3.779 \(\int (a+b x) \left (a^2-b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=100 \[ \frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}+\frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0833223, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{4} a x \left (a^2-b^2 x^2\right )^{3/2}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 b}+\frac{3 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}+\frac{3}{8} a^3 x \sqrt{a^2-b^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3415, size = 83, normalized size = 0.83 \[ \frac{3 a^{5} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{8 b} + \frac{3 a^{3} x \sqrt{a^{2} - b^{2} x^{2}}}{8} + \frac{a x \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{4} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0862777, size = 91, normalized size = 0.91 \[ \frac{15 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-8 a^4+25 a^3 b x+16 a^2 b^2 x^2-10 a b^3 x^3-8 b^4 x^4\right )}{40 b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-8*a^4 + 25*a^3*b*x + 16*a^2*b^2*x^2 - 10*a*b^3*x^3 - 8*b^
4*x^4) + 15*a^5*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(40*b)

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Maple [A]  time = 0.01, size = 91, normalized size = 0.9 \[{\frac{ax}{4} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}x}{8}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{3\,{a}^{5}}{8}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{1}{5\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.755906, size = 112, normalized size = 1.12 \[ \frac{3 \, a^{5} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{8 \, \sqrt{b^{2}}} + \frac{3}{8} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{3} x + \frac{1}{4} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a x - \frac{{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232513, size = 504, normalized size = 5.04 \[ -\frac{8 \, b^{10} x^{10} + 10 \, a b^{9} x^{9} - 120 \, a^{2} b^{8} x^{8} - 155 \, a^{3} b^{7} x^{7} + 440 \, a^{4} b^{6} x^{6} + 605 \, a^{5} b^{5} x^{5} - 640 \, a^{6} b^{4} x^{4} - 860 \, a^{7} b^{3} x^{3} + 320 \, a^{8} b^{2} x^{2} + 400 \, a^{9} b x + 30 \,{\left (5 \, a^{6} b^{4} x^{4} - 20 \, a^{8} b^{2} x^{2} + 16 \, a^{10} -{\left (a^{5} b^{4} x^{4} - 12 \, a^{7} b^{2} x^{2} + 16 \, a^{9}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + 5 \,{\left (8 \, a b^{8} x^{8} + 10 \, a^{2} b^{7} x^{7} - 48 \, a^{3} b^{6} x^{6} - 65 \, a^{4} b^{5} x^{5} + 96 \, a^{5} b^{4} x^{4} + 132 \, a^{6} b^{3} x^{3} - 64 \, a^{7} b^{2} x^{2} - 80 \, a^{8} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{40 \,{\left (5 \, a b^{5} x^{4} - 20 \, a^{3} b^{3} x^{2} + 16 \, a^{5} b -{\left (b^{5} x^{4} - 12 \, a^{2} b^{3} x^{2} + 16 \, a^{4} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(8*b^10*x^10 + 10*a*b^9*x^9 - 120*a^2*b^8*x^8 - 155*a^3*b^7*x^7 + 440*a^4*
b^6*x^6 + 605*a^5*b^5*x^5 - 640*a^6*b^4*x^4 - 860*a^7*b^3*x^3 + 320*a^8*b^2*x^2
+ 400*a^9*b*x + 30*(5*a^6*b^4*x^4 - 20*a^8*b^2*x^2 + 16*a^10 - (a^5*b^4*x^4 - 12
*a^7*b^2*x^2 + 16*a^9)*sqrt(-b^2*x^2 + a^2))*arctan(-(a - sqrt(-b^2*x^2 + a^2))/
(b*x)) + 5*(8*a*b^8*x^8 + 10*a^2*b^7*x^7 - 48*a^3*b^6*x^6 - 65*a^4*b^5*x^5 + 96*
a^5*b^4*x^4 + 132*a^6*b^3*x^3 - 64*a^7*b^2*x^2 - 80*a^8*b*x)*sqrt(-b^2*x^2 + a^2
))/(5*a*b^5*x^4 - 20*a^3*b^3*x^2 + 16*a^5*b - (b^5*x^4 - 12*a^2*b^3*x^2 + 16*a^4
*b)*sqrt(-b^2*x^2 + a^2))

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Sympy [A]  time = 20.254, size = 435, normalized size = 4.35 \[ a^{3} \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + a^{2} b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) - a b^{2} \left (\begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{3}} + \frac{i a^{3} x}{8 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{4} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{3}} - \frac{a^{3} x}{8 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) - b^{3} \left (\begin{cases} - \frac{2 a^{4} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{4}} - \frac{a^{2} x^{2} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a^{2} - b^{2} x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{x^{4} \sqrt{a^{2}}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236326, size = 109, normalized size = 1.09 \[ \frac{3 \, a^{5} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{8 \,{\left | b \right |}} - \frac{1}{40} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (\frac{8 \, a^{4}}{b} -{\left (25 \, a^{3} + 2 \,{\left (8 \, a^{2} b -{\left (4 \, b^{3} x + 5 \, a b^{2}\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*a^5*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/40*sqrt(-b^2*x^2 + a^2)*(8*a^4/
b - (25*a^3 + 2*(8*a^2*b - (4*b^3*x + 5*a*b^2)*x)*x)*x)