Optimal. Leaf size=105 \[ -\frac {2 (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (-\frac {3}{2},m;-\frac {1}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {891, 70, 69} \[ -\frac {2 (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m \, _2F_1\left (-\frac {3}{2},m;-\frac {1}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 891
Rubi steps
\begin {align*} \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^{5/2}} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac {(a e+c d x)^{-m}}{(f+g x)^{5/2}} \, dx\\ &=\left (\left (\frac {g (a e+c d x)}{-c d f+a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac {\left (-\frac {a e g}{c d f-a e g}-\frac {c d g x}{c d f-a e g}\right )^{-m}}{(f+g x)^{5/2}} \, dx\\ &=-\frac {2 \left (-\frac {g (a e+c d x)}{c d f-a e g}\right )^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (-\frac {3}{2},m;-\frac {1}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 93, normalized size = 0.89 \[ -\frac {2 (d+e x)^m ((d+e x) (a e+c d x))^{-m} \left (\frac {g (a e+c d x)}{a e g-c d f}\right )^m \, _2F_1\left (-\frac {3}{2},m;-\frac {1}{2};\frac {c d (f+g x)}{c d f-a e g}\right )}{3 g (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {g x + f} {\left (e x + d\right )}^{m}}{{\left (g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{\frac {5}{2}} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{-m} \left (e x +d \right )^{m}}{\left (g x +f \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{\frac {5}{2}} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (f+g\,x\right )}^{5/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
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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