Optimal. Leaf size=231 \[ \frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}} \]
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Rubi [A]
time = 0.27, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {753, 839, 841,
1180, 214} \begin {gather*} -\frac {\left (3 \sqrt {a} e+2 \sqrt {c} d\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{7/4}}+\frac {(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac {d e \sqrt {d+e x}}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 753
Rule 839
Rule 841
Rule 1180
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx &=\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (-2 c d^2+3 a e^2\right )+\frac {1}{2} c d e x\right )}{a-c x^2} \, dx}{2 a c}\\ &=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\int \frac {c d \left (c d^2-2 a e^2\right )+\frac {1}{2} c e \left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c^2}\\ &=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c d e \left (c d^2-3 a e^2\right )+c d e \left (c d^2-2 a e^2\right )+\frac {1}{2} c e \left (c d^2-3 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^2}\\ &=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^2\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (2 \sqrt {c} d+3 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} c}\\ &=\frac {d e \sqrt {d+e x}}{2 a c}+\frac {(a e+c d x) (d+e x)^{3/2}}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 1.13, size = 257, normalized size = 1.11 \begin {gather*} \frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} \left (c d^2 x+a e (2 d+e x)\right )}{-a+c x^2}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 c d^2-\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (2 c d^2+\sqrt {a} \sqrt {c} d e-3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 313, normalized size = 1.35
method | result | size |
derivativedivides | \(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {\frac {\left (-4 a d \,e^{2} c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (4 a d \,e^{2} c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(313\) |
default | \(2 e^{3} \left (\frac {\frac {\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a c \,e^{2}}+\frac {\left (e^{2} a -c \,d^{2}\right ) d \sqrt {e x +d}}{4 a c \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {\frac {\left (-4 a d \,e^{2} c +2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (4 a d \,e^{2} c -2 c^{2} d^{3}+3 \sqrt {a c \,e^{2}}\, a \,e^{2}-\sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 a \,e^{2}}\right )\) | \(313\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1294 vs.
\(2 (184) = 368\).
time = 3.98, size = 1294, normalized size = 5.60 \begin {gather*} -\frac {{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left (-{\left (20 \, c^{3} d^{6} e^{3} - 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} - 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} + {\left (5 \, a^{2} c^{3} d^{3} e^{4} - 9 \, a^{3} c^{2} d e^{6} - {\left (2 \, a^{3} c^{6} d^{2} - 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) - {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left (-{\left (20 \, c^{3} d^{6} e^{3} - 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} - 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} - {\left (5 \, a^{2} c^{3} d^{3} e^{4} - 9 \, a^{3} c^{2} d e^{6} - {\left (2 \, a^{3} c^{6} d^{2} - 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {\frac {4 \, c^{2} d^{5} + a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) + {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left (-{\left (20 \, c^{3} d^{6} e^{3} - 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} - 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} + {\left (5 \, a^{2} c^{3} d^{3} e^{4} - 9 \, a^{3} c^{2} d e^{6} + {\left (2 \, a^{3} c^{6} d^{2} - 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) - {\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt {\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}} \log \left (-{\left (20 \, c^{3} d^{6} e^{3} - 101 \, a c^{2} d^{4} e^{5} + 162 \, a^{2} c d^{2} e^{7} - 81 \, a^{3} e^{9}\right )} \sqrt {x e + d} - {\left (5 \, a^{2} c^{3} d^{3} e^{4} - 9 \, a^{3} c^{2} d e^{6} + {\left (2 \, a^{3} c^{6} d^{2} - 3 \, a^{4} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}}\right )} \sqrt {\frac {4 \, c^{2} d^{5} - a^{3} c^{3} \sqrt {\frac {25 \, c^{2} d^{4} e^{6} - 90 \, a c d^{2} e^{8} + 81 \, a^{2} e^{10}}{a^{3} c^{7}}} - 15 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}}{a^{3} c^{3}}}\right ) + 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {x e + d}}{8 \, {\left (a c^{2} x^{2} - a^{2} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 493 vs.
\(2 (184) = 368\).
time = 2.10, size = 493, normalized size = 2.13 \begin {gather*} \frac {{\left (2 \, a c^{4} d^{4} - 4 \, a^{2} c^{3} d^{2} e^{2} - {\left (c d^{2} e^{2} - 3 \, a e^{4}\right )} a^{2} c^{2} - {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e - \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} + \frac {{\left (2 \, \sqrt {a c} a c^{4} d^{4} - 4 \, \sqrt {a c} a^{2} c^{3} d^{2} e^{2} - {\left (\sqrt {a c} c d^{2} e^{2} - 3 \, \sqrt {a c} a e^{4}\right )} a^{2} c^{2} + {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} d + \sqrt {a c} a^{2} c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} c d^{2} e - \sqrt {x e + d} c d^{3} e + {\left (x e + d\right )}^{\frac {3}{2}} a e^{3} + \sqrt {x e + d} a d e^{3}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 1988, normalized size = 8.61 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {18\,a\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}+\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {15\,d^2\,e^9}{c}-\frac {43\,d^4\,e^7}{4\,a}-\frac {27\,a\,e^{11}}{4\,c^2}+\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^6\,c^3}}-\frac {10\,c\,d^2\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}+\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {15\,d^2\,e^9}{c}-\frac {43\,d^4\,e^7}{4\,a}-\frac {27\,a\,e^{11}}{4\,c^2}+\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^6\,c^3}}+\frac {18\,d\,e^7\,\sqrt {a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}+\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {5\,a^2\,c^4\,d^6\,e^5}{2}-\frac {27\,a^5\,c\,e^{11}}{4}-\frac {43\,a^3\,c^3\,d^4\,e^7}{4}+15\,a^4\,c^2\,d^2\,e^9+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^2}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a\,c}}-\frac {10\,d^3\,e^5\,\sqrt {a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}-\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}+\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{15\,a^5\,c\,d^2\,e^9-\frac {27\,a^6\,e^{11}}{4}+\frac {5\,a^3\,c^3\,d^6\,e^5}{2}-\frac {43\,a^4\,c^2\,d^4\,e^7}{4}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a\,c}+\frac {9\,a\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,c^3}}\right )\,\sqrt {\frac {4\,a^3\,c^6\,d^5-9\,a\,e^5\,\sqrt {a^9\,c^7}+15\,a^5\,c^4\,d\,e^4-15\,a^4\,c^5\,d^3\,e^2+5\,c\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^7}}-\frac {\frac {\left (c\,d^2\,e+a\,e^3\right )\,{\left (d+e\,x\right )}^{3/2}}{2\,a\,c}+\frac {\left (a\,d\,e^3-c\,d^3\,e\right )\,\sqrt {d+e\,x}}{2\,a\,c}}{c\,{\left (d+e\,x\right )}^2-a\,e^2+c\,d^2-2\,c\,d\,\left (d+e\,x\right )}-2\,\mathrm {atanh}\left (\frac {18\,a\,e^8\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}+\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}-\frac {15\,d^2\,e^9}{c}-\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^6\,c^3}}-\frac {10\,c\,d^2\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}+\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a\,e^{11}}{4\,c^2}+\frac {43\,d^4\,e^7}{4\,a}-\frac {15\,d^2\,e^9}{c}-\frac {5\,c\,d^6\,e^5}{2\,a^2}+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,a^4\,c^5}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a^5\,c^4}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^6\,c^3}}-\frac {18\,d\,e^7\,\sqrt {a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}+\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^5\,c\,e^{11}}{4}-\frac {5\,a^2\,c^4\,d^6\,e^5}{2}+\frac {43\,a^3\,c^3\,d^4\,e^7}{4}-15\,a^4\,c^2\,d^2\,e^9+\frac {9\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a^2}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,a\,c}}+\frac {10\,d^3\,e^5\,\sqrt {a^9\,c^7}\,\sqrt {d+e\,x}\,\sqrt {\frac {d^5}{16\,a^3\,c}+\frac {15\,d\,e^4}{64\,a\,c^3}-\frac {15\,d^3\,e^2}{64\,a^2\,c^2}+\frac {9\,e^5\,\sqrt {a^9\,c^7}}{64\,a^5\,c^7}-\frac {5\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^6}}}{\frac {27\,a^6\,e^{11}}{4}-15\,a^5\,c\,d^2\,e^9-\frac {5\,a^3\,c^3\,d^6\,e^5}{2}+\frac {43\,a^4\,c^2\,d^4\,e^7}{4}-\frac {7\,d^3\,e^8\,\sqrt {a^9\,c^7}}{2\,c^2}+\frac {5\,d^5\,e^6\,\sqrt {a^9\,c^7}}{4\,a\,c}+\frac {9\,a\,d\,e^{10}\,\sqrt {a^9\,c^7}}{4\,c^3}}\right )\,\sqrt {\frac {4\,a^3\,c^6\,d^5+9\,a\,e^5\,\sqrt {a^9\,c^7}+15\,a^5\,c^4\,d\,e^4-15\,a^4\,c^5\,d^3\,e^2-5\,c\,d^2\,e^3\,\sqrt {a^9\,c^7}}{64\,a^6\,c^7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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