Optimal. Leaf size=213 \[ \frac {\sqrt {d+e x} (f+g x)^{n+1} (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \, _2F_1\left (1,n+\frac {3}{2};n+2;\frac {c d (f+g x)}{c d f-a e g}\right )}{c d g (n+1) (2 n+3) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}+\frac {2 e (f+g x)^{n+1} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt {d+e x}} \]
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Rubi [A] time = 0.28, antiderivative size = 222, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {880, 891, 70, 69} \[ \frac {2 e (f+g x)^{n+1} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (f+g x)^n (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (2 n+3) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 880
Rule 891
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \int \frac {\sqrt {d+e x} (f+g x)^n}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d g (3+2 n)}\\ &=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {(f+g x)^n}{\sqrt {a e+c d x}} \, dx}{c d g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {\left (\left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) \sqrt {a e+c d x} \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n}\right ) \int \frac {\left (\frac {c d f}{c d f-a e g}+\frac {c d g x}{c d f-a e g}\right )^n}{\sqrt {a e+c d x}} \, dx}{c d g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {2 e (f+g x)^{1+n} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g (3+2 n) \sqrt {d+e x}}-\frac {2 \left (2 a e^2 g (1+n)+c d (e f-d g (3+2 n))\right ) (a e+c d x) \sqrt {d+e x} (f+g x)^n \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (3+2 n) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 145, normalized size = 0.68 \[ \frac {(f+g x)^n \sqrt {(d+e x) (a e+c d x)} \left (\left (c d (d g (2 n+3)-e f)-2 a e^2 g (n+1)\right ) \left (\frac {c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )+c d e (f+g x)\right )}{c^2 d^2 g \left (n+\frac {3}{2}\right ) \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} {\left (g x + f\right )}^{n}}{c d x + a e}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right )^{n}}{\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{n}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^n\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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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