Optimal. Leaf size=63 \[ -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {860} \[ -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.83 \[ -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 193, normalized size = 3.06 \[ -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} {\left (g x + f\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} c d^{2} e^{2} f - a^{3} d e^{3} g + {\left (c^{3} d^{3} e f - a c^{2} d^{2} e^{2} g\right )} x^{3} + {\left ({\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} g\right )} x^{2} + {\left ({\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f - {\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 1.00 \[ \frac {2 \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (a e g -c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {5}{2}} \sqrt {g x + f}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.32, size = 169, normalized size = 2.68 \[ \frac {\left (\frac {2\,f\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}+\frac {2\,g\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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