Optimal. Leaf size=104 \[ -\frac {(f+g x)^{n+1} (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, _2F_1\left (1,n+\frac {9}{2};n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{(n+1) (d+e x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.11, antiderivative size = 122, normalized size of antiderivative = 1.17, 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} \[ \frac {2 (f+g x)^n (a e+c d x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {7}{2},-n;\frac {9}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{7 c d \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 891
Rubi steps
\begin {align*} \int \frac {(f+g x)^n \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}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \int (a e+c d x)^{5/2} (f+g x)^n \, dx}{\sqrt {a e+c d x} \sqrt {d+e x}}\\ &=\frac {\left ((f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}\right ) \int (a e+c d x)^{5/2} \left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^n \, dx}{\sqrt {a e+c d x} \sqrt {d+e x}}\\ &=\frac {2 (a e+c d x)^3 (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, _2F_1\left (\frac {7}{2},-n;\frac {9}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{7 c d \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 110, normalized size = 1.06 \[ \frac {2 (f+g x)^n (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {7}{2},-n;\frac {9}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{7 c d \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{n}}{\sqrt {e x + d}}, x\right ) \]
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 [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {5}{2}} \left (g x +f \right )^{n}}{\left (e x +d \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^n\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
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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