Optimal. Leaf size=220 \[ -\frac {4 x}{9 b^2}-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{9 b^3}+\frac {a \text {ArcSin}(a+b x)^2}{2 b^3}+\frac {a^3 \text {ArcSin}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.27, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4889, 4827,
4847, 4737, 4767, 8, 4795, 30} \begin {gather*} \frac {a^3 \text {ArcSin}(a+b x)^2}{3 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^3}-\frac {2 a^2 x}{b^2}+\frac {a \text {ArcSin}(a+b x)^2}{2 b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{9 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{9 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x)^2+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}-\frac {4 x}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 4737
Rule 4767
Rule 4795
Rule 4827
Rule 4847
Rule 4889
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(a+b x)^2 \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2}{3} \text {Subst}\left (\int \left (-\frac {a^3 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}+\frac {3 a^2 x \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}-\frac {3 a x^2 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}+\frac {x^3 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int \frac {x^3 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int x^2 \, dx,x,a+b x\right )}{9 b^3}-\frac {4 \text {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{9 b^3}+\frac {a \text {Subst}(\int x \, dx,x,a+b x)}{b^3}+\frac {a \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}(\int 1 \, dx,x,a+b x)}{b^3}\\ &=-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {4 \text {Subst}(\int 1 \, dx,x,a+b x)}{9 b^3}\\ &=-\frac {4 x}{9 b^2}-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 111, normalized size = 0.50 \begin {gather*} \frac {-b x \left (24+66 a^2-15 a b x+4 b^2 x^2\right )+6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right ) \text {ArcSin}(a+b x)+9 \left (3 a+2 a^3+2 b^3 x^3\right ) \text {ArcSin}(a+b x)^2}{54 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 231, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{3}-\frac {2 b x}{3}-\frac {2 a}{3}+\frac {2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}+\frac {2 \arcsin \left (b x +a \right ) \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{27}+a^{2} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )+\arcsin \left (b x +a \right )^{2} \left (b x +a \right )}{b^{3}}\) | \(231\) |
default | \(\frac {-\frac {a \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{3}-\frac {2 b x}{3}-\frac {2 a}{3}+\frac {2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}+\frac {2 \arcsin \left (b x +a \right ) \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{27}+a^{2} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )+\arcsin \left (b x +a \right )^{2} \left (b x +a \right )}{b^{3}}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.54, size = 111, normalized size = 0.50 \begin {gather*} -\frac {4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} + 6 \, {\left (11 \, a^{2} + 4\right )} b x - 9 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{54 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.29, size = 243, normalized size = 1.10 \begin {gather*} \begin {cases} \frac {a^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} x}{9 b^{2}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} + \frac {5 a x^{2}}{18 b} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{2}} + \frac {a \operatorname {asin}^{2}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b} - \frac {4 x}{9 b^{2}} + \frac {4 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}^{2}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 271, normalized size = 1.23 \begin {gather*} \frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {2 \, {\left (b x + a\right )} a^{2}}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} - \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )}{9 \, b^{3}} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{27 \, b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a}{2 \, b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )}{3 \, b^{3}} - \frac {14 \, {\left (b x + a\right )}}{27 \, b^{3}} + \frac {a}{4 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\mathrm {asin}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________