Optimal. Leaf size=129 \[ \frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {b c \left (2 c^2+3\right ) \sin ^{-1}\left (c+d x^2\right )}{12 d^3}+\frac {b x^4 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{18 d}+\frac {b \left (11 c^2-5 c d x^2+4\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{36 d^3} \]
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Rubi [A] time = 0.16, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4842, 12, 1114, 742, 779, 619, 216} \[ \frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {b x^4 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{18 d}+\frac {b \left (11 c^2-5 c d x^2+4\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{36 d^3}+\frac {b c \left (2 c^2+3\right ) \sin ^{-1}\left (c+d x^2\right )}{12 d^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 619
Rule 742
Rule 779
Rule 1114
Rule 4842
Rubi steps
\begin {align*} \int x^5 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{6} b \int \frac {2 d x^7}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{3} (b d) \int \frac {x^7}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{6} (b d) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {b x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{18 d}+\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {b \operatorname {Subst}\left (\int \frac {x \left (-2 \left (1-c^2\right )+5 c d x\right )}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{18 d}\\ &=\frac {b x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{18 d}+\frac {b \left (4+11 c^2-5 c d x^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{36 d^3}+\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {\left (b c \left (3+2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{12 d^2}\\ &=\frac {b x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{18 d}+\frac {b \left (4+11 c^2-5 c d x^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{36 d^3}+\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {\left (b c \left (3+2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 d^2}}} \, dx,x,-2 d \left (c+d x^2\right )\right )}{24 d^4}\\ &=\frac {b x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{18 d}+\frac {b \left (4+11 c^2-5 c d x^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{36 d^3}+\frac {b c \left (3+2 c^2\right ) \sin ^{-1}\left (c+d x^2\right )}{12 d^3}+\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 116, normalized size = 0.90 \[ \frac {a x^6}{6}+\frac {b c \left (2 c^2+3\right ) \sin ^{-1}\left (c+d x^2\right )}{12 d^3}+\frac {1}{2} b \left (\frac {11 c^2+4}{18 d^3}-\frac {5 c x^2}{18 d^2}+\frac {x^4}{9 d}\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}+\frac {1}{6} b x^6 \sin ^{-1}\left (c+d x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 97, normalized size = 0.75 \[ \frac {6 \, a d^{3} x^{6} + 3 \, {\left (2 \, b d^{3} x^{6} + 2 \, b c^{3} + 3 \, b c\right )} \arcsin \left (d x^{2} + c\right ) + {\left (2 \, b d^{2} x^{4} - 5 \, b c d x^{2} + 11 \, b c^{2} + 4 \, b\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}}{36 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 220, normalized size = 1.71 \[ \frac {6 \, a d x^{6} + {\left (\frac {18 \, {\left (d x^{2} + c\right )} c^{2} \arcsin \left (d x^{2} + c\right )}{d^{2}} + \frac {6 \, {\left (d x^{2} + c\right )} {\left ({\left (d x^{2} + c\right )}^{2} - 1\right )} \arcsin \left (d x^{2} + c\right )}{d^{2}} - \frac {18 \, {\left ({\left (d x^{2} + c\right )}^{2} - 1\right )} c \arcsin \left (d x^{2} + c\right )}{d^{2}} - \frac {9 \, {\left (d x^{2} + c\right )} \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1} c}{d^{2}} + \frac {18 \, \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1} c^{2}}{d^{2}} + \frac {6 \, {\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right )}{d^{2}} - \frac {9 \, c \arcsin \left (d x^{2} + c\right )}{d^{2}} - \frac {2 \, {\left (-{\left (d x^{2} + c\right )}^{2} + 1\right )}^{\frac {3}{2}}}{d^{2}} + \frac {6 \, \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}}{d^{2}}\right )} b}{36 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 258, normalized size = 2.00 \[ \frac {x^{6} a}{6}+\frac {b \,x^{6} \arcsin \left (d \,x^{2}+c \right )}{6}+\frac {b \,x^{4} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{18 d}-\frac {5 b c \,x^{2} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{36 d^{2}}+\frac {11 b \,c^{2} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{36 d^{3}}+\frac {b \,c^{3} \arctan \left (\frac {\sqrt {d^{2}}\, \left (x^{2}+\frac {c}{d}\right )}{\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}\right )}{6 d^{2} \sqrt {d^{2}}}+\frac {b c \arctan \left (\frac {\sqrt {d^{2}}\, \left (x^{2}+\frac {c}{d}\right )}{\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}\right )}{4 d^{2} \sqrt {d^{2}}}+\frac {b \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{9 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 245, normalized size = 1.90 \[ \frac {1}{6} \, a x^{6} + \frac {1}{36} \, {\left (6 \, x^{6} \arcsin \left (d x^{2} + c\right ) + {\left (\frac {2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} x^{4}}{d^{2}} - \frac {5 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} c x^{2}}{d^{3}} - \frac {15 \, c^{3} \arcsin \left (-\frac {d^{2} x^{2} + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} + \frac {9 \, {\left (c^{2} - 1\right )} c \arcsin \left (-\frac {d^{2} x^{2} + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{4}} + \frac {15 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )} d\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\left (a+b\,\mathrm {asin}\left (d\,x^2+c\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.05, size = 204, normalized size = 1.58 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b c^{3} \operatorname {asin}{\left (c + d x^{2} \right )}}{6 d^{3}} + \frac {11 b c^{2} \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{36 d^{3}} - \frac {5 b c x^{2} \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{36 d^{2}} + \frac {b c \operatorname {asin}{\left (c + d x^{2} \right )}}{4 d^{3}} + \frac {b x^{6} \operatorname {asin}{\left (c + d x^{2} \right )}}{6} + \frac {b x^{4} \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{18 d} + \frac {b \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{9 d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{6} \left (a + b \operatorname {asin}{\relax (c )}\right )}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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