Optimal. Leaf size=105 \[ -\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662}
\begin {gather*} \frac {2 (d+e x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2 (d+e x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )\right ) \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 )^{3/2}} \, dx}{c d e}\\ &=-\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 51, normalized size = 0.49 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-2 a e^2+c d (d-e x)\right )}{c^2 d^2 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 60, normalized size = 0.57
method | result | size |
default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d e x +2 e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) c^{2} d^{2}}\) | \(60\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (c d e x +2 e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{c^{2} d^{2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.35 \begin {gather*} \frac {2 \, {\left (c d x e - c d^{2} + 2 \, a e^{2}\right )}}{\sqrt {c d x + a e} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.96, size = 99, normalized size = 0.94 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x e - c d^{2} + 2 \, a e^{2}\right )} \sqrt {x e + d}}{c^{3} d^{4} x + a c^{2} d^{2} x e^{2} + {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 130, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} e}{c d} - \frac {c d^{2} e^{2} - a e^{4}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d}\right )} e^{\left (-1\right )}}{c d} + \frac {4 \, {\left (c d^{2} e - a e^{3}\right )}}{\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 116, normalized size = 1.10 \begin {gather*} \frac {\left (\frac {2\,x\,\sqrt {d+e\,x}}{c^2\,d^2}+\frac {\left (4\,a\,e^2-2\,c\,d^2\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (c^3\,d^4+a\,c^2\,d^2\,e^2\right )}{c^3\,d^3\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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