Optimal. Leaf size=129 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874}
\begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 874
Rule 886
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)^{11/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {(2 c d) \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}{9 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 79, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} (-7 a e g+c d (9 f+2 g x))}{63 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 136, normalized size = 1.05
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +7 a e g -9 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (g x +f \right )^{\frac {9}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}\) | \(99\) |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 c^{3} d^{3} g \,x^{3}+3 a \,c^{2} d^{2} e g \,x^{2}-9 c^{3} d^{3} f \,x^{2}+12 a^{2} c d \,e^{2} g x -18 a \,c^{2} d^{2} e f x +7 a^{3} e^{3} g -9 a^{2} c d \,e^{2} f \right ) \left (c d x +a e \right )}{63 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {9}{2}} \left (a e g -c d f \right )^{2}}\) | \(136\) |
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. 677 vs.
\(2 (119) = 238\).
time = 0.81, size = 677, normalized size = 5.25 \begin {gather*} \frac {2 \, {\left (2 \, c^{4} d^{4} g x^{4} + 9 \, c^{4} d^{4} f x^{3} - 7 \, a^{4} g e^{4} - {\left (19 \, a^{3} c d g x - 9 \, a^{3} c d f\right )} e^{3} - 3 \, {\left (5 \, a^{2} c^{2} d^{2} g x^{2} - 9 \, a^{2} c^{2} d^{2} f x\right )} e^{2} - {\left (a c^{3} d^{3} g x^{3} - 27 \, a c^{3} d^{3} f x^{2}\right )} e\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}}{63 \, {\left (c^{2} d^{3} f^{2} g^{5} x^{5} + 5 \, c^{2} d^{3} f^{3} g^{4} x^{4} + 10 \, c^{2} d^{3} f^{4} g^{3} x^{3} + 10 \, c^{2} d^{3} f^{5} g^{2} x^{2} + 5 \, c^{2} d^{3} f^{6} g x + c^{2} d^{3} f^{7} + {\left (a^{2} g^{7} x^{6} + 5 \, a^{2} f g^{6} x^{5} + 10 \, a^{2} f^{2} g^{5} x^{4} + 10 \, a^{2} f^{3} g^{4} x^{3} + 5 \, a^{2} f^{4} g^{3} x^{2} + a^{2} f^{5} g^{2} x\right )} e^{3} - {\left (2 \, a c d f g^{6} x^{6} - a^{2} d f^{5} g^{2} + {\left (10 \, a c d f^{2} g^{5} - a^{2} d g^{7}\right )} x^{5} + 5 \, {\left (4 \, a c d f^{3} g^{4} - a^{2} d f g^{6}\right )} x^{4} + 10 \, {\left (2 \, a c d f^{4} g^{3} - a^{2} d f^{2} g^{5}\right )} x^{3} + 10 \, {\left (a c d f^{5} g^{2} - a^{2} d f^{3} g^{4}\right )} x^{2} + {\left (2 \, a c d f^{6} g - 5 \, a^{2} d f^{4} g^{3}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{5} x^{6} - 2 \, a c d^{2} f^{6} g + {\left (5 \, c^{2} d^{2} f^{3} g^{4} - 2 \, a c d^{2} f g^{6}\right )} x^{5} + 10 \, {\left (c^{2} d^{2} f^{4} g^{3} - a c d^{2} f^{2} g^{5}\right )} x^{4} + 10 \, {\left (c^{2} d^{2} f^{5} g^{2} - 2 \, a c d^{2} f^{3} g^{4}\right )} x^{3} + 5 \, {\left (c^{2} d^{2} f^{6} g - 4 \, a c d^{2} f^{4} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{7} - 10 \, a c d^{2} f^{5} 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 = 4.54, size = 315, normalized size = 2.44 \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\,\left (7\,a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^4\,d^4\,x^4}{63\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (19\,a\,e\,g-27\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (5\,a\,e\,g-9\,c\,d\,f\right )}{21\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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