Optimal. Leaf size=114 \[ \frac {2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2}+\frac {2 (d+e x)^{3/2}}{3 c d}-\frac {2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 52, 65,
214} \begin {gather*} -\frac {2 \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^{5/2} d^{5/2}}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^2 d^2}+\frac {2 (d+e x)^{3/2}}{3 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx\\ &=\frac {2 (d+e x)^{3/2}}{3 c d}+\frac {\left (c d^2-a e^2\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{c d}\\ &=\frac {2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2}+\frac {2 (d+e x)^{3/2}}{3 c d}+\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{c^2 d^2}\\ &=\frac {2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2}+\frac {2 (d+e x)^{3/2}}{3 c d}+\frac {\left (2 \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^2 d^2 e}\\ &=\frac {2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2}+\frac {2 (d+e x)^{3/2}}{3 c d}-\frac {2 \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^{5/2} d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 102, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-3 a e^2+c d (4 d+e x)\right )}{3 c^2 d^2}+\frac {2 \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^{5/2} d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 125, normalized size = 1.10
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+a \,e^{2} \sqrt {e x +d}-c \,d^{2} \sqrt {e x +d}\right )}{c^{2} d^{2}}+\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{2} d^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(125\) |
default | \(-\frac {2 \left (-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+a \,e^{2} \sqrt {e x +d}-c \,d^{2} \sqrt {e x +d}\right )}{c^{2} d^{2}}+\frac {2 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{2} d^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(125\) |
risch | \(-\frac {2 \left (-c d e x +3 e^{2} a -4 c \,d^{2}\right ) \sqrt {e x +d}}{3 c^{2} d^{2}}+\frac {2 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2} e^{4}}{c^{2} d^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {4 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a \,e^{2}}{c \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {2 d^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(199\) |
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 = 2.97, size = 252, normalized size = 2.21 \begin {gather*} \left [\frac {3 \, {\left (c d^{2} - a e^{2}\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 (c d x e + 4 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt {x e + d}}{3 \, c^{2} d^{2}}, -\frac {2 \, {\left (3 \, {\left (c d^{2} - a e^{2}\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 (c d x e + 4 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt {x e + d}\right )}}{3 \, c^{2} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.19, size = 107, normalized size = 0.94 \begin {gather*} \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 c d} + \frac {\sqrt {d + e x} \left (- 2 a e^{2} + 2 c d^{2}\right )}{c^{2} d^{2}} + \frac {2 \left (a e^{2} - c d^{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{c^{3} d^{3} \sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 133, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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^{2} d^{2}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} + 3 \, \sqrt {x e + d} c^{2} d^{3} - 3 \, \sqrt {x e + d} a c d e^{2}\right )}}{3 \, c^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 121, normalized size = 1.06 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}}{3\,c\,d}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,{\left (a\,e^2-c\,d^2\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4}\right )\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}{c^{5/2}\,d^{5/2}}-\frac {2\,\left (a\,e^2-c\,d^2\right )\,\sqrt {d+e\,x}}{c^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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