Optimal. Leaf size=144 \[ \frac {5 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac {5 e \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {5 e (d+e x)^{3/2}}{3 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)^{9/2}}{\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)^{5/2}}{(a e+c d x)^2} \, dx\\ &=-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {(5 e) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {\left (5 e \left (c d^2-a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac {5 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {\left (5 e \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^3 d^3}\\ &=\frac {5 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {\left (5 \left (c d^2-a e^2\right )^2\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^3 d^3}\\ &=\frac {5 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 146, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {d+e x} \left (15 a^2 e^4+10 a c d e^2 (-2 d+e x)+c^2 d^2 \left (3 d^2-14 d e x-2 e^2 x^2\right )\right )}{3 c^3 d^3 (a e+c d x)}+\frac {5 e \left (-c d^2+a e^2\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{7/2} d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 187, normalized size = 1.30
method | result | size |
derivativedivides | \(2 e \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 c \,d^{2} \sqrt {e x +d}}{c^{3} d^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{4}+a c \,d^{2} e^{2}-\frac {1}{2} c^{2} d^{4}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {5 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{3} d^{3}}\right )\) | \(187\) |
default | \(2 e \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 c \,d^{2} \sqrt {e x +d}}{c^{3} d^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{4}+a c \,d^{2} e^{2}-\frac {1}{2} c^{2} d^{4}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {5 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{3} d^{3}}\right )\) | \(187\) |
risch | \(-\frac {2 e \left (-c d e x +6 e^{2} a -7 c \,d^{2}\right ) \sqrt {e x +d}}{3 c^{3} d^{3}}-\frac {e^{5} \sqrt {e x +d}\, a^{2}}{d^{3} c^{3} \left (c d e x +e^{2} a \right )}+\frac {2 e^{3} \sqrt {e x +d}\, a}{d \,c^{2} \left (c d e x +e^{2} a \right )}-\frac {d e \sqrt {e x +d}}{c \left (c d e x +e^{2} a \right )}+\frac {5 e^{5} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2}}{d^{3} c^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {10 e^{3} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a}{d \,c^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {5 d e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(298\) |
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 = 3.11, size = 406, normalized size = 2.82 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{3} x e - a c d x e^{3} + a c d^{2} e^{2} - a^{2} e^{4}\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 (14 \, c^{2} d^{3} x e - 3 \, c^{2} d^{4} - 10 \, a c d x e^{3} - 15 \, a^{2} e^{4} + 2 \, {\left (c^{2} d^{2} x^{2} + 10 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{6 \, {\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}, -\frac {15 \, {\left (c^{2} d^{3} x e - a c d x e^{3} + a c d^{2} e^{2} - a^{2} e^{4}\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 (14 \, c^{2} d^{3} x e - 3 \, c^{2} d^{4} - 10 \, a c d x e^{3} - 15 \, a^{2} e^{4} + 2 \, {\left (c^{2} d^{2} x^{2} + 10 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{4} x + a c^{3} d^{3} 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 [A]
time = 1.13, size = 222, normalized size = 1.54 \begin {gather*} \frac {5 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\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^{3} d^{3}} - \frac {\sqrt {x e + d} c^{2} d^{4} e - 2 \, \sqrt {x e + d} a c d^{2} e^{3} + \sqrt {x e + d} a^{2} e^{5}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{3} d^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e + 6 \, \sqrt {x e + d} c^{4} d^{5} e - 6 \, \sqrt {x e + d} a c^{3} d^{3} e^{3}\right )}}{3 \, c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 200, normalized size = 1.39 \begin {gather*} \frac {2\,e\,{\left (d+e\,x\right )}^{3/2}}{3\,c^2\,d^2}-\frac {\sqrt {d+e\,x}\,\left (a^2\,e^5-2\,a\,c\,d^2\,e^3+c^2\,d^4\,e\right )}{c^4\,d^4\,\left (d+e\,x\right )-c^4\,d^5+a\,c^3\,d^3\,e^2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {5\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^5-2\,a\,c\,d^2\,e^3+c^2\,d^4\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}{c^{7/2}\,d^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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