Optimal. Leaf size=63 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}} \]
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Rubi [A]
time = 0.05, 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 = {874}
\begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 874
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 78, normalized size = 1.24
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {5}{2}}}\) | \(63\) |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c^{2} d^{2} x^{2}+2 a c d e x +a^{2} e^{2}\right ) \left (c d x +a e \right )}{7 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )}\) | \(78\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (58) = 116\).
time = 0.78, size = 314, normalized size = 4.98 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} x^{2} e + 3 \, a^{2} c d x e^{2} + a^{3} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{7 \, {\left (c d^{2} f g^{4} x^{4} + 4 \, c d^{2} f^{2} g^{3} x^{3} + 6 \, c d^{2} f^{3} g^{2} x^{2} + 4 \, c d^{2} f^{4} g x + c d^{2} f^{5} - {\left (a g^{5} x^{5} + 4 \, a f g^{4} x^{4} + 6 \, a f^{2} g^{3} x^{3} + 4 \, a f^{3} g^{2} x^{2} + a f^{4} g x\right )} e^{2} + {\left (c d f g^{4} x^{5} - a d f^{4} g + {\left (4 \, c d f^{2} g^{3} - a d g^{5}\right )} x^{4} + 2 \, {\left (3 \, c d f^{3} g^{2} - 2 \, a d f g^{4}\right )} x^{3} + 2 \, {\left (2 \, c d f^{4} g - 3 \, a d f^{2} g^{3}\right )} x^{2} + {\left (c d f^{5} - 4 \, a d f^{3} g^{2}\right )} x\right )} e\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 [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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Mupad [B]
time = 4.34, size = 325, normalized size = 5.16 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^3\,e^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {2\,c^3\,d^3\,x^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a^2\,c\,d\,e^2\,x}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a\,c^2\,d^2\,e\,x^2}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (7\,c\,d\,f^4-7\,a\,e\,f^3\,g\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {x^2\,\sqrt {f+g\,x}\,\left (21\,a\,e\,f\,g^3-21\,c\,d\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}-\frac {x\,\sqrt {f+g\,x}\,\left (21\,c\,d\,f^3\,g-21\,a\,e\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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