Optimal. Leaf size=205 \[ \frac {(d+e x)^m (-b e+c d-c e x) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]
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Rubi [A] time = 0.31, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {794, 679, 677, 70, 69} \[ \frac {(d+e x)^m (-b e+c d-c e x) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (m+1) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 677
Rule 679
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac {(b e g (1+2 m)-2 c (d g m+e f (1+m))) \int \frac {(d+e x)^m}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c e (1+m)}\\ &=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac {\left ((b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^m}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 c e (1+m)}\\ &=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac {\left ((b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {1}{2}-m} \sqrt {c d^2-b d e-c d e x}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^{-\frac {1}{2}+m}}{\sqrt {c d^2-b d e-c d e x}} \, dx}{2 c e (1+m) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}-\frac {\left ((b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (-\frac {c d e \left (1+\frac {e x}{d}\right )}{-c d e-\frac {e \left (c d^2-b d e\right )}{d}}\right )^{\frac {1}{2}-m} \sqrt {c d^2-b d e-c d e x}\right ) \int \frac {\left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{-\frac {1}{2}+m}}{\sqrt {c d^2-b d e-c d e x}} \, dx}{2 c e (1+m) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}+\frac {(b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (c d-b e-c e x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (1+m) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 155, normalized size = 0.76 \[ -\frac {2 (d+e x)^m \sqrt {(d+e x) (c (d-e x)-b e)} \left (\frac {e \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \, _2F_1\left (\frac {1}{2},-m-\frac {1}{2};\frac {3}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{c}+e (e f-d g)\right )}{e^3 (2 m+1) (b e-2 c d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )} {\left (e x + d\right )}^{m}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (e x +d \right )^{m}}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\, dx \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}\,{d x} \]
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 \[ \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^m}{\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{m} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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