Optimal. Leaf size=129 \[ \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]
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Rubi [A] time = 0.15, 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 = {872, 860} \[ \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 872
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 = 0.08, size = 79, normalized size = 0.61 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} (c d (9 f+2 g x)-7 a e g)}{63 \sqrt {d+e x} (f+g x)^{9/2} (c d f-a e g)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 639, normalized size = 4.95 \[ \frac {2 \, {\left (2 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 7 \, a^{4} e^{4} g + {\left (9 \, c^{4} d^{4} f - a c^{3} d^{3} e g\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{3} e f - 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{2} e^{2} f - 19 \, a^{3} c d e^{3} g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{63 \, {\left (c^{2} d^{3} f^{7} - 2 \, a c d^{2} e f^{6} g + a^{2} d e^{2} f^{5} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{5} - 2 \, a c d e^{2} f g^{6} + a^{2} e^{3} g^{7}\right )} x^{6} + {\left (5 \, c^{2} d^{2} e f^{3} g^{4} + a^{2} d e^{2} g^{7} + {\left (c^{2} d^{3} - 10 \, a c d e^{2}\right )} f^{2} g^{5} - {\left (2 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} f g^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{2} d^{2} e f^{4} g^{3} + a^{2} d e^{2} f g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{3} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{5}\right )} x^{4} + 10 \, {\left (c^{2} d^{2} e f^{5} g^{2} + a^{2} d e^{2} f^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{4}\right )} x^{3} + 5 \, {\left (c^{2} d^{2} e f^{6} g + 2 \, a^{2} d e^{2} f^{3} g^{4} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{5} g^{2} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{7} + 5 \, a^{2} d e^{2} f^{4} g^{3} + {\left (5 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{6} g - {\left (10 \, a c d^{2} e - a^{2} e^{3}\right )} f^{5} g^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.01, size = 99, normalized size = 0.77 \[ -\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +7 a e g -9 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (g x +f \right )^{\frac {9}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^3\,e^3\,\left (7\,a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^4\,d^4\,x^4}{63\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (19\,a\,e\,g-27\,c\,d\,f\right )}{63\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (5\,a\,e\,g-9\,c\,d\,f\right )}{21\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \]
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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