Optimal. Leaf size=88 \[ \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{4 c^8}+\frac {b \sqrt {x}}{4 c^7}+\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6097, 50, 63, 206} \[ \frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{3/2}}{12 c^5}+\frac {b \sqrt {x}}{4 c^7}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{4 c^8}+\frac {b x^{7/2}}{28 c} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 6097
Rubi steps
\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {1}{8} (b c) \int \frac {x^{7/2}}{1-c^2 x} \, dx\\ &=\frac {b x^{7/2}}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {x^{5/2}}{1-c^2 x} \, dx}{8 c}\\ &=\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {x^{3/2}}{1-c^2 x} \, dx}{8 c^3}\\ &=\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {\sqrt {x}}{1-c^2 x} \, dx}{8 c^5}\\ &=\frac {b \sqrt {x}}{4 c^7}+\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx}{8 c^7}\\ &=\frac {b \sqrt {x}}{4 c^7}+\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{4 c^7}\\ &=\frac {b \sqrt {x}}{4 c^7}+\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{4 c^8}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 114, normalized size = 1.30 \[ \frac {a x^4}{4}+\frac {b \log \left (1-c \sqrt {x}\right )}{8 c^8}-\frac {b \log \left (c \sqrt {x}+1\right )}{8 c^8}+\frac {b \sqrt {x}}{4 c^7}+\frac {b x^{3/2}}{12 c^5}+\frac {b x^{5/2}}{20 c^3}+\frac {b x^{7/2}}{28 c}+\frac {1}{4} b x^4 \tanh ^{-1}\left (c \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 89, normalized size = 1.01 \[ \frac {210 \, a c^{8} x^{4} + 105 \, {\left (b c^{8} x^{4} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (15 \, b c^{7} x^{3} + 21 \, b c^{5} x^{2} + 35 \, b c^{3} x + 105 \, b c\right )} \sqrt {x}}{840 \, c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 359, normalized size = 4.08 \[ \frac {1}{4} \, a x^{4} + \frac {2}{105} \, b c {\left (\frac {\frac {105 \, {\left (c \sqrt {x} + 1\right )}^{6}}{{\left (c \sqrt {x} - 1\right )}^{6}} - \frac {315 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {770 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} - \frac {770 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {609 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {203 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 44}{c^{9} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{7}} + \frac {105 \, {\left (\frac {{\left (c \sqrt {x} + 1\right )}^{7}}{{\left (c \sqrt {x} - 1\right )}^{7}} + \frac {7 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {7 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{9} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{8}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 84, normalized size = 0.95 \[ \frac {x^{4} a}{4}+\frac {b \,x^{4} \arctanh \left (c \sqrt {x}\right )}{4}+\frac {b \,x^{\frac {7}{2}}}{28 c}+\frac {b \,x^{\frac {5}{2}}}{20 c^{3}}+\frac {b \,x^{\frac {3}{2}}}{12 c^{5}}+\frac {b \sqrt {x}}{4 c^{7}}+\frac {b \ln \left (c \sqrt {x}-1\right )}{8 c^{8}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{8 c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 86, normalized size = 0.98 \[ \frac {1}{4} \, a x^{4} + \frac {1}{840} \, {\left (210 \, x^{4} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (15 \, c^{6} x^{\frac {7}{2}} + 21 \, c^{4} x^{\frac {5}{2}} + 35 \, c^{2} x^{\frac {3}{2}} + 105 \, \sqrt {x}\right )}}{c^{8}} - \frac {105 \, \log \left (c \sqrt {x} + 1\right )}{c^{9}} + \frac {105 \, \log \left (c \sqrt {x} - 1\right )}{c^{9}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 86, normalized size = 0.98 \[ \frac {\frac {b\,c^3\,x^{3/2}}{12}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{4}+\frac {b\,c^5\,x^{5/2}}{20}+\frac {b\,c^7\,x^{7/2}}{28}+\frac {b\,c\,\sqrt {x}}{4}}{c^8}+\frac {b\,\left (105\,x^4\,\ln \left (c\,\sqrt {x}+1\right )-105\,x^4\,\ln \left (1-c\,\sqrt {x}\right )\right )}{840}+\frac {a\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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