Optimal. Leaf size=210 \[ \frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65,
214} \begin {gather*} -\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}+\frac {9 e \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^{9/2}}{(a e+c d x)^2} \, dx\\ &=-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^2\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^3\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^4 d^4}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 \left (c d^2-a e^2\right )^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 246, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {d+e x} \left (315 a^4 e^8-210 a^3 c d e^6 (5 d-e x)+42 a^2 c^2 d^2 e^4 \left (29 d^2-17 d e x-e^2 x^2\right )+6 a c^3 d^3 e^2 \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+c^4 d^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 c^5 d^5 (a e+c d x)}+\frac {9 e \left (-c d^2+a e^2\right )^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{11/2} d^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs.
\(2(182)=364\).
time = 0.91, size = 375, normalized size = 1.79
method | result | size |
derivativedivides | \(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+2 a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}-c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{6} \sqrt {e x +d}-12 a^{2} e^{4} d^{2} c \sqrt {e x +d}+12 a \,e^{2} d^{4} c^{2} \sqrt {e x +d}-4 d^{6} c^{3} \sqrt {e x +d}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{d^{5} c^{5}}\right )\) | \(375\) |
default | \(2 e \left (-\frac {-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+2 a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}-c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}+4 a^{3} e^{6} \sqrt {e x +d}-12 a^{2} e^{4} d^{2} c \sqrt {e x +d}+12 a \,e^{2} d^{4} c^{2} \sqrt {e x +d}-4 d^{6} c^{3} \sqrt {e x +d}}{c^{5} d^{5}}+\frac {\frac {\left (-\frac {1}{2} a^{4} e^{8}+2 a^{3} c \,d^{2} e^{6}-3 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}-\frac {1}{2} c^{4} d^{8}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {9 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{d^{5} c^{5}}\right )\) | \(375\) |
risch | \(-\frac {2 e \left (-5 c^{3} d^{3} e^{3} x^{3}+14 a \,c^{2} d^{2} e^{4} x^{2}-29 c^{3} d^{4} e^{2} x^{2}-35 a^{2} c d \,e^{5} x +98 a \,c^{2} d^{3} e^{3} x -78 c^{3} d^{5} e x +140 e^{6} a^{3}-455 e^{4} d^{2} a^{2} c +504 d^{4} e^{2} c^{2} a -194 d^{6} c^{3}\right ) \sqrt {e x +d}}{35 c^{5} d^{5}}-\frac {e^{9} \sqrt {e x +d}\, a^{4}}{d^{5} c^{5} \left (c d e x +e^{2} a \right )}+\frac {4 e^{7} \sqrt {e x +d}\, a^{3}}{d^{3} c^{4} \left (c d e x +e^{2} a \right )}-\frac {6 e^{5} \sqrt {e x +d}\, a^{2}}{d \,c^{3} \left (c d e x +e^{2} a \right )}+\frac {4 d \,e^{3} \sqrt {e x +d}\, a}{c^{2} \left (c d e x +e^{2} a \right )}-\frac {d^{3} e \sqrt {e x +d}}{c \left (c d e x +e^{2} a \right )}+\frac {9 e^{9} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{4}}{d^{5} c^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {36 e^{7} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{3}}{d^{3} c^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {54 e^{5} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2}}{d \,c^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {36 d \,e^{3} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a}{c^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {9 d^{3} e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(583\) |
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 = 4.36, size = 766, normalized size = 3.65 \begin {gather*} \left [\frac {315 \, {\left (c^{4} d^{7} x e - 3 \, a c^{3} d^{5} x e^{3} + a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{3} x e^{5} - 3 \, a^{2} c^{2} d^{4} e^{4} - a^{3} c d x e^{7} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, \sqrt {x e + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}} - a e^{2}}{c d x + a e}\right ) + 2 \, {\left (388 \, c^{4} d^{7} x e - 35 \, c^{4} d^{8} - 210 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} + 42 \, {\left (a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{6} - 6 \, {\left (3 \, a c^{3} d^{3} x^{3} - 119 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (5 \, c^{4} d^{4} x^{4} - 69 \, a c^{3} d^{4} x^{2} - 609 \, a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (29 \, c^{4} d^{5} x^{3} - 426 \, a c^{3} d^{5} x\right )} e^{3} + 12 \, {\left (13 \, c^{4} d^{6} x^{2} + 44 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{70 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac {315 \, {\left (c^{4} d^{7} x e - 3 \, a c^{3} d^{5} x e^{3} + a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{3} x e^{5} - 3 \, a^{2} c^{2} d^{4} e^{4} - a^{3} c d x e^{7} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {x e + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (388 \, c^{4} d^{7} x e - 35 \, c^{4} d^{8} - 210 \, a^{3} c d x e^{7} - 315 \, a^{4} e^{8} + 42 \, {\left (a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{6} - 6 \, {\left (3 \, a c^{3} d^{3} x^{3} - 119 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (5 \, c^{4} d^{4} x^{4} - 69 \, a c^{3} d^{4} x^{2} - 609 \, a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (29 \, c^{4} d^{5} x^{3} - 426 \, a c^{3} d^{5} x\right )} e^{3} + 12 \, {\left (13 \, c^{4} d^{6} x^{2} + 44 \, a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {x e + d}}{35 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\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. 428 vs.
\(2 (189) = 378\).
time = 1.84, size = 428, normalized size = 2.04 \begin {gather*} \frac {9 \, {\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{5} d^{5}} - \frac {\sqrt {x e + d} c^{4} d^{8} e - 4 \, \sqrt {x e + d} a c^{3} d^{6} e^{3} + 6 \, \sqrt {x e + d} a^{2} c^{2} d^{4} e^{5} - 4 \, \sqrt {x e + d} a^{3} c d^{2} e^{7} + \sqrt {x e + d} a^{4} e^{9}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{5} d^{5}} + \frac {2 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{12} d^{12} e + 14 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{12} d^{13} e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{12} d^{14} e + 140 \, \sqrt {x e + d} c^{12} d^{15} e - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{11} d^{11} e^{3} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{11} d^{12} e^{3} - 420 \, \sqrt {x e + d} a c^{11} d^{13} e^{3} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c^{10} d^{10} e^{5} + 420 \, \sqrt {x e + d} a^{2} c^{10} d^{11} e^{5} - 140 \, \sqrt {x e + d} a^{3} c^{9} d^{9} e^{7}\right )}}{35 \, c^{14} d^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 443, normalized size = 2.11 \begin {gather*} \frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,c^2\,d^2}-\left (\frac {\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )}{c^2\,d^2}+\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{c^6\,d^6}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e\right )}{c^6\,d^6\,\left (d+e\,x\right )-c^6\,d^7+a\,c^5\,d^5\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{3\,c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{3\,c^6\,d^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^4\,d^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{11/2}\,d^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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