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

Optimal. Leaf size=140 \[ -\frac{7 a^2 \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{7 a (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{20 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}+\frac{7 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}+\frac{7}{8} a^3 x \sqrt{a^2-b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.153562, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{7 a^2 \left (a^2-b^2 x^2\right )^{3/2}}{12 b}-\frac{7 a (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{20 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{5 b}+\frac{7 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{8 b}+\frac{7}{8} a^3 x \sqrt{a^2-b^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5463, size = 119, normalized size = 0.85 \[ \frac{7 a^{5} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{8 b} + \frac{7 a^{3} x \sqrt{a^{2} - b^{2} x^{2}}}{8} - \frac{7 a^{2} \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b} - \frac{7 a \left (a + b x\right ) \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{20 b} - \frac{\left (a + b x\right )^{2} \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.09123, size = 91, normalized size = 0.65 \[ \frac{105 a^5 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-136 a^4+15 a^3 b x+112 a^2 b^2 x^2+90 a b^3 x^3+24 b^4 x^4\right )}{120 b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-136*a^4 + 15*a^3*b*x + 112*a^2*b^2*x^2 + 90*a*b^3*x^3 + 2
4*b^4*x^4) + 105*a^5*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(120*b)

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Maple [A]  time = 0.01, size = 114, normalized size = 0.8 \[{\frac{7\,{a}^{3}x}{8}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{7\,{a}^{5}}{8}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{b{x}^{2}}{5} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{17\,{a}^{2}}{15\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ax}{4} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*a^3*x*(-b^2*x^2+a^2)^(1/2)+7/8*a^5/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(-b^2*x^
2+a^2)^(1/2))-1/5*b*x^2*(-b^2*x^2+a^2)^(3/2)-17/15*a^2*(-b^2*x^2+a^2)^(3/2)/b-3/
4*a*x*(-b^2*x^2+a^2)^(3/2)

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Maxima [A]  time = 0.774183, size = 143, normalized size = 1.02 \[ \frac{7 \, a^{5} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{8 \, \sqrt{b^{2}}} + \frac{7}{8} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{3} x - \frac{1}{5} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} b x^{2} - \frac{3}{4} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a x - \frac{17 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{2}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*a^5*arcsin(b^2*x/sqrt(a^2*b^2))/sqrt(b^2) + 7/8*sqrt(-b^2*x^2 + a^2)*a^3*x -
1/5*(-b^2*x^2 + a^2)^(3/2)*b*x^2 - 3/4*(-b^2*x^2 + a^2)^(3/2)*a*x - 17/15*(-b^2
*x^2 + a^2)^(3/2)*a^2/b

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Fricas [A]  time = 0.228299, size = 504, normalized size = 3.6 \[ \frac{24 \, b^{10} x^{10} + 90 \, a b^{9} x^{9} - 200 \, a^{2} b^{8} x^{8} - 1155 \, a^{3} b^{7} x^{7} - 920 \, a^{4} b^{6} x^{6} + 2325 \, a^{5} b^{5} x^{5} + 3840 \, a^{6} b^{4} x^{4} - 1020 \, a^{7} b^{3} x^{3} - 2880 \, a^{8} b^{2} x^{2} - 240 \, a^{9} b x - 210 \,{\left (5 \, a^{6} b^{4} x^{4} - 20 \, a^{8} b^{2} x^{2} + 16 \, a^{10} -{\left (a^{5} b^{4} x^{4} - 12 \, a^{7} b^{2} x^{2} + 16 \, a^{9}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + 5 \,{\left (24 \, a b^{8} x^{8} + 90 \, a^{2} b^{7} x^{7} + 16 \, a^{3} b^{6} x^{6} - 345 \, a^{4} b^{5} x^{5} - 480 \, a^{5} b^{4} x^{4} + 228 \, a^{6} b^{3} x^{3} + 576 \, a^{7} b^{2} x^{2} + 48 \, a^{8} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{120 \,{\left (5 \, a b^{5} x^{4} - 20 \, a^{3} b^{3} x^{2} + 16 \, a^{5} b -{\left (b^{5} x^{4} - 12 \, a^{2} b^{3} x^{2} + 16 \, a^{4} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(24*b^10*x^10 + 90*a*b^9*x^9 - 200*a^2*b^8*x^8 - 1155*a^3*b^7*x^7 - 920*a^
4*b^6*x^6 + 2325*a^5*b^5*x^5 + 3840*a^6*b^4*x^4 - 1020*a^7*b^3*x^3 - 2880*a^8*b^
2*x^2 - 240*a^9*b*x - 210*(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*(24*a*b^8*x^8 + 90*a^2*b^7*x^7 + 16*a^3*b^6*x^6 - 345*a^4*b^5*
x^5 - 480*a^5*b^4*x^4 + 228*a^6*b^3*x^3 + 576*a^7*b^2*x^2 + 48*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 = 19.1681, size = 439, normalized size = 3.14 \[ a^{3} \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + 3 a^{2} b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) + 3 a b^{2} \left (\begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{8 b^{3}} + \frac{i a^{3} x}{8 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{5}}{4 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{4} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{8 b^{3}} - \frac{a^{3} x}{8 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{5}}{4 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) + b^{3} \left (\begin{cases} - \frac{2 a^{4} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{4}} - \frac{a^{2} x^{2} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a^{2} - b^{2} x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{x^{4} \sqrt{a^{2}}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(-b**2*x**2+a**2)**(1/2),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)) + 3*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)) +
3*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*s
qrt(-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 = 0.228548, size = 109, normalized size = 0.78 \[ \frac{7 \, a^{5} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{8 \,{\left | b \right |}} - \frac{1}{120} \, \sqrt{-b^{2} x^{2} + a^{2}}{\left (\frac{136 \, a^{4}}{b} -{\left (15 \, a^{3} + 2 \,{\left (56 \, a^{2} b + 3 \,{\left (4 \, b^{3} x + 15 \, a b^{2}\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/8*a^5*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/120*sqrt(-b^2*x^2 + a^2)*(136*a
^4/b - (15*a^3 + 2*(56*a^2*b + 3*(4*b^3*x + 15*a*b^2)*x)*x)*x)