Optimal. Leaf size=64 \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{12} b c x^3 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 x \sqrt {1-\frac {c^2}{x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 271, 191} \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{12} b c x^3 \sqrt {1-\frac {c^2}{x^2}}+\frac {1}{6} b c^3 x \sqrt {1-\frac {c^2}{x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 191
Rule 271
Rule 4842
Rubi steps
\begin {align*} \int x^3 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} b \int \frac {c x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{4} (b c) \int \frac {x^2}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}}} \, dx\\ &=\frac {1}{6} b c^3 \sqrt {1-\frac {c^2}{x^2}} x+\frac {1}{12} b c \sqrt {1-\frac {c^2}{x^2}} x^3+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (\frac {c}{x}\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 59, normalized size = 0.92 \[ \frac {a x^4}{4}+b \sqrt {\frac {x^2-c^2}{x^2}} \left (\frac {c^3 x}{6}+\frac {c x^3}{12}\right )+\frac {1}{4} b x^4 \sin ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.47, size = 51, normalized size = 0.80 \[ \frac {1}{4} \, b x^{4} \arcsin \left (\frac {c}{x}\right ) + \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (2 \, b c^{3} x + b c x^{3}\right )} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.31, size = 340, normalized size = 5.31 \[ \frac {3 \, b c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {c}{x}\right ) + 3 \, a c x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4} + 2 \, b c^{2} x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3} + 12 \, b c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {c}{x}\right ) + 12 \, a c^{3} x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2} + 18 \, b c^{4} x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )} + 18 \, b c^{5} \arcsin \left (\frac {c}{x}\right ) + 18 \, a c^{5} - \frac {18 \, b c^{6}}{x {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}} + \frac {12 \, b c^{7} \arcsin \left (\frac {c}{x}\right )}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a c^{7}}{x^{2} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b c^{8}}{x^{3} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b c^{9} \arcsin \left (\frac {c}{x}\right )}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a c^{9}}{x^{4} {\left (\sqrt {-\frac {c^{2}}{x^{2}} + 1} + 1\right )}^{4}}}{192 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 71, normalized size = 1.11 \[ -c^{4} \left (-\frac {a \,x^{4}}{4 c^{4}}+b \left (-\frac {x^{4} \arcsin \left (\frac {c}{x}\right )}{4 c^{4}}-\frac {x^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{12 c^{3}}-\frac {x \sqrt {1-\frac {c^{2}}{x^{2}}}}{6 c}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.09, size = 59, normalized size = 0.92 \[ \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arcsin \left (\frac {c}{x}\right ) + {\left (x^{3} {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, c^{2} x \sqrt {-\frac {c^{2}}{x^{2}} + 1}\right )} c\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (\frac {c}{x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.69, size = 107, normalized size = 1.67 \[ \frac {a x^{4}}{4} + \frac {b c \left (\begin {cases} \frac {2 c^{3} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} + \frac {c x^{2} \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\\frac {2 i c^{3} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} + \frac {i c x^{2} \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3} & \text {otherwise} \end {cases}\right )}{4} + \frac {b x^{4} \operatorname {asin}{\left (\frac {c}{x} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________