Optimal. Leaf size=101 \[ -\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 44, 65,
214} \begin {gather*} \frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}}-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {\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 d^2-a e^2}\\ &=-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 100, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d+e x}}{\left (-c d^2+a e^2\right ) (a e+c d x)}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (-c d^2+a e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 111, normalized size = 1.10
method | result | size |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (e^{2} a -c \,d^{2}\right ) \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \left (e^{2} a -c \,d^{2}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )\) | \(111\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (e^{2} a -c \,d^{2}\right ) \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \left (e^{2} a -c \,d^{2}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )\) | \(111\) |
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 = 2.88, size = 347, normalized size = 3.44 \begin {gather*} \left [-\frac {\sqrt {c^{2} d^{3} - a c d e^{2}} {\left (c d x e + a e^{2}\right )} \log \left (\frac {c d x e + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {x e + d}}{c d x + a e}\right ) + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (c^{4} d^{6} x - 2 \, a c^{3} d^{4} x e^{2} + a c^{3} d^{5} e + a^{2} c^{2} d^{2} x e^{4} - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )}}, -\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d x e + a e^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {x e + d}}{c d x e + c d^{2}}\right ) + {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {x e + d}}{c^{4} d^{6} x - 2 \, a c^{3} d^{4} x e^{2} + a c^{3} d^{5} e + a^{2} c^{2} d^{2} x e^{4} - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}}\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 = 1.24, size = 111, normalized size = 1.10 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d^{2} - a e^{2}\right )}} - \frac {\sqrt {x e + d} e}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} {\left (c d^{2} - a e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 97, normalized size = 0.96 \begin {gather*} \frac {e\,\mathrm {atan}\left (\frac {c\,d\,\sqrt {d+e\,x}}{\sqrt {c\,d}\,\sqrt {a\,e^2-c\,d^2}}\right )}{\sqrt {c\,d}\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}+\frac {e\,\sqrt {d+e\,x}}{\left (a\,e^2-c\,d^2\right )\,\left (a\,e^2-c\,d^2+c\,d\,\left (d+e\,x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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