3.8.80 \(\int (a+b x) \sqrt {a^2-b^2 x^2} \, dx\) [780]

Optimal. Leaf size=76 \[ \frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {655, 201, 223, 209} \begin {gather*} \frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \text {ArcTan}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Sqrt[a^2 - b^2*x^2],x]

[Out]

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

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (a+b x) \sqrt {a^2-b^2 x^2} \, dx &=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+a \int \sqrt {a^2-b^2 x^2} \, dx\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {1}{2} a^3 \int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {1}{2} a^3 \text {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right )\\ &=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 89, normalized size = 1.17 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2} \left (-2 a^2+3 a b x+2 b^2 x^2\right )}{6 b}-\frac {a^3 \log \left (-\sqrt {-b^2} x+\sqrt {a^2-b^2 x^2}\right )}{2 \sqrt {-b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-2*a^2 + 3*a*b*x + 2*b^2*x^2))/(6*b) - (a^3*Log[-(Sqrt[-b^2]*x) + Sqrt[a^2 - b^2*x^2]])/
(2*Sqrt[-b^2])

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Maple [A]
time = 0.42, size = 73, normalized size = 0.96

method	result	size
risch	\(-\frac {\left (-2 b^{2} x^{2}-3 a b x +2 a^{2}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{6 b}+\frac {a^{3} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\)	\(72\)
default	\(-\frac {\left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{3 b}+a \left (\frac {x \sqrt {-b^{2} x^{2}+a^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\right )\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b^2*x^2+a^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 53, normalized size = 0.70 \begin {gather*} \frac {a^{3} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} x^{2} + a^{2}} a x - \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b^2*x^2+a^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.46, size = 73, normalized size = 0.96 \begin {gather*} -\frac {6 \, a^{3} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - {\left (2 \, b^{2} x^{2} + 3 \, a b x - 2 \, a^{2}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b^2*x^2+a^2)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(6*a^3*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) - (2*b^2*x^2 + 3*a*b*x - 2*a^2)*sqrt(-b^2*x^2 + a^2))/b

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Sympy [C] Result contains complex when optimal does not.
time = 1.82, size = 144, normalized size = 1.89 \begin {gather*} a \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 ) + 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.72, size = 56, normalized size = 0.74 \begin {gather*} \frac {a^{3} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{2 \, {\left | b \right |}} + \frac {1}{6} \, \sqrt {-b^{2} x^{2} + a^{2}} {\left ({\left (2 \, b x + 3 \, a\right )} x - \frac {2 \, a^{2}}{b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b^2*x^2+a^2)^(1/2),x, algorithm="giac")

[Out]

1/2*a^3*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) + 1/6*sqrt(-b^2*x^2 + a^2)*((2*b*x + 3*a)*x - 2*a^2/b)

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Mupad [B]
time = 0.85, size = 75, normalized size = 0.99 \begin {gather*} \frac {a^3\,\ln \left (x\,\sqrt {-b^2}+\sqrt {a^2-b^2\,x^2}\right )}{2\,\sqrt {-b^2}}-\frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{3\,b}+\frac {a\,x\,\sqrt {a^2-b^2\,x^2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*log(x*(-b^2)^(1/2) + (a^2 - b^2*x^2)^(1/2)))/(2*(-b^2)^(1/2)) - (a^2 - b^2*x^2)^(3/2)/(3*b) + (a*x*(a^2 -
b^2*x^2)^(1/2))/2

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