Optimal. Leaf size=115 \[ -\frac {3 b c \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d^2}+\frac {b x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d}-\frac {b \left (1+2 c^2\right ) \text {ArcSin}\left (c+d x^2\right )}{8 d^2}+\frac {1}{4} x^4 \left (a+b \text {ArcSin}\left (c+d x^2\right )\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 115, 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 = {4926, 12, 1128,
756, 654, 633, 222} \begin {gather*} \frac {1}{4} x^4 \left (a+b \text {ArcSin}\left (c+d x^2\right )\right )-\frac {b \left (2 c^2+1\right ) \text {ArcSin}\left (c+d x^2\right )}{8 d^2}+\frac {b x^2 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{8 d}-\frac {3 b c \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{8 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 633
Rule 654
Rule 756
Rule 1128
Rule 4926
Rubi steps
\begin {align*} \int x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{4} b \int \frac {2 d x^5}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{2} (b d) \int \frac {x^5}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{4} (b d) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=\frac {b x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {b \text {Subst}\left (\int \frac {-1+c^2+3 c d x}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{8 d}\\ &=-\frac {3 b c \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d^2}+\frac {b x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {\left (b \left (1+2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{8 d}\\ &=-\frac {3 b c \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d^2}+\frac {b x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {\left (b \left (1+2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 d^2}}} \, dx,x,-2 d \left (c+d x^2\right )\right )}{16 d^3}\\ &=-\frac {3 b c \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d^2}+\frac {b x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{8 d}-\frac {b \left (1+2 c^2\right ) \sin ^{-1}\left (c+d x^2\right )}{8 d^2}+\frac {1}{4} x^4 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 98, normalized size = 0.85 \begin {gather*} \frac {a x^4}{4}+\frac {1}{2} b \left (-\frac {3 c}{4 d^2}+\frac {x^2}{4 d}\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}-\frac {b \left (1+2 c^2\right ) \text {ArcSin}\left (c+d x^2\right )}{8 d^2}+\frac {1}{4} b x^4 \text {ArcSin}\left (c+d x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 191, normalized size = 1.66
method | result | size |
default | \(\frac {x^{4} a}{4}+\frac {b \,x^{4} \arcsin \left (d \,x^{2}+c \right )}{4}+\frac {b \,x^{2} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{8 d}-\frac {3 b c \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{8 d^{2}}-\frac {b \,c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, \left (x^{2}+\frac {c}{d}\right )}{\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}\right )}{4 d \sqrt {d^{2}}}-\frac {b \arctan \left (\frac {\sqrt {d^{2}}\, \left (x^{2}+\frac {c}{d}\right )}{\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}\right )}{8 d \sqrt {d^{2}}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 174, normalized size = 1.51 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \arcsin \left (d x^{2} + c\right ) + d {\left (\frac {\sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} x^{2}}{d^{2}} + \frac {3 \, c^{2} \arcsin \left (-\frac {d^{2} x^{2} + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {{\left (c^{2} - 1\right )} \arcsin \left (-\frac {d^{2} x^{2} + c d}{\sqrt {c^{2} d^{2} - {\left (c^{2} - 1\right )} d^{2}}}\right )}{d^{3}} - \frac {3 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} c}{d^{3}}\right )}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 79, normalized size = 0.69 \begin {gather*} \frac {2 \, a d^{2} x^{4} + {\left (2 \, b d^{2} x^{4} - 2 \, b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (b d x^{2} - 3 \, b c\right )}}{8 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 133, normalized size = 1.16 \begin {gather*} \begin {cases} \frac {a x^{4}}{4} - \frac {b c^{2} \operatorname {asin}{\left (c + d x^{2} \right )}}{4 d^{2}} - \frac {3 b c \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{8 d^{2}} + \frac {b x^{4} \operatorname {asin}{\left (c + d x^{2} \right )}}{4} + \frac {b x^{2} \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{8 d} - \frac {b \operatorname {asin}{\left (c + d x^{2} \right )}}{8 d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{4} \left (a + b \operatorname {asin}{\left (c \right )}\right )}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 128, normalized size = 1.11 \begin {gather*} -\frac {{\left (d x^{2} + c\right )} a c + {\left ({\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}\right )} b c}{2 \, d^{2}} + \frac {2 \, {\left ({\left (d x^{2} + c\right )}^{2} - 1\right )} b \arcsin \left (d x^{2} + c\right ) + {\left (d x^{2} + c\right )} \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1} b + 2 \, {\left ({\left (d x^{2} + c\right )}^{2} - 1\right )} a + b \arcsin \left (d x^{2} + c\right )}{8 \, d^{2}} \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 = -0.01 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {asin}\left (d\,x^2+c\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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