Optimal. Leaf size=109 \[ \frac {4 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
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Rubi [A] time = 0.06, antiderivative size = 109, 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 = {656, 648} \[ \frac {4 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.50 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (c d (3 d+e x)-2 a e^2\right )}{3 c^2 d^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 73, normalized size = 0.67 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 0.63 \[ -\frac {2 \left (c d x +a e \right ) \left (-c d e x +2 a \,e^{2}-3 c \,d^{2}\right ) \sqrt {e x +d}}{3 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 65, normalized size = 0.60 \[ \frac {2 \, {\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} + {\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )}}{3 \, \sqrt {c d x + a e} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 85, normalized size = 0.78 \[ \frac {\left (\frac {2\,x\,\sqrt {d+e\,x}}{3\,c\,d}-\frac {\left (4\,a\,e^2-6\,c\,d^2\right )\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x+\frac {d}{e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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