Optimal. Leaf size=61 \[ \frac {2 (b x)^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (\frac {7}{2};-n,2;\frac {9}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2} \]
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Rubi [A] time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {130, 511, 510} \[ \frac {2 (b x)^{7/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} F_1\left (\frac {7}{2};-n,2;\frac {9}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {(b x)^{5/2} (c+d x)^n}{(e+f x)^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^6 \left (c+\frac {d x^2}{b}\right )^n}{\left (e+\frac {f x^2}{b}\right )^2} \, dx,x,\sqrt {b x}\right )}{b}\\ &=\frac {\left (2 (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (1+\frac {d x^2}{b c}\right )^n}{\left (e+\frac {f x^2}{b}\right )^2} \, dx,x,\sqrt {b x}\right )}{b}\\ &=\frac {2 (b x)^{7/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (\frac {7}{2};-n,2;\frac {9}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{7 b e^2}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 133, normalized size = 2.18 \[ \frac {2 b^2 \sqrt {b x} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (9 e F_1\left (\frac {1}{2};-n,1;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )-3 e F_1\left (\frac {1}{2};-n,2;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )-6 e \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {d x}{c}\right )+f x \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {d x}{c}\right )\right )}{3 f^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x} {\left (d x + c\right )}^{n} b^{2} x^{2}}{f^{2} x^{2} + 2 \, e f x + e^{2}}, x\right ) \]
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 \[ \int \frac {\left (b x\right )^{\frac {5}{2}} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x \right )^{\frac {5}{2}} \left (d x +c \right )^{n}}{\left (f x +e \right )^{2}}\, dx \]
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 \[ \int \frac {\left (b x\right )^{\frac {5}{2}} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^2} \,d x \]
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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