Optimal. Leaf size=208 \[ \frac {(d+e x)^m (-b e+c d-c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (b e g (2 m+3)-2 c (d g m+e f (m+3))) \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 c^2 e^2 (m+3)}-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (m+3)} \]
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Rubi [A] time = 0.31, antiderivative size = 208, 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} \[ \frac {(d+e x)^m (-b e+c d-c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (b e g (2 m+3)-2 c (d g m+e f (m+3))) \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 c^2 e^2 (m+3)}-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (m+3)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 677
Rule 679
Rule 794
Rubi steps
\begin {align*} \int (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 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (3+m)}-\frac {(b e g (3+2 m)-2 c (d g m+e f (3+m))) \int (d+e x)^m \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{2 c e (3+m)}\\ &=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (3+m)}-\frac {\left ((b e g (3+2 m)-2 c (d g m+e f (3+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \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 (3+m)}\\ &=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (3+m)}-\frac {\left ((b e g (3+2 m)-2 c (d g m+e f (3+m))) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-m} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}\right ) \int \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 (3+m) \sqrt {c d^2-b d e-c d e x}}\\ &=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (3+m)}-\frac {\left ((b e g (3+2 m)-2 c (d g m+e f (3+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-b e^2 x-c e^2 x^2}\right ) \int \sqrt {c d^2-b d e-c d e x} \left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{\frac {1}{2}+m} \, dx}{2 c e (3+m) \sqrt {c d^2-b d e-c d e x}}\\ &=-\frac {g (d+e x)^m \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c e^2 (3+m)}+\frac {(b e g (3+2 m)-2 c (d g m+e f (3+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) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 c^2 e^2 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 149, normalized size = 0.72 \[ \frac {(d+e x)^{m-1} ((d+e x) (c (d-e x)-b e))^{3/2} \left (-e \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-\frac {1}{2}} (2 c (d g m+e f (m+3))-b e g (2 m+3)) \, _2F_1\left (\frac {3}{2},-m-\frac {1}{2};\frac {5}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )-3 c e g (d+e x)\right )}{3 c^2 e^3 (m+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.37, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (e x +d \right )^{m}\, dx \]
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 \[ \int \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )} {\left (e x + d\right )}^{m}\,{d x} \]
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 \[ \int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^m\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x} \,d x \]
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 \[ \int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{m} \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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