-b^2*x^2 + a^2))*arctan(-(a - sqrt(-b^2*x^2 + a
^2))/(b*x)) - (6*b^7*x^7 + 16*a*b^6*x^6 - 39*a^2*b^5*x^5 - 144*a^3*b^4*x^4 - 24*
a^4*b^3*x^3 + 192*a^5*b^2*x^2 + 72*a^6*b*x)*sqrt(-b^2*x^2 + a^2))/(b^5*x^4 - 8*a
^2*b^3*x^2 + 8*a^4*b + 4*(a*b^3*x^2 - 2*a^3*b)*sqrt(-b^2*x^2 + a^2))

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Sympy [A]  time = 18.3423, size = 350, normalized size = 3.27 \[ a^{2} \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + 2 a b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) + b^{2} \left (\begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{3}} + \frac{i a^{3} x}{8 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{4} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{3}} - \frac{a^{3} x}{8 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227382, size = 93, normalized size = 0.87 \[ \frac{5 \, a^{4} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{8 \,{\left | b \right |}} - \frac{1}{24} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (\frac{16 \, a^{3}}{b} -{\left (9 \, a^{2} + 2 \,{\left (3 \, b^{2} x + 8 \, a b\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*a^4*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/24*sqrt(-b^2*x^2 + a^2)*(16*a^3
/b - (9*a^2 + 2*(3*b^2*x + 8*a*b)*x)*x)