Optimal. Leaf size=142 \[ -\frac {d (b c-3 a d) \sqrt {c+d x}}{a b^2}+\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {(b c-a d)^{3/2} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{5/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {100, 159, 162,
65, 214} \begin {gather*} \frac {(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{5/2}}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}-\frac {d \sqrt {c+d x} (b c-3 a d)}{a b^2}+\frac {(c+d x)^{3/2} (b c-a d)}{a b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 159
Rule 162
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x (a+b x)^2} \, dx &=\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}+\frac {\int \frac {\sqrt {c+d x} \left (b c^2-\frac {1}{2} d (b c-3 a d) x\right )}{x (a+b x)} \, dx}{a b}\\ &=-\frac {d (b c-3 a d) \sqrt {c+d x}}{a b^2}+\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}+\frac {2 \int \frac {\frac {b^2 c^3}{2}+\frac {1}{4} d \left (b^2 c^2+4 a b c d-3 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a b^2}\\ &=-\frac {d (b c-3 a d) \sqrt {c+d x}}{a b^2}+\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}+\frac {c^3 \int \frac {1}{x \sqrt {c+d x}} \, dx}{a^2}-\frac {\left ((b c-a d)^2 (2 b c+3 a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^2 b^2}\\ &=-\frac {d (b c-3 a d) \sqrt {c+d x}}{a b^2}+\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 d}-\frac {\left ((b c-a d)^2 (2 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^2 b^2 d}\\ &=-\frac {d (b c-3 a d) \sqrt {c+d x}}{a b^2}+\frac {(b c-a d) (c+d x)^{3/2}}{a b (a+b x)}-\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {(b c-a d)^{3/2} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 145, normalized size = 1.02 \begin {gather*} \frac {\frac {a \sqrt {c+d x} \left (b^2 c^2+3 a^2 d^2+2 a b d (-c+d x)\right )}{b^2 (a+b x)}+\frac {\sqrt {-b c+a d} \left (2 b^2 c^2+a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{5/2}}-2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 179, normalized size = 1.26
method | result | size |
derivativedivides | \(2 d^{2} \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{b \left (d x +c \right )+a d -b c}+\frac {\left (3 a^{3} d^{3}-4 a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{2} a^{2} d^{2}}-\frac {c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{2}}\right )\) | \(179\) |
default | \(2 d^{2} \left (\frac {\sqrt {d x +c}}{b^{2}}-\frac {\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{b \left (d x +c \right )+a d -b c}+\frac {\left (3 a^{3} d^{3}-4 a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}}{b^{2} a^{2} d^{2}}-\frac {c^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{2} d^{2}}\right )\) | \(179\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.12, size = 865, normalized size = 6.09 \begin {gather*} \left [-\frac {{\left (2 \, a b^{2} c^{2} + a^{2} b c d - 3 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (2 \, a^{2} b d^{2} x + a b^{2} c^{2} - 2 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, \frac {{\left (2 \, a b^{2} c^{2} + a^{2} b c d - 3 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + {\left (2 \, a^{2} b d^{2} x + a b^{2} c^{2} - 2 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} \sqrt {d x + c}}{a^{2} b^{3} x + a^{3} b^{2}}, \frac {4 \, {\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - {\left (2 \, a b^{2} c^{2} + a^{2} b c d - 3 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (2 \, a^{2} b d^{2} x + a b^{2} c^{2} - 2 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{3} x + a^{3} b^{2}\right )}}, \frac {{\left (2 \, a b^{2} c^{2} + a^{2} b c d - 3 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + 2 \, {\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a^{2} b d^{2} x + a b^{2} c^{2} - 2 \, a^{2} b c d + 3 \, a^{3} d^{2}\right )} \sqrt {d x + c}}{a^{2} b^{3} x + a^{3} b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1294 vs.
\(2 (126) = 252\).
time = 127.76, size = 1294, normalized size = 9.11 \begin {gather*} \frac {2 a^{2} d^{4} \sqrt {c + d x}}{2 a^{2} b^{2} d^{2} - 2 a b^{3} c d + 2 a b^{3} d^{2} x - 2 b^{4} c d x} - \frac {a^{2} d^{4} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{2}} + \frac {a^{2} d^{4} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{2}} - \frac {6 a c d^{3} \sqrt {c + d x}}{2 a^{2} b d^{2} - 2 a b^{2} c d + 2 a b^{2} d^{2} x - 2 b^{3} c d x} + \frac {3 a c d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {3 a c d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {4 a d^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{3} \sqrt {\frac {a d}{b} - c}} - \frac {2 b c^{3} d \sqrt {c + d x}}{2 a^{3} d^{2} - 2 a^{2} b c d + 2 a^{2} b d^{2} x - 2 a b^{2} c d x} - \frac {3 c^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2} + \frac {3 c^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2} + \frac {6 c^{2} d^{2} \sqrt {c + d x}}{2 a^{2} d^{2} - 2 a b c d + 2 a b d^{2} x - 2 b^{2} c d x} + \frac {6 c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 d^{2} \sqrt {c + d x}}{b^{2}} + \frac {b c^{3} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {b c^{3} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a} - \frac {2 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{a^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a^{2} \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 193, normalized size = 1.36 \begin {gather*} \frac {2 \, c^{3} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} + \frac {2 \, \sqrt {d x + c} d^{2}}{b^{2}} - \frac {{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2} b^{2}} + \frac {\sqrt {d x + c} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a b c d^{2} + \sqrt {d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 1295, normalized size = 9.12 \begin {gather*} \frac {2\,d^2\,\sqrt {c+d\,x}}{b^2}+\frac {\mathrm {atan}\left (\frac {a^2\,d^8\,\sqrt {c^5}\,\sqrt {c+d\,x}\,36{}\mathrm {i}}{36\,a^2\,c^3\,d^8+40\,b^2\,c^5\,d^6+\frac {80\,b^3\,c^6\,d^5}{a}-\frac {60\,b^4\,c^7\,d^4}{a^2}-96\,a\,b\,c^4\,d^7}+\frac {c^2\,d^6\,\sqrt {c^5}\,\sqrt {c+d\,x}\,40{}\mathrm {i}}{40\,c^5\,d^6-\frac {96\,a\,c^4\,d^7}{b}+\frac {80\,b\,c^6\,d^5}{a}-\frac {60\,b^2\,c^7\,d^4}{a^2}+\frac {36\,a^2\,c^3\,d^8}{b^2}}+\frac {c^3\,d^5\,\sqrt {c^5}\,\sqrt {c+d\,x}\,80{}\mathrm {i}}{80\,c^6\,d^5+\frac {40\,a\,c^5\,d^6}{b}-\frac {60\,b\,c^7\,d^4}{a}-\frac {96\,a^2\,c^4\,d^7}{b^2}+\frac {36\,a^3\,c^3\,d^8}{b^3}}-\frac {b\,c^4\,d^4\,\sqrt {c^5}\,\sqrt {c+d\,x}\,60{}\mathrm {i}}{80\,a\,c^6\,d^5-60\,b\,c^7\,d^4+\frac {40\,a^2\,c^5\,d^6}{b}-\frac {96\,a^3\,c^4\,d^7}{b^2}+\frac {36\,a^4\,c^3\,d^8}{b^3}}-\frac {a\,c\,d^7\,\sqrt {c^5}\,\sqrt {c+d\,x}\,96{}\mathrm {i}}{40\,b\,c^5\,d^6-96\,a\,c^4\,d^7+\frac {80\,b^2\,c^6\,d^5}{a}+\frac {36\,a^2\,c^3\,d^8}{b}-\frac {60\,b^3\,c^7\,d^4}{a^2}}\right )\,\sqrt {c^5}\,2{}\mathrm {i}}{a^2}+\frac {\sqrt {c+d\,x}\,\left (a^2\,d^3-2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{a\,\left (b^3\,\left (c+d\,x\right )-b^3\,c+a\,b^2\,d\right )}-\frac {\mathrm {atan}\left (\frac {c^4\,d^5\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}\,70{}\mathrm {i}}{72\,a^3\,b\,c^3\,d^8-50\,b^4\,c^6\,d^5-170\,a\,b^3\,c^5\,d^6-162\,a^4\,c^2\,d^9+\frac {54\,a^5\,c\,d^{10}}{b}+196\,a^2\,b^2\,c^4\,d^7+\frac {60\,b^5\,c^7\,d^4}{a}}-\frac {c^3\,d^6\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}\,90{}\mathrm {i}}{54\,a^4\,c\,d^{10}-170\,b^4\,c^5\,d^6+196\,a\,b^3\,c^4\,d^7-162\,a^3\,b\,c^2\,d^9+72\,a^2\,b^2\,c^3\,d^8-\frac {50\,b^5\,c^6\,d^5}{a}+\frac {60\,b^6\,c^7\,d^4}{a^2}}+\frac {c^5\,d^4\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}\,60{}\mathrm {i}}{72\,a^4\,c^3\,d^8+60\,b^4\,c^7\,d^4-50\,a\,b^3\,c^6\,d^5+196\,a^3\,b\,c^4\,d^7+\frac {54\,a^6\,c\,d^{10}}{b^2}-170\,a^2\,b^2\,c^5\,d^6-\frac {162\,a^5\,c^2\,d^9}{b}}-\frac {a\,c^2\,d^7\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}\,54{}\mathrm {i}}{196\,a\,b^4\,c^4\,d^7-170\,b^5\,c^5\,d^6+72\,a^2\,b^3\,c^3\,d^8-162\,a^3\,b^2\,c^2\,d^9-\frac {50\,b^6\,c^6\,d^5}{a}+\frac {60\,b^7\,c^7\,d^4}{a^2}+54\,a^4\,b\,c\,d^{10}}+\frac {a^2\,c\,d^8\,\sqrt {c+d\,x}\,\sqrt {-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}\,54{}\mathrm {i}}{196\,a\,b^5\,c^4\,d^7-170\,b^6\,c^5\,d^6+54\,a^4\,b^2\,c\,d^{10}+72\,a^2\,b^4\,c^3\,d^8-162\,a^3\,b^3\,c^2\,d^9-\frac {50\,b^7\,c^6\,d^5}{a}+\frac {60\,b^8\,c^7\,d^4}{a^2}}\right )\,\sqrt {-b^5\,{\left (a\,d-b\,c\right )}^3}\,\left (3\,a\,d+2\,b\,c\right )\,1{}\mathrm {i}}{a^2\,b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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