Optimal. Leaf size=371 \[ -\frac {14 \sqrt {1-(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1-(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1-(a+b x)^2}}{4 b^3}+\frac {2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \text {ArcSin}(a+b x)}{4 b^3}-\frac {4 (a+b x) \text {ArcSin}(a+b x)}{3 b^3}-\frac {6 a^2 (a+b x) \text {ArcSin}(a+b x)}{b^3}+\frac {3 a (a+b x)^2 \text {ArcSin}(a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \text {ArcSin}(a+b x)}{9 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{3 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{3 b^3}+\frac {a \text {ArcSin}(a+b x)^3}{2 b^3}+\frac {a^3 \text {ArcSin}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x)^3 \]
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Rubi [A]
time = 0.31, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {4889, 4827,
4857, 3398, 3377, 2718, 3392, 30, 2715, 8, 2713} \begin {gather*} \frac {a^3 \text {ArcSin}(a+b x)^3}{3 b^3}-\frac {6 a^2 (a+b x) \text {ArcSin}(a+b x)}{b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{b^3}-\frac {6 a^2 \sqrt {1-(a+b x)^2}}{b^3}-\frac {2 (a+b x)^3 \text {ArcSin}(a+b x)}{9 b^3}+\frac {\sqrt {1-(a+b x)^2} (a+b x)^2 \text {ArcSin}(a+b x)^2}{3 b^3}+\frac {3 a (a+b x)^2 \text {ArcSin}(a+b x)}{2 b^3}-\frac {3 a \sqrt {1-(a+b x)^2} (a+b x) \text {ArcSin}(a+b x)^2}{2 b^3}-\frac {4 (a+b x) \text {ArcSin}(a+b x)}{3 b^3}+\frac {a \text {ArcSin}(a+b x)^3}{2 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{3 b^3}-\frac {3 a \text {ArcSin}(a+b x)}{4 b^3}+\frac {1}{3} x^3 \text {ArcSin}(a+b x)^3+\frac {3 a \sqrt {1-(a+b x)^2} (a+b x)}{4 b^3}+\frac {2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {14 \sqrt {1-(a+b x)^2}}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2713
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 4827
Rule 4857
Rule 4889
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(a+b x)^3 \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3-\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3-\text {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sin (x)}{b}\right )^3 \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3-\text {Subst}\left (\int \left (-\frac {a^3 x^2}{b^3}+\frac {3 a^2 x^2 \sin (x)}{b^3}-\frac {3 a x^2 \sin ^2(x)}{b^3}+\frac {x^2 \sin ^3(x)}{b^3}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=\frac {a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3-\frac {\text {Subst}\left (\int x^2 \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}+\frac {(3 a) \text {Subst}\left (\int x^2 \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int x^2 \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3+\frac {2 \text {Subst}\left (\int \sin ^3(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{9 b^3}-\frac {2 \text {Subst}\left (\int x^2 \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}+\frac {(3 a) \text {Subst}\left (\int x^2 \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {(3 a) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {3 a (a+b x) \sqrt {1-(a+b x)^2}}{4 b^3}-\frac {6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac {3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3-\frac {2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1-(a+b x)^2}\right )}{9 b^3}-\frac {4 \text {Subst}\left (\int x \cos (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}-\frac {(3 a) \text {Subst}\left (\int 1 \, dx,x,\sin ^{-1}(a+b x)\right )}{4 b^3}+\frac {\left (6 a^2\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {2 \sqrt {1-(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1-(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1-(a+b x)^2}}{4 b^3}+\frac {2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \sin ^{-1}(a+b x)}{4 b^3}-\frac {4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}-\frac {6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac {3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3+\frac {4 \text {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{3 b^3}\\ &=-\frac {14 \sqrt {1-(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1-(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1-(a+b x)^2}}{4 b^3}+\frac {2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \sin ^{-1}(a+b x)}{4 b^3}-\frac {4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}-\frac {6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac {3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^3\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 181, normalized size = 0.49 \begin {gather*} \frac {-\sqrt {1-a^2-2 a b x-b^2 x^2} \left (160+575 a^2-65 a b x+8 b^2 x^2\right )-3 \left (170 a^3+132 a^2 b x+a \left (75-30 b^2 x^2\right )+8 b x \left (6+b^2 x^2\right )\right ) \text {ArcSin}(a+b x)+18 \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)^2+18 \left (3 a+2 a^3+2 b^3 x^3\right ) \text {ArcSin}(a+b x)^3}{108 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 344, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (4 \arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{4}+\frac {\arcsin \left (b x +a \right )^{3} \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{3}+\arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-\frac {14 \sqrt {1-\left (b x +a \right )^{2}}}{9}-2 \left (b x +a \right ) \arcsin \left (b x +a \right )+\frac {\arcsin \left (b x +a \right )^{2} \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {2 \arcsin \left (b x +a \right ) \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{9}-\frac {2 \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{27}+a^{2} \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arcsin \left (b x +a \right )\right )+\arcsin \left (b x +a \right )^{3} \left (b x +a \right )}{b^{3}}\) | \(344\) |
default | \(\frac {-\frac {a \left (4 \arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{4}+\frac {\arcsin \left (b x +a \right )^{3} \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{3}+\arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-\frac {14 \sqrt {1-\left (b x +a \right )^{2}}}{9}-2 \left (b x +a \right ) \arcsin \left (b x +a \right )+\frac {\arcsin \left (b x +a \right )^{2} \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {2 \arcsin \left (b x +a \right ) \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{9}-\frac {2 \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{27}+a^{2} \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arcsin \left (b x +a \right )\right )+\arcsin \left (b x +a \right )^{3} \left (b x +a \right )}{b^{3}}\) | \(344\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 2.28, size = 152, normalized size = 0.41 \begin {gather*} \frac {18 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (8 \, b^{3} x^{3} - 30 \, a b^{2} x^{2} + 170 \, a^{3} + 12 \, {\left (11 \, a^{2} + 4\right )} b x + 75 \, a\right )} \arcsin \left (b x + a\right ) - {\left (8 \, b^{2} x^{2} - 65 \, a b x - 18 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \arcsin \left (b x + a\right )^{2} + 575 \, a^{2} + 160\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{108 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 432, normalized size = 1.16 \begin {gather*} \begin {cases} \frac {a^{3} \operatorname {asin}^{3}{\left (a + b x \right )}}{3 b^{3}} - \frac {85 a^{3} \operatorname {asin}{\left (a + b x \right )}}{18 b^{3}} - \frac {11 a^{2} x \operatorname {asin}{\left (a + b x \right )}}{3 b^{2}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{6 b^{3}} - \frac {575 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{3}} + \frac {5 a x^{2} \operatorname {asin}{\left (a + b x \right )}}{6 b} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{6 b^{2}} + \frac {65 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{2}} + \frac {a \operatorname {asin}^{3}{\left (a + b x \right )}}{2 b^{3}} - \frac {25 a \operatorname {asin}{\left (a + b x \right )}}{12 b^{3}} + \frac {x^{3} \operatorname {asin}^{3}{\left (a + b x \right )}}{3} - \frac {2 x^{3} \operatorname {asin}{\left (a + b x \right )}}{9} + \frac {x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b} - \frac {2 x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b} - \frac {4 x \operatorname {asin}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {40 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 389, normalized size = 1.05 \begin {gather*} \frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{3}}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{3}}{b^{3}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )^{3}}{2 \, b^{3}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )}{2 \, b^{3}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{4 \, b^{3}} - \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {14 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac {3 \, a \arcsin \left (b x + a\right )}{4 \, b^{3}} + \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{27 \, b^{3}} - \frac {14 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{9 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\mathrm {asin}\left (a+b\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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